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Add picture of platform and sections summary

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Thomas Dehaeze 2020-11-06 11:59:09 +01:00
parent 33b8583f62
commit ff0e0bf73e
12 changed files with 28936 additions and 620 deletions
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18.957 c 174.625 18.957 174.594 18.707 174.594 18.504 c 174.594 18.285
174.609 18.207 174.719 17.77 c 174.938 16.879 l 175.297 15.488 l 175.359
15.207 175.359 15.191 175.359 15.16 c 175.359 14.988 175.25 14.879 175.078
14.879 c 174.844 14.879 174.688 15.098 174.656 15.316 c 174.484 14.957
174.203 14.691 173.75 14.691 c 172.594 14.691 171.359 16.145 171.359 17.598
c 171.359 18.52 171.906 19.176 172.688 19.176 c 172.875 19.176 173.375
19.129 173.969 18.426 c 174.047 18.848 174.391 19.176 174.875 19.176 c 175.219
19.176 175.453 18.941 175.609 18.629 c 175.766 18.27 175.906 17.645 175.906
17.645 c h
174.516 15.941 m 174.016 17.895 l 173.969 18.066 173.969 18.082 173.828
18.254 c 173.391 18.801 172.984 18.957 172.703 18.957 c 172.203 18.957
172.062 18.41 172.062 18.02 c 172.062 17.535 172.391 16.316 172.609 15.863
c 172.922 15.27 173.359 14.91 173.766 14.91 c 174.406 14.91 174.547 15.723
174.547 15.785 c 174.547 15.832 174.531 15.895 174.516 15.941 c h
174.516 15.941 m f
Q q
0 g
0.990354 w
0 J
0 j
[] 0.0 d
10 M q 1 0 0 -1 0 0 cm
165.91 -22.859 m 165.91 -75.98 l 148.965 -75.98 l S Q
145.77 75.98 m 150.188 77.652 l 148.719 75.98 l 150.188 74.309 l h
145.77 75.98 m f
Q q
131.176 60.199 34 30.801 re W n
q
131 60 35 31 re W n
[ 1 0 0 1 0 0 ] concat
q
0 g
0.990354 w
0 J
0 j
[] 0.0 d
10 M q -1 0 0 1 0 0 cm
-145.77 75.98 m -150.188 77.652 l -148.719 75.98 l -150.188 74.309 l h
-145.77 75.98 m S Q
Q
Q
Q q
0 g
167.891 22.859 m 167.891 21.766 167.004 20.879 165.91 20.879 c 164.816
20.879 163.93 21.766 163.93 22.859 c 163.93 23.953 164.816 24.84 165.91
24.84 c 167.004 24.84 167.891 23.953 167.891 22.859 c h
167.891 22.859 m f
0.990354 w
0 J
0 j
[] 0.0 d
10 M q 1 0 0 -1 0 0 cm
109.555 -75.98 m 102.891 -75.98 l S Q
99.695 75.98 m 104.113 77.652 l 102.641 75.98 l 104.113 74.309 l h
99.695 75.98 m f
Q q
85.176 60.199 33 30.801 re W n
q
85 60 34 31 re W n
[ 1 0 0 1 0 0 ] concat
q
0 g
0.990354 w
0 J
0 j
[] 0.0 d
10 M q -1 0 0 1 0 0 cm
-99.695 75.98 m -104.113 77.652 l -102.641 75.98 l -104.113 74.309 l h
-99.695 75.98 m S Q
Q
Q
Q q
0 g
0.990354 w
0 J
0 j
[] 0.0 d
10 M q 1 0 0 -1 0 0 cm
63.48 -75.98 m 56.813 -75.98 l S Q
53.617 75.98 m 58.035 77.652 l 56.566 75.98 l 58.035 74.309 l h
53.617 75.98 m f
Q q
39.176 60.199 33 30.801 re W n
q
39 60 34 31 re W n
[ 1 0 0 1 0 0 ] concat
q
0 g
0.990354 w
0 J
0 j
[] 0.0 d
10 M q -1 0 0 1 0 0 cm
-53.617 75.98 m -58.035 77.652 l -56.566 75.98 l -58.035 74.309 l h
-53.617 75.98 m S Q
Q
Q
Q q
0 32.199 48.176 45 re W n
q
0 32 49 46 re W n
[ 1 0 0 1 0 0 ] concat
q
0 g
0.990354 w
0 J
0 j
[] 0.0 d
10 M q 1 0 0 -1 0 0 cm
17.402 -75.98 m 0.496 -75.98 l 0.496 -33.676 l 47.602 -33.676 l S Q
Q
Q
Q q
0 g
50.797 33.676 m 46.379 32 l 47.852 33.676 l 46.379 35.348 l h
50.797 33.676 m f
0.990354 w
0 J
0 j
[] 0.0 d
10 M q 1 0 0 -1 0 0 cm
50.797 -33.676 m 46.379 -32 l 47.852 -33.676 l 46.379 -35.348 l h
50.797 -33.676 m S Q
48.023 25.848 m 48.023 25.613 47.82 25.613 47.633 25.613 c 44.852 25.613
l 44.648 25.613 44.273 25.613 43.836 26.082 c 43.508 26.426 43.227 26.91
43.227 26.973 c 43.227 26.973 43.227 27.066 43.352 27.066 c 43.43 27.066
43.445 27.02 43.508 26.941 c 43.992 26.191 44.555 26.191 44.758 26.191
c 45.586 26.191 l 44.617 29.363 l 44.57 29.488 44.523 29.691 44.523 29.723
c 44.523 29.832 44.586 30.004 44.805 30.004 c 45.133 30.004 45.18 29.723
45.211 29.566 c 45.867 26.191 l 47.539 26.191 l 47.664 26.191 48.023 26.191
48.023 25.848 c h
48.023 25.848 m f
Q Q
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@ -689,6 +689,24 @@ This Matlab function is accessible [[file:gravimeter/pzmap_testCL.m][here]].
