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Add damping analysis

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Thomas Dehaeze 2020-12-10 13:51:57 +01:00
parent 404c78505a
commit bcdfc62052
6 changed files with 3672 additions and 307 deletions
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@ -1346,18 +1346,17 @@ Ideally, the mechanical system should be designed in order to have a decoupled s
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.
Or it can be decoupled at high frequency if the Jacobians are evaluated at the CoM.
** SVD decoupling performances :noexport:
** SVD decoupling performances
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.
#+begin_src matlab
la = l/2; % Position of Act. [m]
ha = 0; % Position of Act. [m]
#+end_src
Let's do the SVD decoupling on a plant that is mostly real (low damping) and one with a large imaginary part (larger damping).
Start with small damping, the obtained diagonal and off-diagonal terms are shown in Figure [[fig:gravimeter_svd_low_damping]].
#+begin_src matlab
c = 2e1; % Actuator Damping [N/(m/s)]
#+end_src
#+begin_src matlab
#+begin_src matlab :exports none
%% Name of the Simulink File
mdl = 'gravimeter';
@ -1374,9 +1373,7 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
#+end_src
#+begin_src matlab
wc = 2*pi*10; % Decoupling frequency [rad/s]
H1 = evalfr(G, j*wc);
D = pinv(real(H1'*H1));
@ -1385,11 +1382,45 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
Gsvd = inv(U)*G*inv(V');
#+end_src
#+begin_src matlab :exports none
figure;
% Magnitude
hold on;
for i_in = 1:3
for i_out = [1:i_in-1, i_in+1:3]
plot(freqs, abs(squeeze(freqresp(Gsvd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(Gsvd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'DisplayName', '$G_{svd}(i,j)\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for i_in_out = 1:3
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));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
legend('location', 'northwest');
ylim([1e-8, 1e0]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/gravimeter_svd_low_damping.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:gravimeter_svd_low_damping
#+caption: Diagonal and off-diagonal term when decoupling with SVD on the gravimeter with small damping
#+RESULTS:
[[file:figs/gravimeter_svd_low_damping.png]]
Now take a larger damping, the obtained diagonal and off-diagonal terms are shown in Figure [[fig:gravimeter_svd_high_damping]].
#+begin_src matlab
c = 5e2; % Actuator Damping [N/(m/s)]
#+end_src
#+begin_src matlab
#+begin_src matlab :exports none
%% Name of the Simulink File
mdl = 'gravimeter';
@ -1406,9 +1437,7 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
#+end_src
#+begin_src matlab
wc = 2*pi*10; % Decoupling frequency [rad/s]
H1 = evalfr(G, j*wc);
D = pinv(real(H1'*H1));
@ -1417,71 +1446,6 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
Gsvdd = inv(U)*G*inv(V');
#+end_src
#+begin_src matlab
JMa = [1 0 -h/2
0 1 l/2
1 0 h/2
0 1 0];
JMt = [1 0 -ha
0 1 la
0 1 -la];
#+end_src
#+begin_src matlab
GM = pinv(JMa)*G*pinv(JMt');
GM.InputName = {'Fx', 'Fy', 'Mz'};
GM.OutputName = {'Dx', 'Dy', 'Rz'};
#+end_src
#+begin_src matlab :exports none
figure;
% Magnitude
hold on;
for i_in = 1:3
for i_out = [1:i_in-1, i_in+1:3]
plot(freqs, abs(squeeze(freqresp(GM(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(GM(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'DisplayName', '$G_x(i,j)\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for i_in_out = 1:3
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));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
legend('location', 'southeast');
ylim([1e-8, 1e0]);
#+end_src
#+begin_src matlab :exports none
figure;
% Magnitude
hold on;
for i_in = 1:3
for i_out = [1:i_in-1, i_in+1:3]
plot(freqs, abs(squeeze(freqresp(Gsvd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(Gsvd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'DisplayName', '$G_x(i,j)\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for i_in_out = 1:3
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));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
legend('location', 'southeast');
ylim([1e-8, 1e0]);
#+end_src
#+begin_src matlab :exports none
figure;
@ -1494,18 +1458,27 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
end
end
plot(freqs, abs(squeeze(freqresp(Gsvdd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'DisplayName', '$G_x(i,j)\ i \neq j$');
'DisplayName', '$G_{svd}(i,j)\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for i_in_out = 1:3
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));
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));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
legend('location', 'southeast');
legend('location', 'northwest');
ylim([1e-8, 1e0]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/gravimeter_svd_high_damping.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:gravimeter_svd_high_damping
#+caption: Diagonal and off-diagonal term when decoupling with SVD on the gravimeter with high damping
#+RESULTS:
[[file:figs/gravimeter_svd_high_damping.png]]
* Stewart Platform - Simscape Model
:PROPERTIES:
:header-args:matlab+: :tangle stewart_platform/script.m