Change gravimeter axis

gravimeter/script.m

@@ -20,6 +20,10 @@ open('gravimeter.slx')
 % #+caption: Model of the gravimeter
 % [[file:figs/gravimeter_model.png]]
 
+% #+name: fig:leg_model
+% #+caption: Model of the struts
+% [[file:figs/leg_model.png]]
+
 % The parameters used for the simulation are the following:
 
 l  = 1.0; % Length of the mass [m]
@@ -57,7 +61,7 @@ io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
 
 G = linearize(mdl, io);
 G.InputName  = {'F1', 'F2', 'F3'};
-G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
+G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
 
 
 
@@ -125,22 +129,22 @@ end
 
 % The Jacobian corresponding to the sensors and actuators are defined below:
 
-Ja = [1 0  h/2
+Ja = [1 0 -h/2
-      0 1 -l/2
+      0 1  l/2
-      1 0 -h/2
+      1 0  h/2
       0 1  0];
 
-Jt = [1 0  ha
+Jt = [1 0 -ha
-      0 1 -la
+      0 1  la
-      0 1  la];
+      0 1 -la];
 
 
 
 % And the plant $\bm{G}_x$ is computed:
 
 Gx = pinv(Ja)*G*pinv(Jt');
-Gx.InputName  = {'Fx', 'Fz', 'My'};
+Gx.InputName  = {'Fx', 'Fy', 'Mz'};
-Gx.OutputName  = {'Dx', 'Dz', 'Ry'};
+Gx.OutputName  = {'Dx', 'Dy', 'Rz'};
 
 size(Gx)
 
@@ -385,6 +389,43 @@ legend('location', 'southwest');
 linkaxes([ax1,ax2],'y');
 ylim([1e-5, 1e1]);
 
+
+
+% #+name: fig:gravimeter_rga
+% #+caption: Obtained norm of RGA elements for the SVD decoupled plant and the Jacobian decoupled plant
+% #+RESULTS:
+% [[file:figs/gravimeter_rga.png]]
+
+% The RGA-number is also a measure of diagonal dominance:
+% \begin{equation}
+%   \text{RGA-number} = \| \Lambda(G) - I \|_\text{sum}
+% \end{equation}
+
+
+% Relative Gain Array for the decoupled plant using SVD:
+RGA_svd = zeros(size(Gsvd,1), size(Gsvd,2), length(freqs));
+Gsvd_inv = inv(Gsvd);
+for f_i = 1:length(freqs)
+  RGA_svd(:, :, f_i) = abs(evalfr(Gsvd, j*2*pi*freqs(f_i)).*evalfr(Gsvd_inv, j*2*pi*freqs(f_i))');
+end
+
+% Relative Gain Array for the decoupled plant using the Jacobian:
+RGA_x = zeros(size(Gx,1), size(Gx,2), length(freqs));
+Gx_inv = inv(Gx);
+for f_i = 1:length(freqs)
+  RGA_x(:, :, f_i) = abs(evalfr(Gx, j*2*pi*freqs(f_i)).*evalfr(Gx_inv, j*2*pi*freqs(f_i))');
+end
+
+RGA_num_svd = squeeze(sum(sum(RGA_svd - eye(3))));
+RGA_num_x = squeeze(sum(sum(RGA_x - eye(3))));
+
+figure;
+hold on;
+plot(freqs, RGA_num_svd)
+plot(freqs, RGA_num_x)
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('RGA-Number');
+
 % Obtained Decoupled Plants
 % <<sec:gravimeter_decoupled_plant>>
 
@@ -457,7 +498,7 @@ plot(freqs, abs(squeeze(freqresp(Gx(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5],
 set(gca,'ColorOrderIndex',1)
 plot(freqs, abs(squeeze(freqresp(Gx(1, 1), freqs, 'Hz'))), 'DisplayName', '$G_x(1,1) = A_x/F_x$');
 plot(freqs, abs(squeeze(freqresp(Gx(2, 2), freqs, 'Hz'))), 'DisplayName', '$G_x(2,2) = A_y/F_y$');
-plot(freqs, abs(squeeze(freqresp(Gx(3, 3), freqs, 'Hz'))), 'DisplayName', '$G_x(3,3) = R_y/M_y$');
+plot(freqs, abs(squeeze(freqresp(Gx(3, 3), freqs, 'Hz'))), 'DisplayName', '$G_x(3,3) = R_z/M_z$');
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
@@ -605,7 +646,7 @@ plot(freqs, abs(squeeze(freqresp(G_svd(2,2)/s^2, freqs, 'Hz'))), '--');
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
-title('$D_z/D_{w,z}$');
+title('$D_y/D_{w,y}$');
 
