Table of Contents
-
-
- 1. Gravimeter - Simscape Model +
- 1. Gravimeter - Simscape Model
-
-
- 1.1. Introduction -
- 1.2. Gravimeter Model - Parameters -
- 1.3. System Identification -
- 1.4. Decoupling using the Jacobian -
- 1.5. Decoupling using the SVD -
- 1.6. Verification of the decoupling using the “Gershgorin Radii” -
- 1.7. Verification of the decoupling using the “Relative Gain Array” -
- 1.8. Obtained Decoupled Plants -
- 1.9. Diagonal Controller -
- 1.10. Closed-Loop system Performances -
- 1.11. Robustness to a change of actuator position -
- 1.12. Combined / comparison of K and M decoupling +
- 1.1. Introduction +
- 1.2. Gravimeter Model - Parameters +
- 1.3. System Identification +
- 1.4. Decoupling using the Jacobian +
- 1.5. Decoupling using the SVD +
- 1.6. Verification of the decoupling using the “Gershgorin Radii” +
- 1.7. Verification of the decoupling using the “Relative Gain Array” +
- 1.8. Obtained Decoupled Plants +
- 1.9. Diagonal Controller +
- 1.10. Closed-Loop system Performances +
- 1.11. Robustness to a change of actuator position +
- 1.12. Combined / comparison of K and M decoupling
-
-
- 1.12.1. Decoupling of the mass matrix -
- 1.12.2. Decoupling of the stiffness matrix -
- 1.12.3. Combined decoupling of the mass and stiffness matrices -
- 1.12.4. Conclusion +
- 1.12.1. Decoupling of the mass matrix +
- 1.12.2. Decoupling of the stiffness matrix +
- 1.12.3. Combined decoupling of the mass and stiffness matrices +
- 1.12.4. Conclusion
- - 1.13. SVD decoupling performances +
- 1.13. SVD decoupling performances
- - 2. Stewart Platform - Simscape Model +
- 2. Stewart Platform - Simscape Model
-
-
- 2.1. Simscape Model - Parameters -
- 2.2. Identification of the plant -
- 2.3. Decoupling using the Jacobian -
- 2.4. Decoupling using the SVD -
- 2.5. Verification of the decoupling using the “Gershgorin Radii” -
- 2.6. Verification of the decoupling using the “Relative Gain Array” -
- 2.7. Obtained Decoupled Plants -
- 2.8. Diagonal Controller -
- 2.9. Closed-Loop system Performances +
- 2.1. Simscape Model - Parameters +
- 2.2. Identification of the plant +
- 2.3. Decoupling using the Jacobian +
- 2.4. Decoupling using the SVD +
- 2.5. Verification of the decoupling using the “Gershgorin Radii” +
- 2.6. Verification of the decoupling using the “Relative Gain Array” +
- 2.7. Obtained Decoupled Plants +
- 2.8. Diagonal Controller +
- 2.9. Closed-Loop system Performances
-
-
- In Section 1 on a 3-DoF gravimeter -
- In Section 2 on a 6-DoF Stewart platform +
- In Section 1 on a 3-DoF gravimeter +
- In Section 2 on a 6-DoF Stewart platform
-
1 Gravimeter - Simscape Model
+
+
1 Gravimeter - Simscape Model
-
-
1.1 Introduction
+
+
-1.1 Introduction
In this part, diagonal control using both the SVD and the Jacobian matrices are applied on a gravimeter model:
-
-
- Section 1.2: the model is described and its parameters are defined. -
- Section 1.3: the plant dynamics from the actuators to the sensors is computed from a Simscape model. -
- Section 1.4: the plant is decoupled using the Jacobian matrices. -
- Section 1.5: the Singular Value Decomposition is performed on a real approximation of the plant transfer matrix and further use to decouple the system. -
- Section 1.6: the effectiveness of the decoupling is computed using the Gershorin radii -
- Section 1.7: the effectiveness of the decoupling is computed using the Relative Gain Array -
- Section 1.8: the obtained decoupled plants are compared -
- Section 1.9: the diagonal controller is developed -
- Section 1.10: the obtained closed-loop performances for the two methods are compared +
- Section 1.2: the model is described and its parameters are defined. +
- Section 1.3: the plant dynamics from the actuators to the sensors is computed from a Simscape model. +
- Section 1.4: the plant is decoupled using the Jacobian matrices. +
- Section 1.5: the Singular Value Decomposition is performed on a real approximation of the plant transfer matrix and further use to decouple the system. +
- Section 1.6: the effectiveness of the decoupling is computed using the Gershorin radii +
- Section 1.7: the effectiveness of the decoupling is computed using the Relative Gain Array +
- Section 1.8: the obtained decoupled plants are compared +
- Section 1.9: the diagonal controller is developed +
- Section 1.10: the obtained closed-loop performances for the two methods are compared
-
1.2 Gravimeter Model - Parameters
+
+
1.2 Gravimeter Model - Parameters
-The model of the gravimeter is schematically shown in Figure 1. +The model of the gravimeter is schematically shown in Figure 1.