:PROPERTIES: :PROPERTIES:
:header-args:matlab+: :tangle stewart_platform/simscape_model.m :header-args:matlab+: :tangle stewart_platform/simscape_model.m
:END: :END:
** Introduction :ignore:
In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure [[fig:SP_assembly]].
#+name: fig:SP_assembly
#+caption: Stewart Platform CAD View
[[file:figs/SP_assembly.png]]
The analysis of the SVD control applied to the Stewart platform is performed in the following sections:
- Section [[sec:stewart_simscape]]: The parameters of the Simscape model of the Stewart platform are defined
- Section [[sec:stewart_identification]]: The plant is identified from the Simscape model and the centralized plant is computed thanks to the Jacobian
- Section [[sec:stewart_dynamics]]: The identified Dynamics is shown
- Section [[sec:stewart_real_approx]]: A real approximation of the plant is computed for further decoupling using the Singular Value Decomposition (SVD)
- Section [[sec:stewart_svd_decoupling]]: The decoupling is performed thanks to the SVD. The effectiveness of the decoupling is verified using the Gershorin Radii
- Section [[sec:stewart_decoupled_plant]]: The dynamics of the decoupled plant is shown
- Section [[sec:stewart_diagonal_control]]: A diagonal controller is defined to control the decoupled plant
- Section [[sec:stewart_closed_loop_results]]: Finally, the closed loop system properties are studied
** Matlab Init :noexport: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) #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>> <<matlab-dir>>
@ -707,7 +725,7 @@ This Matlab function is accessible [[file:gravimeter/pzmap_testCL.m][here]].
addpath('STEP'); addpath('STEP');
#+end_src #+end_src
** Jacobian ** Jacobian :noexport:
First, the position of the "joints" (points of force application) are estimated and the Jacobian computed. First, the position of the "joints" (points of force application) are estimated and the Jacobian computed.
#+begin_src matlab #+begin_src matlab
open('drone_platform_jacobian.slx'); open('drone_platform_jacobian.slx');
@ -741,7 +759,8 @@ First, the position of the "joints" (points of force application) are estimated
save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J'); save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
#+end_src #+end_src
** Simscape Model ** Simscape Model - Parameters
<<sec:stewart_simscape>>
#+begin_src matlab #+begin_src matlab
open('drone_platform.slx'); open('drone_platform.slx');
#+end_src #+end_src
@ -757,16 +776,19 @@ Definition of spring parameters
cz = 0.025; cz = 0.025;
#+end_src #+end_src
Gravity:
#+begin_src matlab #+begin_src matlab
g = 0; g = 0;
#+end_src #+end_src
We load the Jacobian. We load the Jacobian (previously computed from the geometry).
#+begin_src matlab #+begin_src matlab
load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J'); load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
#+end_src #+end_src
** Identification of the plant ** Identification of the plant
<<sec:stewart_identification>>
The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform. The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform.
#+begin_src matlab #+begin_src matlab
%% Name of the Simulink File %% Name of the Simulink File
@ -792,39 +814,41 @@ There are 24 states (6dof for the bottom platform + 6dof for the top platform).
#+RESULTS: #+RESULTS:
: State-space model with 6 outputs, 12 inputs, and 24 states. : State-space model with 6 outputs, 12 inputs, and 24 states.