 ax3 = nexttile;
 hold on;
@@ -615,8 +656,421 @@ plot(freqs, abs(squeeze(freqresp(G_svd(3,3)/s^2, freqs, 'Hz'))), '--');
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
-title('$R_y/R_{w,y}$');
+title('$R_z/R_{w,z}$');
 
 linkaxes([ax1,ax2,ax3],'xy');
 xlim([freqs(1), freqs(end)]);
 xlim([1e-2, 5e1]); ylim([1e-7, 1e-2]);
+
+
+
+% #+name: fig:gravimeter_platform_simscape_cl_transmissibility
+% #+caption: Obtained Transmissibility
+% #+RESULTS:
+% [[file:figs/gravimeter_platform_simscape_cl_transmissibility.png]]
+
+
+freqs = logspace(-2, 2, 1000);
+
+figure;
+hold on;
+for out_i = 1:3
+    for in_i = out_i+1:3
+        set(gca,'ColorOrderIndex',1)
+        plot(freqs, abs(squeeze(freqresp(G(    out_i,in_i), freqs, 'Hz'))));
+        set(gca,'ColorOrderIndex',2)
+        plot(freqs, abs(squeeze(freqresp(G_cen(out_i,in_i), freqs, 'Hz'))));
+        set(gca,'ColorOrderIndex',3)
+        plot(freqs, abs(squeeze(freqresp(G_svd(out_i,in_i), freqs, 'Hz'))), '--');
+    end
+end
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Transmissibility'); xlabel('Frequency [Hz]');
+
+% Robustness to a change of actuator position
+
+% Let say we change the position of the actuators:
+
+la = l/2*0.7; % Position of Act. [m]
+ha = h/2*0.7; % Position of Act. [m]
+
+%% Name of the Simulink File
+mdl = 'gravimeter';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
+
+G = linearize(mdl, io);
+G.InputName  = {'F1', 'F2', 'F3'};
+G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
+
+G_cen_b = feedback(G, pinv(Jt')*K_cen*pinv(Ja));
+G_svd_b = feedback(G, inv(V')*K_svd*U_inv(1:3, :));
+
+
+
+% The new plant is computed, and the centralized and SVD control architectures are applied using the previsouly computed Jacobian matrices and $U$ and $V$ matrices.
+
+% The closed-loop system are still stable, and their
+
+
+freqs = logspace(-2, 2, 1000);
+
+figure;
+tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile;
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_cen(1,1)/s^2, freqs, 'Hz'))), 'DisplayName', 'Initial');
+plot(freqs, abs(squeeze(freqresp(G_cen_b(1,1)/s^2, freqs, 'Hz'))), 'DisplayName', 'Jacobian');
+plot(freqs, abs(squeeze(freqresp(G_svd_b(1,1)/s^2, freqs, 'Hz'))), '--', 'DisplayName', 'SVD');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Transmissibility'); xlabel('Frequency [Hz]');
+title('$D_x/D_{w,x}$');
+legend('location', 'southwest');
+
+ax2 = nexttile;
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_cen(2,2)/s^2, freqs, 'Hz'))));
+plot(freqs, abs(squeeze(freqresp(G_cen_b(2,2)/s^2, freqs, 'Hz'))));
+plot(freqs, abs(squeeze(freqresp(G_svd_b(2,2)/s^2, freqs, 'Hz'))), '--');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
+title('$D_y/D_{w,y}$');
+
+ax3 = nexttile;
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_cen(3,3)/s^2, freqs, 'Hz'))));
+plot(freqs, abs(squeeze(freqresp(G_cen_b(3,3)/s^2, freqs, 'Hz'))));
+plot(freqs, abs(squeeze(freqresp(G_svd_b(3,3)/s^2, freqs, 'Hz'))), '--');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
+title('$R_z/R_{w,z}$');
+
+linkaxes([ax1,ax2,ax3],'xy');
+xlim([freqs(1), freqs(end)]);
+xlim([1e-2, 5e1]); ylim([1e-7, 3e-4]);
+
+% Decoupling of the mass matrix
+
+% #+name: fig:gravimeter_model_M
+% #+caption: Choice of {O} such that the Mass Matrix is Diagonal
+% [[file:figs/gravimeter_model_M.png]]
+
+
+la = l/2; % Position of Act. [m]
+ha = h/2; % Position of Act. [m]
+
+%% Name of the Simulink File
+mdl = 'gravimeter';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
+
+G = linearize(mdl, io);
+G.InputName  = {'F1', 'F2', 'F3'};
+G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
+
+
+
+% Decoupling at the CoM (Mass decoupled)
+
+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];
+
+GM = pinv(JMa)*G*pinv(JMt');
+GM.InputName  = {'Fx', 'Fy', 'Mz'};
+GM.OutputName  = {'Dx', 'Dy', 'Rz'};
+
+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]);
+