-
+
-
-
Figure 1: Model of the gravimeter
+
Figure 2: Model of the struts
@@ -153,55 +153,55 @@ The model of the gravimeter is schematically shown in Figure -l = 1.0; % Length of the mass [m] - h = 1.7; % Height of the mass [m] +l = 1.0; % Length of the mass [m] +h = 1.7; % Height of the mass [m] - la = l/2; % Position of Act. [m] - ha = h/2; % Position of Act. [m] +la = l/2; % Position of Act. [m] +ha = h/2; % Position of Act. [m] - m = 400; % Mass [kg] - I = 115; % Inertia [kg m^2] +m = 400; % Mass [kg] +I = 115; % Inertia [kg m^2] - k = 15e3; % Actuator Stiffness [N/m] - c = 2e1; % Actuator Damping [N/(m/s)] +k = 15e3; % Actuator Stiffness [N/m] +c = 2e1; % Actuator Damping [N/(m/s)] - deq = 0.2; % Length of the actuators [m] +deq = 0.2; % Length of the actuators [m] - g = 0; % Gravity [m/s2] +g = 0; % Gravity [m/s2]
-
1.3 System Identification
+
+
1.3 System Identification
-
%% Name of the Simulink File - mdl = 'gravimeter'; +%% 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; +%% 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 = linearize(mdl, io); +G.InputName = {'F1', 'F2', 'F3'}; +G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
-The inputs and outputs of the plant are shown in Figure 3. +The inputs and outputs of the plant are shown in Figure 3.
@@ -218,7 +218,7 @@ And 4 outputs (the two 2-DoF accelerometers): \end{equation} -
+
Figure 3: Schematic of the gravimeter plant
@@ -264,7 +264,7 @@ We can check the poles of the plant: As expected, the plant as 6 states (2 translations + 1 rotation)
-
@@ -274,11 +274,11 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
size(G) +size(G)
-The bode plot of all elements of the plant are shown in Figure 4. +The bode plot of all elements of the plant are shown in Figure 4.
-
+
Figure 4: Open Loop Transfer Function from 3 Actuators to 4 Accelerometers
@@ -286,15 +286,15 @@ The bode plot of all elements of the plant are shown in Figure -1.4 Decoupling using the Jacobian
+
+
1.4 Decoupling using the Jacobian
-Consider the control architecture shown in Figure 5. +Consider the control architecture shown in Figure 5.
@@ -312,16 +312,16 @@ The Jacobian matrix \(J_{a}\) is used to compute the vertical acceleration, hori \end{equation}
-We thus define a new plant as defined in Figure 5. +We thus define a new plant as defined in Figure 5. \[ \bm{G}_x(s) = J_a^{-1} \bm{G}(s) J_{\tau}^{-T} \]
-\(\bm{G}_x(s)\) correspond to the $3 × 3$transfer function matrix from forces and torques applied to the gravimeter at its center of mass to the absolute acceleration of the gravimeter’s center of mass (Figure 5). +\(\bm{G}_x(s)\) correspond to the $3 × 3$transfer function matrix from forces and torques applied to the gravimeter at its center of mass to the absolute acceleration of the gravimeter’s center of mass (Figure 5).
