Analysis + Control of Simscape Model
This commit is contained in:
617
index.org
617
index.org
@@ -34,14 +34,16 @@
|
||||
#+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 raw replace :buffer no
|
||||
#+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 both
|
||||
#+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:
|
||||
|
||||
* Simscape Model - Gravimeter
|
||||
* Gravimeter - Simscape Model
|
||||
** 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>>
|
||||
@@ -380,7 +382,7 @@ The plant as 6 states as expected (2 translations + 1 rotation)
|
||||
rot = PHI(:,11,11);
|
||||
#+end_src
|
||||
|
||||
* Simscape Model - Stewart Platform
|
||||
* Stewart Platform - Simscape Model
|
||||
** 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>>
|
||||
@@ -392,7 +394,6 @@ The plant as 6 states as expected (2 translations + 1 rotation)
|
||||
|
||||
** Jacobian
|
||||
First, the position of the "joints" (points of force application) are estimated and the Jacobian computed.
|
||||
|
||||
#+begin_src matlab
|
||||
open('stewart_platform/drone_platform_jacobian.slx');
|
||||
#+end_src
|
||||
@@ -425,7 +426,7 @@ First, the position of the "joints" (points of force application) are estimated
|
||||
save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
|
||||
#+end_src
|
||||
|
||||
** Simulink
|
||||
** Simscape Model
|
||||
#+begin_src matlab
|
||||
open('stewart_platform/drone_platform.slx');
|
||||
#+end_src
|
||||
@@ -446,7 +447,7 @@ We load the Jacobian.
|
||||
load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
|
||||
#+end_src
|
||||
|
||||
|
||||
** Identification of the plant
|
||||
The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform.
|
||||
#+begin_src matlab
|
||||
%% Name of the Simulink File
|
||||
@@ -454,30 +455,41 @@ The dynamics is identified from forces applied by each legs to the measured acce
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
|
||||
G = linearize(mdl, io);
|
||||
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
|
||||
'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
|
||||
#+end_src
|
||||
|
||||
There are 24 states (6dof for the bottom platform + 6dof for the top platform).
|
||||
#+begin_src matlab :results output replace
|
||||
size(G)
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
: State-space model with 6 outputs, 6 inputs, and 12 states.
|
||||
: State-space model with 6 outputs, 12 inputs, and 24 states.
|
||||
|
||||
#+begin_src matlab
|
||||
% 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.
|
||||
#+begin_src matlab
|
||||
Gx = -G*inv(J');
|
||||
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
||||
Gx = G*blkdiag(eye(6), inv(J'));
|
||||
Gx.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
|
||||
'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
||||
|
||||
Gl = -J*G;
|
||||
Gl = J*G;
|
||||
Gl.OutputName = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
|
||||
#+end_src
|
||||
|
||||
** Obtained Dynamics
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
@@ -485,9 +497,9 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Gx(1, 1), freqs, 'Hz'))), 'DisplayName', '$A_x/F_x$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx(2, 2), freqs, 'Hz'))), 'DisplayName', '$A_y/F_y$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx(3, 3), freqs, 'Hz'))), 'DisplayName', '$A_z/F_z$');
|
||||
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('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$A_z/F_z$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
@@ -495,13 +507,13 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
for i = 1:3
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gx(i, i), freqs, 'Hz')))));
|
||||
end
|
||||
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('Az', 'Fz'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-270, 90]);
|
||||
ylim([-180, 180]);
|
||||
yticks([-360:90:360]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
@@ -523,9 +535,9 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Gx(4, 4), freqs, 'Hz'))), 'DisplayName', '$A_{R_x}/M_x$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx(5, 5), freqs, 'Hz'))), 'DisplayName', '$A_{R_y}/M_y$');
|
||||
plot(freqs, abs(squeeze(freqresp(Gx(6, 6), freqs, 'Hz'))), 'DisplayName', '$A_{R_z}/M_z$');
|
||||
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('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$A_{R_z}/M_z$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [rad/(Nm)]'); set(gca, 'XTickLabel',[]);
|
||||
@@ -533,13 +545,13 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