+% Decoupling of the stiffness matrix
+
+% #+name: fig:gravimeter_model_K
+% #+caption: Choice of {O} such that the Stiffness Matrix is Diagonal
+% [[file:figs/gravimeter_model_K.png]]
+
+% Decoupling at the point where K is diagonal (x = 0, y = -h/2 from the schematic {O} frame):
+
+JKa = [1 0  0
+       0 1 -l/2
+       1 0 -h
+       0 1  0];
+
+JKt = [1 0  0
+       0 1 -la
+       0 1  la];
+
+
+
+% And the plant $\bm{G}_x$ is computed:
+
+GK = pinv(JKa)*G*pinv(JKt');
+GK.InputName  = {'Fx', 'Fy', 'Mz'};
+GK.OutputName  = {'Dx', 'Dy', 'Rz'};
+
+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(GK(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+             'HandleVisibility', 'off');
+    end
+end
+plot(freqs, abs(squeeze(freqresp(GK(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(GK(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]);
+
+% Combined decoupling of the mass and stiffness matrices
+
+% #+name: fig:gravimeter_model_KM
+% #+caption: Ideal location of the actuators such that both the mass and stiffness matrices are diagonal
+% [[file:figs/gravimeter_model_KM.png]]
+
+% To do so, the actuator position should be modified
+
+
+la = l/2; % Position of Act. [m]
+ha = 0; % Position of Act. [m]
+
+%% Name of the Simulink File
+mdl = 'gravimeter';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
+
+G = linearize(mdl, io);
+G.InputName  = {'F1', 'F2', 'F3'};
+G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
+
+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];
+
+GKM = pinv(JMa)*G*pinv(JMt');
+GKM.InputName  = {'Fx', 'Fy', 'Mz'};
+GKM.OutputName  = {'Dx', 'Dy', 'Rz'};
+
+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(GKM(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+             'HandleVisibility', 'off');
+    end
+end
+plot(freqs, abs(squeeze(freqresp(GKM(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(GKM(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]);
+
+% SVD decoupling performances                                     :noexport:
+
+
+la = l/2; % Position of Act. [m]
+ha = 0; % Position of Act. [m]
+
+c = 2e1; % Actuator Damping [N/(m/s)]
+
+%% Name of the Simulink File
+mdl = 'gravimeter';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
+
+G = linearize(mdl, io);
+G.InputName  = {'F1', 'F2', 'F3'};
+G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
+
+wc = 2*pi*10; % Decoupling frequency [rad/s]
+H1 = evalfr(G, j*wc);
+D = pinv(real(H1'*H1));
+H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
+[U,S,V] = svd(H1);
+Gsvd = inv(U)*G*inv(V');
+
+c = 5e2; % Actuator Damping [N/(m/s)]
+
+%% Name of the Simulink File
+mdl = 'gravimeter';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
+
+G = linearize(mdl, io);
+G.InputName  = {'F1', 'F2', 'F3'};
+G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
+
+wc = 2*pi*10; % Decoupling frequency [rad/s]
+H1 = evalfr(G, j*wc);
+D = pinv(real(H1'*H1));
+H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
+[U,S,V] = svd(H1);
+Gsvdd = inv(U)*G*inv(V');
+
+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];
+
+GM = pinv(JMa)*G*pinv(JMt');
+GM.InputName  = {'Fx', 'Fy', 'Mz'};
+GM.OutputName  = {'Dx', 'Dy', 'Rz'};
+
+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]);
+
+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]);
+
+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(Gsvdd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+             'HandleVisibility', 'off');
+    end
+end
+plot(freqs, abs(squeeze(freqresp(Gsvdd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+     'DisplayName', '$G_x(i,j)\ i \neq j$');
+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));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Magnitude');
+legend('location', 'southeast');
+ylim([1e-8, 1e0]);

index.org

@@ -138,7 +138,7 @@ The parameters used for the simulation are the following:
 