-
+
Figure 5: Decoupled plant \(\bm{G}_x\) using the Jacobian matrix \(J\)
@@ -331,14 +331,14 @@ We thus define a new plant as defined in Figure 5. The Jacobian corresponding to the sensors and actuators are defined below:
-
@@ -346,9 +346,9 @@ The Jacobian corresponding to the sensors and actuators are defined below:
And the plant \(\bm{G}_x\) is computed:
Ja = [1 0 -h/2 - 0 1 l/2 - 1 0 h/2 - 0 1 0]; +Ja = [1 0 -h/2 + 0 1 l/2 + 1 0 h/2 + 0 1 0]; - Jt = [1 0 -ha - 0 1 la - 0 1 -la]; +Jt = [1 0 -ha + 0 1 la + 0 1 -la];
-
@@ -359,11 +359,11 @@ State-space model with 3 outputs, 3 inputs, and 6 states.
Gx = pinv(Ja)*G*pinv(Jt'); - Gx.InputName = {'Fx', 'Fy', 'Mz'}; - Gx.OutputName = {'Dx', 'Dy', 'Rz'}; +Gx = pinv(Ja)*G*pinv(Jt'); +Gx.InputName = {'Fx', 'Fy', 'Mz'}; +Gx.OutputName = {'Dx', 'Dy', 'Rz'};
-The diagonal and off-diagonal elements of \(G_x\) are shown in Figure 6. +The diagonal and off-diagonal elements of \(G_x\) are shown in Figure 6.
-
+
Figure 6: Diagonal and off-diagonal elements of \(G_x\)
@@ -371,11 +371,11 @@ The diagonal and off-diagonal elements of \(G_x\) are shown in Figure -1.5 Decoupling using the SVD
+
+
1.5 Decoupling using the SVD
@@ -386,9 +386,9 @@ In order to decouple the plant using the SVD, first a real approximation of the Let’s compute a real approximation of the complex matrix \(H_1\) which corresponds to the the transfer function \(G(j\omega_c)\) from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency \(\omega_c\).
-
@@ -396,8 +396,8 @@ Let’s compute a real approximation of the complex matrix \(H_1\) which cor
The real approximation is computed as follows:
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);
-
@@ -445,7 +445,7 @@ Now, the Singular Value Decomposition of \(H_1\) is performed:
D = pinv(real(H1'*H1)); - H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2)))); +D = pinv(real(H1'*H1)); +H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
-
@@ -524,11 +524,11 @@ Now, the Singular Value Decomposition of \(H_1\) is performed:
[U,S,V] = svd(H1); +[U,S,V] = svd(H1);
-The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 7. +The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 7.
-
+
Figure 7: Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition
@@ -540,7 +540,7 @@ The decoupled plant is then:
-
@@ -554,15 +554,15 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
The 4th output (corresponding to the null singular value) is discarded, and we only keep the \(3 \times 3\) plant:
Gsvd = inv(U)*G*inv(V'); +Gsvd = inv(U)*G*inv(V');
-
Gsvd = Gsvd(1:3, 1:3); +Gsvd = Gsvd(1:3, 1:3);
-The diagonal and off-diagonal elements of the “SVD” plant are shown in Figure 8. +The diagonal and off-diagonal elements of the “SVD” plant are shown in Figure 8.