for i = 4:6
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gx(i, i), freqs, 'Hz')))));
|
||||
end
|
||||
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('Arz', 'Mz'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-270, 90]);
|
||||
ylim([-180, 180]);
|
||||
yticks([-360:90:360]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
@@ -562,11 +574,11 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
for i = 1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gl(i, i), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', i), sprintf('F%i', i)), freqs, 'Hz'))));
|
||||
end
|
||||
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]);
|
||||
plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', i), sprintf('F%i', j)), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
|
||||
end
|
||||
end
|
||||
hold off;
|
||||
@@ -576,12 +588,12 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
for i = 1:6
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gl(i, i), freqs, 'Hz')))));
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(sprintf('A%i', i), sprintf('F%i', i)), freqs, 'Hz'))));
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-270, 90]);
|
||||
ylim([-180, 180]);
|
||||
yticks([-360:90:360]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
@@ -595,3 +607,548 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le
|
||||
#+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', '$D_z/D_{w,z}$');
|
||||
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', '$R_z/R_{w,z}$');
|
||||
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
|
||||
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
|
||||
wc = 2*pi*20; % Decoupling frequency [rad/s]
|
||||
Gc = G({'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); % Transfer function to find a real approximation
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
H1 = evalfr(Gc, j*wc);
|
||||
#+end_src
|
||||
|
||||
The real approximation is computed as follows:
|
||||
#+begin_src matlab
|
||||
D = pinv(real(H1'*H1));
|
||||
H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
|
||||
#+end_src
|
||||
|
||||
** Verification of the decoupling using the "Gershgorin Radii"
|
||||
First, the Singular Value Decomposition of $H_1$ is performed:
|
||||
\[ H_1 = U \Sigma V^H \]
|
||||
|
||||
#+begin_src matlab
|
||||
[U,S,V] = svd(H1);
|
||||
#+end_src
|
||||
|
||||
Then, the "Gershgorin Radii" is computed for the plant $G_c(s)$ and the "SVD Decoupled Plant" $G_d(s)$:
|
||||
\[ G_d(s) = U^T G_c(s) V \]
|
||||
|
||||
It is done over the following frequencies.
|
||||
#+begin_src matlab
|
||||
freqs = logspace(-1,2,1000); % [Hz]
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
for i = 1:length(freqs)
|
||||
H = abs(evalfr(Gc, j*2*pi*freqs(i)));
|
||||
for j = 1:size(H,2)
|
||||
g_r1(i,j) = (sum(H(j,:)) - H(j,j))/H(j,j);
|
||||
end
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
Gd = U'*Gc*V;
|
||||
|
||||
for i = 1:length(freqs)
|
||||
H_dec = abs(evalfr(Gd, j*2*pi*freqs(i)));
|
||||
for j = 1:size(H,2)
|
||||
g_r2(i,j) = (sum(H_dec(j,:)) - H_dec(j,j))/H_dec(j,j);
|
||||
end
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports results
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, g_r1(:,1), 'DisplayName', '$a_x$')
|
||||
plot(freqs, g_r1(:,2), 'DisplayName', '$a_y$')
|
||||
plot(freqs, g_r1(:,3), 'DisplayName', '$a_z$')
|
||||
plot(freqs, g_r1(:,4), 'DisplayName', '$a_{R_x}$')
|
||||
plot(freqs, g_r1(:,5), 'DisplayName', '$a_{R_y}$')
|
||||
plot(freqs, g_r1(:,6), 'DisplayName', '$a_{R_z}$')
|
||||
plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit')
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
hold off;
|
||||
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports results
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, g_r2(:,1), 'DisplayName', '$a_x$')
|
||||
plot(freqs, g_r2(:,2), 'DisplayName', '$a_y$')
|
||||
plot(freqs, g_r2(:,3), 'DisplayName', '$a_z$')
|
||||
plot(freqs, g_r2(:,4), 'DisplayName', '$a_{R_x}$')
|
||||
plot(freqs, g_r2(:,5), 'DisplayName', '$a_{R_y}$')
|
||||
plot(freqs, g_r2(:,6), 'DisplayName', '$a_{R_z}$')
|
||||
plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit')
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||
#+end_src
|
||||
|
||||
** Decoupled Plant
|
||||
Let's see the bode plot of the decoupled plant $G_d(s)$.
|
||||
\[ G_d(s) = U^T G_c(s) V \]
|
||||
|
||||
#+begin_src matlab :exports results
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gd(i, i), freqs, 'Hz'))), ...