   G = linearize(mdl, io);
   G.InputName  = {'F1', 'F2', 'F3'};
-  G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
+  G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
 #+end_src
 
 The inputs and outputs of the plant are shown in Figure [[fig:gravimeter_plant_schematic]].
@@ -149,7 +149,7 @@ More precisely there are three inputs (the three actuator forces):
 \end{equation}
 And 4 outputs (the two 2-DoF accelerometers):
 \begin{equation}
-  \bm{a} = \begin{bmatrix} a_{1x} \\ a_{1z} \\ a_{2x} \\ a_{2z} \end{bmatrix}
+  \bm{a} = \begin{bmatrix} a_{1x} \\ a_{1y} \\ a_{2x} \\ a_{2y} \end{bmatrix}
 \end{equation}
 
 #+begin_src latex :file gravimeter_plant_schematic.pdf :tangle no :exports results
@@ -158,7 +158,7 @@ And 4 outputs (the two 2-DoF accelerometers):
 
     % Connections and labels
     \draw[<-] (G.west) -- ++(-2.0, 0) node[above right]{$\bm{\tau} = \begin{bmatrix}\tau_1 \\ \tau_2 \\ \tau_2 \end{bmatrix}$};
-    \draw[->] (G.east) -- ++( 2.0, 0)  node[above left]{$\bm{a} = \begin{bmatrix} a_{1x} \\ a_{1z} \\ a_{2x} \\ a_{2z} \end{bmatrix}$};
+    \draw[->] (G.east) -- ++( 2.0, 0)  node[above left]{$\bm{a} = \begin{bmatrix} a_{1x} \\ a_{1y} \\ a_{2x} \\ a_{2y} \end{bmatrix}$};
   \end{tikzpicture}
 #+end_src
 
@@ -232,12 +232,12 @@ Consider the control architecture shown in Figure [[fig:gravimeter_decouple_jaco
 
 The Jacobian matrix $J_{\tau}$ is used to transform forces applied by the three actuators into forces/torques applied on the gravimeter at its center of mass:
 \begin{equation}
-  \begin{bmatrix} \tau_1 \\ \tau_2 \\ \tau_3 \end{bmatrix} = J_{\tau}^{-T} \begin{bmatrix} F_x \\ F_z \\ M_y \end{bmatrix}
+  \begin{bmatrix} \tau_1 \\ \tau_2 \\ \tau_3 \end{bmatrix} = J_{\tau}^{-T} \begin{bmatrix} F_x \\ F_y \\ M_z \end{bmatrix}
 \end{equation}
 
 The Jacobian matrix $J_{a}$ is used to compute the vertical acceleration, horizontal acceleration and rotational acceleration of the mass with respect to its center of mass:
 \begin{equation}
-  \begin{bmatrix} a_x \\ a_z \\ a_{R_y} \end{bmatrix} = J_{a}^{-1} \begin{bmatrix} a_{x1} \\ a_{z1} \\ a_{x2} \\ a_{z2} \end{bmatrix}
+  \begin{bmatrix} a_x \\ a_y \\ a_{R_z} \end{bmatrix} = J_{a}^{-1} \begin{bmatrix} a_{x1} \\ a_{y1} \\ a_{x2} \\ a_{y2} \end{bmatrix}
 \end{equation}
 
 We thus define a new plant as defined in Figure [[fig:gravimeter_decouple_jacobian]].
@@ -252,10 +252,10 @@ $\bm{G}_x(s)$ correspond to the $3 \times 3$transfer function matrix from forces
     \node[block, right=0.6 of G] (Ja) {$J_{a}^{-1}$};
 
