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. 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. 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 Let's do the SVD decoupling on a plant that is mostly real (low damping) and one with a large imaginary part (larger damping).
la = l/2; % Position of Act. [m]
ha = 0; % Position of Act. [m]
#+end_src
Start with small damping, the obtained diagonal and off-diagonal terms are shown in Figure [[fig:gravimeter_svd_low_damping]].
#+begin_src matlab #+begin_src matlab
c = 2e1; % Actuator Damping [N/(m/s)] c = 2e1; % Actuator Damping [N/(m/s)]
#+end_src #+end_src
#+begin_src matlab #+begin_src matlab :exports none
%% Name of the Simulink File %% Name of the Simulink File
mdl = 'gravimeter'; 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 = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'}; G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'}; G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
#+end_src
#+begin_src matlab
wc = 2*pi*10; % Decoupling frequency [rad/s] wc = 2*pi*10; % Decoupling frequency [rad/s]
H1 = evalfr(G, j*wc); H1 = evalfr(G, j*wc);
D = pinv(real(H1'*H1)); 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'); Gsvd = inv(U)*G*inv(V');
#+end_src #+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 #+begin_src matlab
c = 5e2; % Actuator Damping [N/(m/s)] c = 5e2; % Actuator Damping [N/(m/s)]
#+end_src #+end_src
#+begin_src matlab #+begin_src matlab :exports none
%% Name of the Simulink File %% Name of the Simulink File
mdl = 'gravimeter'; 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 = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'}; G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'}; G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
#+end_src
#+begin_src matlab
wc = 2*pi*10; % Decoupling frequency [rad/s] wc = 2*pi*10; % Decoupling frequency [rad/s]
H1 = evalfr(G, j*wc); H1 = evalfr(G, j*wc);
D = pinv(real(H1'*H1)); 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'); Gsvdd = inv(U)*G*inv(V');
#+end_src #+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 #+begin_src matlab :exports none
figure; figure;
@ -1494,18 +1458,27 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
end end
end end
plot(freqs, abs(squeeze(freqresp(Gsvdd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... 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) set(gca,'ColorOrderIndex',1)
for i_in_out = 1:3 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 end
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude'); xlabel('Frequency [Hz]'); ylabel('Magnitude');
legend('location', 'southeast'); legend('location', 'northwest');
ylim([1e-8, 1e0]); ylim([1e-8, 1e0]);
#+end_src #+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 * Stewart Platform - Simscape Model
:PROPERTIES: :PROPERTIES:
:header-args:matlab+: :tangle stewart_platform/script.m :header-args:matlab+: :tangle stewart_platform/script.m