|
||||
'DisplayName', sprintf('$G(%i, %i)$', i, i));
|
||||
end
|
||||
for i = 1:5
|
||||
for j = i+1:6
|
||||
plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude'); xlabel('Frequency [Hz]');
|
||||
legend('location', 'southeast');
|
||||
#+end_src
|
||||
|
||||
** Diagonal Controller
|
||||
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
|
||||
wc = 2*pi*0.1; % Crossover Frequency [rad/s]
|
||||
C_g = 50; % DC Gain
|
||||
|
||||
K = eye(6)*C_g/(s+wc);
|
||||
#+end_src
|
||||
|
||||
** Centralized Control
|
||||
The control diagram for the centralized control is shown below.
|
||||
|
||||
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.
|
||||
|
||||
#+begin_src latex :file centralized_control.pdf
|
||||
\begin{tikzpicture}
|
||||
\node[block={2cm}{1.5cm}] (G) {$G$};
|
||||
\node[block, below right=0.6 and -0.5 of G] (K) {$K_c$};
|
||||
\node[block, below left= 0.6 and -0.5 of G] (J) {$J^{-T}$};
|
||||
|
||||
% Inputs of the controllers
|
||||
\coordinate[] (inputd) at ($(G.south west)!0.75!(G.north west)$);
|
||||
\coordinate[] (inputu) at ($(G.south west)!0.25!(G.north west)$);
|
||||
|
||||
% Connections and labels
|
||||
\draw[<-] (inputd) -- ++(-0.8, 0) node[above right]{$D_w$};
|
||||
\draw[->] (G.east) -- ++(2.0, 0) node[above left]{$a$};
|
||||
\draw[->] ($(G.east)+(1.4, 0)$)node[branch]{} |- (K.east);
|
||||
\draw[->] (K.west) -- (J.east) node[above right]{$\mathcal{F}$};
|
||||
\draw[->] (J.west) -- ++(-0.6, 0) |- (inputu) node[above left]{$\tau$};
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
[[file:figs/centralized_control.png]]
|
||||
|
||||
#+begin_src matlab
|
||||
G_cen = feedback(G, inv(J')*K, [7:12], [1:6]);
|
||||
#+end_src
|
||||
|
||||
** SVD Control
|
||||
The SVD control architecture is shown below.
|
||||
The matrices $U$ and $V$ are used to decoupled the plant $G$.
|
||||
|
||||
#+begin_src latex :file svd_control.pdf
|
||||
\begin{tikzpicture}
|
||||
\node[block={2cm}{1.5cm}] (G) {$G$};
|
||||
\node[block, below right=0.6 and 0 of G] (U) {$U^{-1}$};
|
||||
\node[block, below=0.6 of G] (K) {$K_{\text{SVD}}$};
|
||||
\node[block, below left= 0.6 and 0 of G] (V) {$V^{-T}$};
|
||||
|
||||
% Inputs of the controllers
|
||||
\coordinate[] (inputd) at ($(G.south west)!0.75!(G.north west)$);
|
||||
\coordinate[] (inputu) at ($(G.south west)!0.25!(G.north west)$);
|
||||
|
||||
% Connections and labels
|
||||
\draw[<-] (inputd) -- ++(-0.8, 0) node[above right]{$D_w$};
|
||||
\draw[->] (G.east) -- ++(2.5, 0) node[above left]{$a$};
|
||||
\draw[->] ($(G.east)+(2.0, 0)$) node[branch]{} |- (U.east);
|
||||
\draw[->] (U.west) -- (K.east);
|
||||
\draw[->] (K.west) -- (V.east);
|
||||
\draw[->] (V.west) -- ++(-0.6, 0) |- (inputu) node[above left]{$\tau$};
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
[[file:figs/svd_control.png]]
|
||||
|
||||
SVD Control
|
||||
#+begin_src matlab
|
||||
G_svd = feedback(G, pinv(V')*K*pinv(U), [7:12], [1:6]);
|
||||
#+end_src
|
||||
|
||||
** Results
|
||||
The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure [[fig:stewart_platform_simscape_cl_transmissibility]].