#+begin_src matlab The "centralized" plant $\bm{G}_x$ is now computed (Figure [[fig:centralized_control]]).
% G = G*blkdiag(inv(J), eye(6));
% G.InputName = {'Dw1', 'Dw2', 'Dw3', 'Dw4', 'Dw5', 'Dw6', ...
% 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
#+end_src
Thanks to the Jacobian, we compute the transfer functions in the frame of the legs and in an inertial frame. #+name: fig:centralized_control
#+caption: Centralized control architecture
[[file:figs/centralized_control.png]]
Thanks to the Jacobian, we compute the transfer functions in the inertial frame (transfer function from forces and torques applied to the top platform to the absolute acceleration of the top platform).
#+begin_src matlab #+begin_src matlab
Gx = G*blkdiag(eye(6), inv(J')); Gx = G*blkdiag(eye(6), inv(J'));
Gx.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ... Gx.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; 'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
% Gl = J*G;
% Gl.OutputName = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
#+end_src #+end_src
** Obtained Dynamics ** Obtained Dynamics
<<sec:stewart_dynamics>>
#+begin_src matlab :exports none #+begin_src matlab :exports none
freqs = logspace(-1, 2, 1000); freqs = logspace(-1, 2, 1000);
figure; figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = subplot(2, 1, 1); % Magnitude
ax1 = nexttile([2, 1]);
hold on; hold on;
plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))), 'DisplayName', '$A_x/F_x$'); plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))), 'DisplayName', '$A_x/F_x$');
plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))), 'DisplayName', '$A_y/F_y$'); plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))), 'DisplayName', '$A_y/F_y$');
plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$A_z/F_z$'); plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$A_z/F_z$');
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast'); legend('location', 'southeast');
ax2 = subplot(2, 1, 2); % Phase
ax2 = nexttile;
hold on; hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz')))); plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz')))); plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))));
@ -839,7 +863,7 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file replace #+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/stewart_platform_translations.pdf', 'width', 'full', 'height', 'full'); exportFig('figs/stewart_platform_translations.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
#+end_src #+end_src
#+name: fig:stewart_platform_translations #+name: fig:stewart_platform_translations
@ -851,18 +875,21 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le
freqs = logspace(-1, 2, 1000); freqs = logspace(-1, 2, 1000);
figure; figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = subplot(2, 1, 1); % Magnitude
ax1 = nexttile([2, 1]);
hold on; hold on;
plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))), 'DisplayName', '$A_{R_x}/M_x$'); plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))), 'DisplayName', '$A_{R_x}/M_x$');
plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))), 'DisplayName', '$A_{R_y}/M_y$'); plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))), 'DisplayName', '$A_{R_y}/M_y$');
plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$A_{R_z}/M_z$'); plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$A_{R_z}/M_z$');
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [rad/(Nm)]'); set(gca, 'XTickLabel',[]); ylabel('Magnitude [rad/(Nm)]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast'); legend('location', 'southeast');
ax2 = subplot(2, 1, 2); % Phase
ax2 = nexttile;
hold on; hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz')))); plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz')))); plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))));
@ -877,7 +904,7 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file replace #+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/stewart_platform_rotations.pdf', 'width', 'full', 'height', 'full'); exportFig('figs/stewart_platform_rotations.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
#+end_src #+end_src
#+name: fig:stewart_platform_rotations #+name: fig:stewart_platform_rotations
@ -885,94 +912,9 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le
#+RESULTS: #+RESULTS:
[[file:figs/stewart_platform_rotations.png]] [[file:figs/stewart_platform_rotations.png]]
#+begin_src matlab :exports none
freqs = logspace(-1, 2, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for out_i = 1:5
for in_i = i+1:6
plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', out_i), sprintf('F%i', in_i)), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
end
end
for ch_i = 1:6
plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', ch_i), sprintf('F%i', ch_i)), 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 ch_i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(sprintf('A%i', ch_i), sprintf('F%i', ch_i)), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-360:90:360]);
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/stewart_platform_legs.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:stewart_platform_legs
#+caption: Stewart Platform Plant from forces applied by the legs to displacement of the legs
#+RESULTS:
[[file:figs/stewart_platform_legs.png]]
#+begin_src matlab :exports none
freqs = logspace(-1, 2, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_x/D_{w,x}$');
plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_y/D_{w,y}$');
plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Dwz')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_z/D_{w,z}$');
% set(gca,'ColorOrderIndex',1)
% plot(freqs, abs(squeeze(freqresp(TR(1,1), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
% plot(freqs, abs(squeeze(freqresp(TR(2,2), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
% plot(freqs, abs(squeeze(freqresp(TR(3,3), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - Translations'); xlabel('Frequency [Hz]');
legend('location', 'northeast');
ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Rwx')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_x/R_{w,x}$');
plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'Rwy')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_y/R_{w,y}$');
plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Rwz')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_z/R_{w,z}$');
% set(gca,'ColorOrderIndex',1)
% plot(freqs, abs(squeeze(freqresp(TR(4,4), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
% plot(freqs, abs(squeeze(freqresp(TR(5,5), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
% plot(freqs, abs(squeeze(freqresp(TR(6,6), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility - Rotations'); xlabel('Frequency [Hz]');
legend('location', 'northeast');
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/stewart_platform_transmissibility.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:stewart_platform_transmissibility
#+caption: Transmissibility
#+RESULTS:
[[file:figs/stewart_platform_transmissibility.png]]
** Real Approximation of $G$ at the decoupling frequency ** Real Approximation of $G$ at the decoupling frequency
<<sec:stewart_real_approx>>
Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_c(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$. Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_c(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$.