     % Connections and labels
-    \draw[<-] (Jt.west) -- ++(-2.5, 0) node[above right]{$\bm{\mathcal{F}} = \begin{bmatrix}F_x \\ F_z \\ M_y \end{bmatrix}$};
+    \draw[<-] (Jt.west) -- ++(-2.5, 0) node[above right]{$\bm{\mathcal{F}} = \begin{bmatrix}F_x \\ F_y \\ M_z \end{bmatrix}$};
     \draw[->] (Jt.east) -- (G.west)  node[above left]{$\bm{\tau}$};
     \draw[->] (G.east) -- (Ja.west)  node[above left]{$\bm{a}$};
-    \draw[->] (Ja.east) -- ++( 2.6, 0)  node[above left]{$\bm{\mathcal{A}} = \begin{bmatrix}a_x \\ a_z \\ a_{R_y} \end{bmatrix}$};
+    \draw[->] (Ja.east) -- ++( 2.6, 0)  node[above left]{$\bm{\mathcal{A}} = \begin{bmatrix}a_x \\ a_y \\ a_{R_z} \end{bmatrix}$};
 
     \begin{scope}[on background layer]
       \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gx) {};
@@ -284,8 +284,8 @@ The Jacobian corresponding to the sensors and actuators are defined below:
 And the plant $\bm{G}_x$ is computed:
 #+begin_src matlab
   Gx = pinv(Ja)*G*pinv(Jt');
-  Gx.InputName  = {'Fx', 'Fz', 'My'};
+  Gx.InputName  = {'Fx', 'Fy', 'Mz'};
-  Gx.OutputName  = {'Dx', 'Dz', 'Ry'};
+  Gx.OutputName  = {'Dx', 'Dy', 'Rz'};
 #+end_src
 
 #+begin_src matlab :results output replace :exports results
@@ -736,7 +736,7 @@ Similarly, the bode plots of the diagonal elements and off-diagonal elements of
   set(gca,'ColorOrderIndex',1)
   plot(freqs, abs(squeeze(freqresp(Gx(1, 1), freqs, 'Hz'))), 'DisplayName', '$G_x(1,1) = A_x/F_x$');
   plot(freqs, abs(squeeze(freqresp(Gx(2, 2), freqs, 'Hz'))), 'DisplayName', '$G_x(2,2) = A_y/F_y$');
-  plot(freqs, abs(squeeze(freqresp(Gx(3, 3), freqs, 'Hz'))), 'DisplayName', '$G_x(3,3) = R_y/M_y$');
+  plot(freqs, abs(squeeze(freqresp(Gx(3, 3), freqs, 'Hz'))), 'DisplayName', '$G_x(3,3) = R_z/M_z$');
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
@@ -961,7 +961,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
-  title('$D_z/D_{w,z}$');
+  title('$D_y/D_{w,y}$');
 
   ax3 = nexttile;
   hold on;
@@ -971,7 +971,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
-  title('$R_y/R_{w,y}$');
+  title('$R_z/R_{w,z}$');
 
   linkaxes([ax1,ax2,ax3],'xy');
   xlim([freqs(1), freqs(end)]);
@@ -1040,7 +1040,7 @@ Let say we change the position of the actuators:
 
   G = linearize(mdl, io);
   G.InputName  = {'F1', 'F2', 'F3'};
-  G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
+  G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
 #+end_src
 
 #+begin_src matlab :exports none
@@ -1077,7 +1077,7 @@ The closed-loop system are still stable, and their
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
-  title('$D_z/D_{w,z}$');
+  title('$D_y/D_{w,y}$');
 
   ax3 = nexttile;
   hold on;
@@ -1087,7 +1087,7 @@ The closed-loop system are still stable, and their
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
-  title('$R_y/R_{w,y}$');
+  title('$R_z/R_{w,z}$');
 
   linkaxes([ax1,ax2,ax3],'xy');
   xlim([freqs(1), freqs(end)]);
@@ -1145,7 +1145,7 @@ To do so, the actuators (springs) should be positioned such that the stiffness m
 
   G = linearize(mdl, io);
   G.InputName  = {'F1', 'F2', 'F3'};
-  G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
+  G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
 #+end_src
 
 Decoupling at the CoM (Mass decoupled)
@@ -1162,8 +1162,8 @@ Decoupling at the CoM (Mass decoupled)
 
 #+begin_src matlab
   GM = pinv(JMa)*G*pinv(JMt');
-  GM.InputName  = {'Fx', 'Fz', 'My'};
+  GM.InputName  = {'Fx', 'Fy', 'Mz'};
-  GM.OutputName  = {'Dx', 'Dz', 'Ry'};
+  GM.OutputName  = {'Dx', 'Dy', 'Rz'};
 #+end_src
 