|
||||
|
||||
#+begin_src matlab :exports results
|
||||
freqs = logspace(-3, 3, 1000);
|
||||
|
||||
figure
|
||||
|
||||
ax1 = subplot(2, 3, 1);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G( 'Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Open-Loop');
|
||||
plot(freqs, abs(squeeze(freqresp(G_cen('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Centralized');
|
||||
plot(freqs, abs(squeeze(freqresp(G_svd('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'SVD');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Transmissibility - $D_x/D_{w,x}$'); xlabel('Frequency [Hz]');
|
||||
legend('location', 'southwest');
|
||||
|
||||
ax2 = subplot(2, 3, 2);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G( 'Ay', 'Dwy')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_cen('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_svd('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Transmissibility - $D_y/D_{w,y}$'); xlabel('Frequency [Hz]');
|
||||
|
||||
ax3 = subplot(2, 3, 3);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G( 'Az', 'Dwz')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_cen('Az', 'Dwz')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_svd('Az', 'Dwz')/s^2, freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Transmissibility - $D_z/D_{w,z}$'); xlabel('Frequency [Hz]');
|
||||
|
||||
ax4 = subplot(2, 3, 4);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G( 'Arx', 'Rwx')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_cen('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_svd('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Transmissibility - $R_x/R_{w,x}$'); xlabel('Frequency [Hz]');
|
||||
|
||||
ax5 = subplot(2, 3, 5);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G( 'Ary', 'Rwy')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_cen('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_svd('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Transmissibility - $R_y/R_{w,y}$'); xlabel('Frequency [Hz]');
|
||||
|
||||
ax6 = subplot(2, 3, 6);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G( 'Arz', 'Rwz')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_cen('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(G_svd('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Transmissibility - $R_z/R_{w,z}$'); xlabel('Frequency [Hz]');
|
||||
|
||||
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/stewart_platform_simscape_cl_transmissibility.pdf', 'width', 1600, 'height', 'full');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:stewart_platform_simscape_cl_transmissibility
|
||||
#+caption: Obtained Transmissibility
|
||||
#+RESULTS:
|
||||
[[file:figs/stewart_platform_simscape_cl_transmissibility.png]]
|
||||
|
||||
* Stewart Platform - Analytical Model
|
||||
** 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
|
||||
%% Bode plot options
|
||||
opts = bodeoptions('cstprefs');
|
||||
opts.FreqUnits = 'Hz';
|
||||
opts.MagUnits = 'abs';
|
||||
opts.MagScale = 'log';
|
||||
opts.PhaseWrapping = 'on';
|
||||
opts.xlim = [1 1000];
|
||||
#+end_src
|
||||
|
||||
** Characteristics
|
||||
#+begin_src matlab
|
||||
L = 0.055;
|
||||
Zc = 0;
|
||||
m = 0.2;
|
||||
k = 1e3;
|
||||
c = 2*0.1*sqrt(k*m);
|
||||
|
||||
Rx = 0.04;
|
||||
Rz = 0.04;
|
||||
Ix = m*Rx^2;
|
||||
Iy = m*Rx^2;
|
||||
Iz = m*Rz^2;
|
||||
#+end_src
|
||||
|
||||
** Mass Matrix
|
||||
#+begin_src matlab
|
||||
M = m*[1 0 0 0 Zc 0;
|
||||
0 1 0 -Zc 0 0;
|
||||
0 0 1 0 0 0;
|
||||
0 -Zc 0 Rx^2+Zc^2 0 0;
|
||||
Zc 0 0 0 Rx^2+Zc^2 0;
|
||||
0 0 0 0 0 Rz^2];
|
||||
#+end_src
|
||||
|
||||
** Jacobian Matrix
|
||||
#+begin_src matlab
|
||||
Bj=1/sqrt(6)*[ 1 1 -2 1 1 -2;
|
||||
sqrt(3) -sqrt(3) 0 sqrt(3) -sqrt(3) 0;
|
||||
sqrt(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2) sqrt(2);
|
||||
0 0 L L -L -L;
|
||||
-L*2/sqrt(3) -L*2/sqrt(3) L/sqrt(3) L/sqrt(3) L/sqrt(3) L/sqrt(3);
|
||||
L*sqrt(2) -L*sqrt(2) L*sqrt(2) -L*sqrt(2) L*sqrt(2) -L*sqrt(2)];
|
||||
#+end_src
|
||||
|
||||
** Stifnness matrix and Damping matrix
|
||||
#+begin_src matlab
|
||||
kv = k/3; % [N/m]
|
||||
kh = 0.5*k/3; % [N/m]
|
||||
|
||||
K = diag([3*kh,3*kh,3*kv,3*kv*Rx^2/2,3*kv*Rx^2/2,3*kh*Rx^2]); % Stiffness Matrix
|
||||
|
||||
C = c*K/100000; % Damping Matrix
|
||||
#+end_src
|
||||
|
||||
** State Space System
|
||||
#+begin_src matlab
|
||||
A = [zeros(6) eye(6); -M\K -M\C];
|
||||
Bw = [zeros(6); -eye(6)];
|
||||
Bu = [zeros(6); M\Bj];
|
||||
Co = [-M\K -M\C];
|
||||
D = [zeros(6) M\Bj];
|
||||
|
||||
ST = ss(A,[Bw Bu],Co,D);
|
||||
#+end_src
|
||||
|
||||
- OUT 1-6: 6 dof
|
||||
- IN 1-6 : ground displacement in the directions of the legs
|
||||
- IN 7-12: forces in the actuators.