#+begin_src matlab #+begin_src matlab
wc = 2*pi*30; % Decoupling frequency [rad/s] wc = 2*pi*30; % Decoupling frequency [rad/s]
@ -989,7 +931,39 @@ The real approximation is computed as follows:
H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2)))); H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
#+end_src #+end_src
#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(H1, {}, {}, ' %.1f ');
#+end_src
#+caption: Real approximate of $G$ at the decoupling frequency $\omega_c$
#+RESULTS:
| 4.4 | -2.1 | -2.1 | 4.4 | -2.4 | -2.4 |
| -0.2 | -3.9 | 3.9 | 0.2 | -3.8 | 3.8 |
| 3.4 | 3.4 | 3.4 | 3.4 | 3.4 | 3.4 |
| -367.1 | -323.8 | 323.8 | 367.1 | 43.3 | -43.3 |
| -162.0 | -237.0 | -237.0 | -162.0 | 398.9 | 398.9 |
| 220.6 | -220.6 | 220.6 | -220.6 | 220.6 | -220.6 |
Please not that the plant $G$ at $\omega_c$ is already an almost real matrix.
This can be seen on the Bode plots where the phase is close to 1.
This can be verified below where only the real value of $G(\omega_c)$ is shown
#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(real(evalfr(Gc, j*wc)), {}, {}, ' %.1f ');
#+end_src
#+RESULTS:
| 4.4 | -2.1 | -2.1 | 4.4 | -2.4 | -2.4 |
| -0.2 | -3.9 | 3.9 | 0.2 | -3.8 | 3.8 |
| 3.4 | 3.4 | 3.4 | 3.4 | 3.4 | 3.4 |
| -367.1 | -323.8 | 323.8 | 367.1 | 43.3 | -43.3 |
| -162.0 | -237.0 | -237.0 | -162.0 | 398.9 | 398.9 |
| 220.6 | -220.6 | 220.6 | -220.6 | 220.6 | -220.6 |
** Verification of the decoupling using the "Gershgorin Radii" ** Verification of the decoupling using the "Gershgorin Radii"
<<sec:stewart_svd_decoupling>>
First, the Singular Value Decomposition of $H_1$ is performed: First, the Singular Value Decomposition of $H_1$ is performed:
\[ H_1 = U \Sigma V^H \] \[ H_1 = U \Sigma V^H \]
@ -1069,6 +1043,8 @@ Gershgorin Radii for the decoupled plant using the Jacobian:
[[file:figs/simscape_model_gershgorin_radii.png]] [[file:figs/simscape_model_gershgorin_radii.png]]
** Decoupled Plant ** Decoupled Plant
<<sec:stewart_decoupled_plant>>
Let's see the bode plot of the decoupled plant $G_d(s)$. Let's see the bode plot of the decoupled plant $G_d(s)$.
\[ G_d(s) = U^T G_c(s) V \] \[ G_d(s) = U^T G_c(s) V \]
@ -1089,7 +1065,7 @@ Let's see the bode plot of the decoupled plant $G_d(s)$.
end end
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude'); xlabel('Frequency [Hz]'); ylabel('Magnitude'); xlabel('Frequency [Hz]');
legend('location', 'southeast'); legend('location', 'southeast');
#+end_src #+end_src
@ -1119,7 +1095,7 @@ Let's see the bode plot of the decoupled plant $G_d(s)$.
end end
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude'); xlabel('Frequency [Hz]'); ylabel('Magnitude'); xlabel('Frequency [Hz]');
legend('location', 'southeast'); legend('location', 'southeast');
#+end_src #+end_src
@ -1133,6 +1109,8 @@ Let's see the bode plot of the decoupled plant $G_d(s)$.