 #+begin_src matlab :exports none
@@ -1195,7 +1195,7 @@ Decoupling at the CoM (Mass decoupled)
 #+end_src
 
 #+name: fig:jac_decoupling_M
-#+caption:
+#+caption: Diagonal and off-diagonal elements of the decoupled plant
 #+RESULTS:
 [[file:figs/jac_decoupling_M.png]]
 
@@ -1220,8 +1220,8 @@ Decoupling at the point where K is diagonal (x = 0, y = -h/2 from the schematic
 And the plant $\bm{G}_x$ is computed:
 #+begin_src matlab
   GK = pinv(JKa)*G*pinv(JKt');
-  GK.InputName  = {'Fx', 'Fz', 'My'};
+  GK.InputName  = {'Fx', 'Fy', 'Mz'};
-  GK.OutputName  = {'Dx', 'Dz', 'Ry'};
+  GK.OutputName  = {'Dx', 'Dy', 'Rz'};
 #+end_src
 
 #+begin_src matlab :exports none
@@ -1253,7 +1253,7 @@ And the plant $\bm{G}_x$ is computed:
 #+end_src
 
 #+name: fig:jac_decoupling_K
-#+caption:
+#+caption: Diagonal and off-diagonal elements of the decoupled plant
 #+RESULTS:
 [[file:figs/jac_decoupling_K.png]]
 
@@ -1286,7 +1286,7 @@ To do so, the actuator position should be modified
 
   G = linearize(mdl, io);
   G.InputName  = {'F1', 'F2', 'F3'};
-  G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
+  G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
 #+end_src
 
 #+begin_src matlab
@@ -1302,8 +1302,8 @@ To do so, the actuator position should be modified
 
 #+begin_src matlab
   GKM = pinv(JMa)*G*pinv(JMt');
-  GKM.InputName  = {'Fx', 'Fz', 'My'};
+  GKM.InputName  = {'Fx', 'Fy', 'Mz'};
-  GKM.OutputName  = {'Dx', 'Dz', 'Ry'};
+  GKM.OutputName  = {'Dx', 'Dy', 'Rz'};
 #+end_src
 
 #+begin_src matlab :exports none
@@ -1335,7 +1335,7 @@ To do so, the actuator position should be modified
 #+end_src
 
 #+name: fig:jac_decoupling_KM
-#+caption:
+#+caption: Diagonal and off-diagonal elements of the decoupled plant
 #+RESULTS:
 [[file:figs/jac_decoupling_KM.png]]
 
@@ -1373,7 +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', 'Az1', 'Ax2', 'Az2'};
+  G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
 #+end_src
 
 #+begin_src matlab
@@ -1405,7 +1405,7 @@ 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', 'Az1', 'Ax2', 'Az2'};
+  G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
 #+end_src
 
 #+begin_src matlab
@@ -1430,8 +1430,8 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
 
 #+begin_src matlab
   GM = pinv(JMa)*G*pinv(JMt');
-  GM.InputName  = {'Fx', 'Fz', 'My'};
+  GM.InputName  = {'Fx', 'Fy', 'Mz'};
-  GM.OutputName  = {'Dx', 'Dz', 'Ry'};
+  GM.OutputName  = {'Dx', 'Dy', 'Rz'};
 #+end_src
 
 #+begin_src matlab :exports none
@@ -1506,36 +1506,6 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
   ylim([1e-8, 1e0]);
 #+end_src
 
-** SVD U and V matrices                                            :noexport:
-
-#+begin_src matlab
-  la = l/2; % Position of Act. [m]
-  ha = 0; % Position of Act. [m]
-#+end_src
-
-#+begin_src matlab
-  c = 2e1; % Actuator Damping [N/(m/s)]
-#+end_src
-
-#+begin_src matlab
-  %% Name of the Simulink File
-  mdl = 'gravimeter';
-
-  %% Input/Output definition
-  clear io; io_i = 1;
-  io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
-  io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
-  io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
-  io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
-  io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
-  io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
-  io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
-
-  G = linearize(mdl, io);
-  G.InputName  = {'F1', 'F2', 'F3'};
-  G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
-#+end_src
-
 * Stewart Platform - Simscape Model
 :PROPERTIES:
 :header-args:matlab+: :tangle stewart_platform/script.m