|
||||
#+begin_src matlab
|
||||
ST.StateName = {'x';'y';'z';'theta_x';'theta_y';'theta_z';...
|
||||
'dx';'dy';'dz';'dtheta_x';'dtheta_y';'dtheta_z'};
|
||||
ST.InputName = {'w1';'w2';'w3';'w4';'w5';'w6';...
|
||||
'u1';'u2';'u3';'u4';'u5';'u6'};
|
||||
ST.OutputName = {'ax';'ay';'az';'atheta_x';'atheta_y';'atheta_z'};
|
||||
#+end_src
|
||||
|
||||
** Transmissibility
|
||||
#+begin_src matlab
|
||||
TR=ST*[eye(6); zeros(6)];
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
figure
|
||||
subplot(231)
|
||||
bodemag(TR(1,1),opts);
|
||||
subplot(232)
|
||||
bodemag(TR(2,2),opts);
|
||||
subplot(233)
|
||||
bodemag(TR(3,3),opts);
|
||||
subplot(234)
|
||||
bodemag(TR(4,4),opts);
|
||||
subplot(235)
|
||||
bodemag(TR(5,5),opts);
|
||||
subplot(236)
|
||||
bodemag(TR(6,6),opts);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/stewart_platform_analytical_transmissibility.pdf', 'width', 'full', 'height', 'full');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:stewart_platform_analytical_transmissibility
|
||||
#+caption: Transmissibility
|
||||
#+RESULTS:
|
||||
[[file:figs/stewart_platform_analytical_transmissibility.png]]
|
||||
|
||||
** Real approximation of $G(j\omega)$ at decoupling frequency
|
||||
#+begin_src matlab
|
||||
sys1 = ST*[zeros(6); eye(6)]; % take only the forces inputs
|
||||
|
||||
dec_fr = 20;
|
||||
H1 = evalfr(sys1,j*2*pi*dec_fr);
|
||||
H2 = H1;
|
||||
D = pinv(real(H2'*H2));
|
||||
H1 = inv(D*real(H2'*diag(exp(j*angle(diag(H2*D*H2.'))/2)))) ;
|
||||
[U,S,V] = svd(H1);
|
||||
|
||||
wf = logspace(-1,2,1000);
|
||||
for i = 1:length(wf)
|
||||
H = abs(evalfr(sys1,j*2*pi*wf(i)));
|
||||
H_dec = abs(evalfr(U'*sys1*V,j*2*pi*wf(i)));
|
||||
for j = 1:size(H,2)
|
||||
g_r1(i,j) = (sum(H(j,:))-H(j,j))/H(j,j);
|
||||
g_r2(i,j) = (sum(H_dec(j,:))-H_dec(j,j))/H_dec(j,j);
|
||||
% keyboard
|
||||
end
|
||||
g_lim(i) = 0.5;
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Coupled and Decoupled Plant "Gershgorin Radii"
|
||||
#+begin_src matlab
|
||||
figure;
|
||||
title('Coupled plant')
|
||||
loglog(wf,g_r1(:,1),wf,g_r1(:,2),wf,g_r1(:,3),wf,g_r1(:,4),wf,g_r1(:,5),wf,g_r1(:,6),wf,g_lim,'--');
|
||||
legend('$a_x$','$a_y$','$a_z$','$\theta_x$','$\theta_y$','$\theta_z$','Limit');
|
||||
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/gershorin_raddii_coupled_analytical.pdf', 'width', 'full', 'height', 'full');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:gershorin_raddii_coupled_analytical
|
||||
#+caption: Gershorin Raddi for the coupled plant
|
||||
#+RESULTS:
|
||||
[[file:figs/gershorin_raddii_coupled_analytical.png]]
|
||||
|
||||
#+begin_src matlab
|
||||
figure;
|
||||
title('Decoupled plant (10 Hz)')
|
||||
loglog(wf,g_r2(:,1),wf,g_r2(:,2),wf,g_r2(:,3),wf,g_r2(:,4),wf,g_r2(:,5),wf,g_r2(:,6),wf,g_lim,'--');
|
||||