[[file:figs/simscape_model_decoupled_plant_jacobian.png]] [[file:figs/simscape_model_decoupled_plant_jacobian.png]]
** Diagonal Controller ** Diagonal Controller
<<sec:stewart_diagonal_control>>
The controller $K$ is a diagonal controller consisting a low pass filters with a crossover frequency $\omega_c$ and a DC gain $C_g$. The controller $K$ is a diagonal controller consisting a low pass filters with a crossover frequency $\omega_c$ and a DC gain $C_g$.
#+begin_src matlab #+begin_src matlab
@ -1142,13 +1120,12 @@ The controller $K$ is a diagonal controller consisting a low pass filters with a
K = eye(6)*C_g/(s+wc); K = eye(6)*C_g/(s+wc);
#+end_src #+end_src
** Centralized Control The control diagram for the centralized control is shown in Figure [[fig:centralized_control]].
The control diagram for the centralized control is shown below.
The controller $K_c$ is "working" in an cartesian frame. The controller $K_c$ is "working" in an cartesian frame.
The Jacobian is used to convert forces in the cartesian frame to forces applied by the actuators. The Jacobian is used to convert forces in the cartesian frame to forces applied by the actuators.
#+begin_src latex :file centralized_control.pdf :tangle no #+begin_src latex :file centralized_control.pdf :tangle no :exports results
\begin{tikzpicture} \begin{tikzpicture}
\node[block={2cm}{1.5cm}] (G) {$G$}; \node[block={2cm}{1.5cm}] (G) {$G$};
\node[block, below right=0.6 and -0.5 of G] (K) {$K_c$}; \node[block, below right=0.6 and -0.5 of G] (K) {$K_c$};
@ -1167,18 +1144,19 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied
\end{tikzpicture} \end{tikzpicture}
#+end_src #+end_src
#+name: fig:centralized_control
#+caption: Control Diagram for the Centralized control
#+RESULTS: #+RESULTS:
[[file:figs/centralized_control.png]] [[file:figs/centralized_control.png]]
The feedback system is computed as shown below.
#+begin_src matlab #+begin_src matlab
G_cen = feedback(G, inv(J')*K, [7:12], [1:6]); G_cen = feedback(G, inv(J')*K, [7:12], [1:6]);
#+end_src #+end_src
** SVD Control The SVD control architecture is shown in Figure [[fig:svd_control]].
The SVD control architecture is shown below.
The matrices $U$ and $V$ are used to decoupled the plant $G$. The matrices $U$ and $V$ are used to decoupled the plant $G$.
#+begin_src latex :file svd_control.pdf :tangle no :exports results
#+begin_src latex :file svd_control.pdf :tangle no
\begin{tikzpicture} \begin{tikzpicture}
\node[block={2cm}{1.5cm}] (G) {$G$}; \node[block={2cm}{1.5cm}] (G) {$G$};
\node[block, below right=0.6 and 0 of G] (U) {$U^{-1}$}; \node[block, below right=0.6 and 0 of G] (U) {$U^{-1}$};
@ -1199,15 +1177,19 @@ The matrices $U$ and $V$ are used to decoupled the plant $G$.
\end{tikzpicture} \end{tikzpicture}
#+end_src #+end_src
#+name: fig:svd_control
#+caption: Control Diagram for the SVD control
#+RESULTS: #+RESULTS:
[[file:figs/svd_control.png]] [[file:figs/svd_control.png]]
SVD Control The feedback system is computed as shown below.
#+begin_src matlab #+begin_src matlab
G_svd = feedback(G, pinv(V')*K*pinv(U), [7:12], [1:6]); G_svd = feedback(G, pinv(V')*K*pinv(U), [7:12], [1:6]);
#+end_src #+end_src
** Results ** Closed-Loop system Performances
<<sec:stewart_closed_loop_results>>
Let's first verify the stability of the closed-loop systems: Let's first verify the stability of the closed-loop systems:
#+begin_src matlab :results output replace text #+begin_src matlab :results output replace text
isstable(G_cen) isstable(G_cen)
@ -1302,7 +1284,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well
#+RESULTS: #+RESULTS:
[[file:figs/stewart_platform_simscape_cl_transmissibility.png]] [[file:figs/stewart_platform_simscape_cl_transmissibility.png]]
* Stewart Platform - Analytical Model * Stewart Platform - Analytical Model :noexport:
:PROPERTIES: :PROPERTIES:
:header-args:matlab+: :tangle stewart_platform/analytical_model.m :header-args:matlab+: :tangle stewart_platform/analytical_model.m
:END: :END: