Add damping analysis
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figs/gravimeter_svd_high_damping.pdf
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figs/gravimeter_svd_high_damping.pdf
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figs/gravimeter_svd_high_damping.png
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figs/gravimeter_svd_low_damping.pdf
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figs/gravimeter_svd_low_damping.pdf
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figs/gravimeter_svd_low_damping.png
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@ -1346,18 +1346,17 @@ Ideally, the mechanical system should be designed in order to have a decoupled s
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If not the case, the system can either be decoupled as low frequency if the Jacobian are evaluated at a point where the stiffness matrix is decoupled.
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Or it can be decoupled at high frequency if the Jacobians are evaluated at the CoM.
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** SVD decoupling performances :noexport:
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** SVD decoupling performances
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As the SVD is applied on a *real approximation* of the plant dynamics at a frequency $\omega_0$, it is foreseen that the effectiveness of the decoupling depends on the validity of the real approximation.
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#+begin_src matlab
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la = l/2; % Position of Act. [m]
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ha = 0; % Position of Act. [m]
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#+end_src
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Let's do the SVD decoupling on a plant that is mostly real (low damping) and one with a large imaginary part (larger damping).
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Start with small damping, the obtained diagonal and off-diagonal terms are shown in Figure [[fig:gravimeter_svd_low_damping]].
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#+begin_src matlab
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c = 2e1; % Actuator Damping [N/(m/s)]
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#+end_src
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#+begin_src matlab
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#+begin_src matlab :exports none
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%% Name of the Simulink File
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mdl = 'gravimeter';
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@ -1374,9 +1373,7 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
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G = linearize(mdl, io);
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G.InputName = {'F1', 'F2', 'F3'};
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G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
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#+end_src
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#+begin_src matlab
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wc = 2*pi*10; % Decoupling frequency [rad/s]
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H1 = evalfr(G, j*wc);
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D = pinv(real(H1'*H1));
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@ -1385,79 +1382,6 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
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Gsvd = inv(U)*G*inv(V');
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#+end_src
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#+begin_src matlab
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c = 5e2; % Actuator Damping [N/(m/s)]
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#+end_src
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#+begin_src matlab
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%% Name of the Simulink File
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mdl = 'gravimeter';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/F1'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/F2'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/F3'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
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G = linearize(mdl, io);
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G.InputName = {'F1', 'F2', 'F3'};
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G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
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#+end_src
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#+begin_src matlab
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wc = 2*pi*10; % Decoupling frequency [rad/s]
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H1 = evalfr(G, j*wc);
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D = pinv(real(H1'*H1));
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H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
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[U,S,V] = svd(H1);
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Gsvdd = inv(U)*G*inv(V');
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#+end_src
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#+begin_src matlab
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JMa = [1 0 -h/2
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0 1 l/2
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1 0 h/2
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0 1 0];
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JMt = [1 0 -ha
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0 1 la
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0 1 -la];
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#+end_src
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#+begin_src matlab
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GM = pinv(JMa)*G*pinv(JMt');
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GM.InputName = {'Fx', 'Fy', 'Mz'};
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GM.OutputName = {'Dx', 'Dy', 'Rz'};
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#+end_src
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#+begin_src matlab :exports none
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figure;
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% Magnitude
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hold on;
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for i_in = 1:3
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for i_out = [1:i_in-1, i_in+1:3]
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plot(freqs, abs(squeeze(freqresp(GM(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
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'HandleVisibility', 'off');
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end
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end
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plot(freqs, abs(squeeze(freqresp(GM(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
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'DisplayName', '$G_x(i,j)\ i \neq j$');
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set(gca,'ColorOrderIndex',1)
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for i_in_out = 1:3
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plot(freqs, abs(squeeze(freqresp(GM(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_x(%d,%d)$', i_in_out, i_in_out));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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legend('location', 'southeast');
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ylim([1e-8, 1e0]);
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#+end_src
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#+begin_src matlab :exports none
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figure;
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@ -1470,18 +1394,58 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
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end
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end
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plot(freqs, abs(squeeze(freqresp(Gsvd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
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'DisplayName', '$G_x(i,j)\ i \neq j$');
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'DisplayName', '$G_{svd}(i,j)\ i \neq j$');
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set(gca,'ColorOrderIndex',1)
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for i_in_out = 1:3
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plot(freqs, abs(squeeze(freqresp(Gsvd(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_x(%d,%d)$', i_in_out, i_in_out));
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plot(freqs, abs(squeeze(freqresp(Gsvd(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{svd}(%d,%d)$', i_in_out, i_in_out));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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legend('location', 'southeast');
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legend('location', 'northwest');
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ylim([1e-8, 1e0]);
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/gravimeter_svd_low_damping.pdf', 'width', 'wide', 'height', 'normal');
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#+end_src
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#+name: fig:gravimeter_svd_low_damping
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#+caption: Diagonal and off-diagonal term when decoupling with SVD on the gravimeter with small damping
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#+RESULTS:
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[[file:figs/gravimeter_svd_low_damping.png]]
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Now take a larger damping, the obtained diagonal and off-diagonal terms are shown in Figure [[fig:gravimeter_svd_high_damping]].
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#+begin_src matlab
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c = 5e2; % Actuator Damping [N/(m/s)]
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#+end_src
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#+begin_src matlab :exports none
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%% Name of the Simulink File
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mdl = 'gravimeter';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/F1'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/F2'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/F3'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
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G = linearize(mdl, io);
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G.InputName = {'F1', 'F2', 'F3'};
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G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
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wc = 2*pi*10; % Decoupling frequency [rad/s]
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H1 = evalfr(G, j*wc);
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D = pinv(real(H1'*H1));
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H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
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[U,S,V] = svd(H1);
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Gsvdd = inv(U)*G*inv(V');
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#+end_src
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#+begin_src matlab :exports none
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figure;
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@ -1494,18 +1458,27 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
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end
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end
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plot(freqs, abs(squeeze(freqresp(Gsvdd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
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'DisplayName', '$G_x(i,j)\ i \neq j$');
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'DisplayName', '$G_{svd}(i,j)\ i \neq j$');
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set(gca,'ColorOrderIndex',1)
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for i_in_out = 1:3
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plot(freqs, abs(squeeze(freqresp(Gsvdd(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_x(%d,%d)$', i_in_out, i_in_out));
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plot(freqs, abs(squeeze(freqresp(Gsvdd(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{svd}(%d,%d)$', i_in_out, i_in_out));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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legend('location', 'southeast');
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legend('location', 'northwest');
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ylim([1e-8, 1e0]);
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/gravimeter_svd_high_damping.pdf', 'width', 'wide', 'height', 'normal');
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#+end_src
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#+name: fig:gravimeter_svd_high_damping
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#+caption: Diagonal and off-diagonal term when decoupling with SVD on the gravimeter with high damping
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#+RESULTS:
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[[file:figs/gravimeter_svd_high_damping.png]]
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* Stewart Platform - Simscape Model
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:PROPERTIES:
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:header-args:matlab+: :tangle stewart_platform/script.m
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