legend('$S_1$','$S_2$','$S_3$','$S_4$','$S_5$','$S_6$','Limit');
|
||||
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/gershorin_raddii_decoupled_analytical.pdf', 'width', 'full', 'height', 'full');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:gershorin_raddii_decoupled_analytical
|
||||
#+caption: Gershorin Raddi for the decoupled plant
|
||||
#+RESULTS:
|
||||
[[file:figs/gershorin_raddii_decoupled_analytical.png]]
|
||||
|
||||
** Decoupled Plant
|
||||
#+begin_src matlab
|
||||
figure;
|
||||
bodemag(U'*sys1*V,opts)
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/stewart_platform_analytical_decoupled_plant.pdf', 'width', 'full', 'height', 'full');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:stewart_platform_analytical_decoupled_plant
|
||||
#+caption: Decoupled Plant
|
||||
#+RESULTS:
|
||||
[[file:figs/stewart_platform_analytical_decoupled_plant.png]]
|
||||
|
||||
** Controller
|
||||
#+begin_src matlab
|
||||
fc = 2*pi*0.1; % Crossover Frequency [rad/s]
|
||||
c_gain = 50; %
|
||||
|
||||
cont = eye(6)*c_gain/(s+fc);
|
||||
#+end_src
|
||||
|
||||
** Closed Loop System
|
||||
#+begin_src matlab
|
||||
FEEDIN = [7:12]; % Input of controller
|
||||
FEEDOUT = [1:6]; % Output of controller
|
||||
#+end_src
|
||||
|
||||
Centralized Control
|
||||
#+begin_src matlab
|
||||
STcen = feedback(ST, inv(Bj)*cont, FEEDIN, FEEDOUT);
|
||||
TRcen = STcen*[eye(6); zeros(6)];
|
||||
#+end_src
|
||||
|
||||
SVD Control
|
||||
#+begin_src matlab
|
||||
STsvd = feedback(ST, pinv(V')*cont*pinv(U), FEEDIN, FEEDOUT);
|
||||
TRsvd = STsvd*[eye(6); zeros(6)];
|
||||
#+end_src
|
||||
|
||||
** Results
|
||||
#+begin_src matlab
|
||||
figure
|
||||
subplot(231)
|
||||
bodemag(TR(1,1),TRcen(1,1),TRsvd(1,1),opts)
|
||||
legend('OL','Centralized','SVD')
|
||||
subplot(232)
|
||||
bodemag(TR(2,2),TRcen(2,2),TRsvd(2,2),opts)
|
||||
legend('OL','Centralized','SVD')
|
||||
subplot(233)
|
||||
bodemag(TR(3,3),TRcen(3,3),TRsvd(3,3),opts)
|
||||
legend('OL','Centralized','SVD')
|
||||
subplot(234)
|
||||
bodemag(TR(4,4),TRcen(4,4),TRsvd(4,4),opts)
|
||||
legend('OL','Centralized','SVD')
|
||||
subplot(235)
|
||||
bodemag(TR(5,5),TRcen(5,5),TRsvd(5,5),opts)
|
||||
legend('OL','Centralized','SVD')
|
||||
subplot(236)
|
||||
bodemag(TR(6,6),TRcen(6,6),TRsvd(6,6),opts)
|
||||
legend('OL','Centralized','SVD')
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/stewart_platform_analytical_svd_cen_comp.pdf', 'width', 'full', 'height', 'full');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:stewart_platform_analytical_svd_cen_comp
|
||||
#+caption: Comparison of the obtained transmissibility for the centralized control and the SVD control
|
||||
#+RESULTS:
|
||||
[[file:figs/stewart_platform_analytical_svd_cen_comp.png]]
|
||||
|
||||
Reference in New Issue
Block a user