-
+
-
Figure 8: Diagonal and off-diagonal elements of \(G_{svd}\)
@@ -570,11 +570,11 @@ The diagonal and off-diagonal elements of the “SVD” plant are shown
-
1.6 Verification of the decoupling using the “Gershgorin Radii”
+
+
1.6 Verification of the decoupling using the “Gershgorin Radii”
@@ -587,7 +587,7 @@ The “Gershgorin Radii” of a matrix \(S\) is defined by:
-
+
-
Figure 9: Gershgorin Radii of the Coupled and Decoupled plants
@@ -595,11 +595,11 @@ The “Gershgorin Radii” of a matrix \(S\) is defined by:
-
1.7 Verification of the decoupling using the “Relative Gain Array”
+
+
1.7 Verification of the decoupling using the “Relative Gain Array”
@@ -613,11 +613,11 @@ where \(\times\) denotes an element by element multiplication and \(G(s)\) is an
-The obtained RGA elements are shown in Figure 10. +The obtained RGA elements are shown in Figure 10.
-
+
Figure 10: Obtained norm of RGA elements for the SVD decoupled plant and the Jacobian decoupled plant
@@ -631,7 +631,7 @@ The RGA-number is also a measure of diagonal dominance: \end{equation} -
+
-
Figure 11: RGA-Number for the Gravimeter
@@ -639,30 +639,30 @@ The RGA-number is also a measure of diagonal dominance:
-
1.8 Obtained Decoupled Plants
+
+
1.8 Obtained Decoupled Plants
-The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 12. +The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 12.
-
+
Figure 12: Decoupled Plant using SVD
-Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure 13. +Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure 13.
-
+
-
Figure 13: Gravimeter Platform Plant from forces (resp. torques) applied by the legs to the acceleration (resp. angular acceleration) of the platform as well as all the coupling terms between the two (non-diagonal terms of the transfer function matrix)
@@ -670,12 +670,12 @@ Similarly, the bode plots of the diagonal elements and off-diagonal elements of
-
1.9 Diagonal Controller
+
+
1.9 Diagonal Controller
- -The control diagram for the centralized control is shown in Figure 14. + +The control diagram for the centralized control is shown in Figure 14.
@@ -684,19 +684,19 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied
-
+
Figure 14: Control Diagram for the Centralized control
-The SVD control architecture is shown in Figure 15. +The SVD control architecture is shown in Figure 15. The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).
-
+
Figure 15: Control Diagram for the SVD control
@@ -713,32 +713,32 @@ We choose the controller to be a low pass filter:
-
wc = 2*pi*10; % Crossover Frequency [rad/s] - w0 = 2*pi*0.1; % Controller Pole [rad/s] +wc = 2*pi*10; % Crossover Frequency [rad/s] +w0 = 2*pi*0.1; % Controller Pole [rad/s]
-
K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0); - L_cen = K_cen*Gx; - G_cen = feedback(G, pinv(Jt')*K_cen*pinv(Ja)); +K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0); +L_cen = K_cen*Gx; +G_cen = feedback(G, pinv(Jt')*K_cen*pinv(Ja));
-
K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0); - L_svd = K_svd*Gsvd; - U_inv = inv(U); - G_svd = feedback(G, inv(V')*K_svd*U_inv(1:3, :)); +K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0); +L_svd = K_svd*Gsvd; +U_inv = inv(U); +G_svd = feedback(G, inv(V')*K_svd*U_inv(1:3, :));
-The obtained diagonal elements of the loop gains are shown in Figure 16. +The obtained diagonal elements of the loop gains are shown in Figure 16.
-
+
Figure 16: Comparison of the diagonal elements of the loop gains for the SVD control architecture and the Jacobian one
@@ -746,18 +746,18 @@ The obtained diagonal elements of the loop gains are shown in Figure -1.10 Closed-Loop system Performances
+
+
1.10 Closed-Loop system Performances
Let’s first verify the stability of the closed-loop systems:
-
@@ -769,7 +769,7 @@ ans =
isstable(G_cen) +isstable(G_cen)
-
@@ -781,18 +781,18 @@ ans =
isstable(G_svd) +isstable(G_svd)
-The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 17. +The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 17.
-
+
-
Figure 17: Obtained Transmissibility
+
-
Figure 18: Obtain coupling terms of the transmissibility matrix
@@ -801,15 +801,15 @@ The obtained transmissibility in Open-loop, for the centralized control as well
-
1.11 Robustness to a change of actuator position
+
+
1.11 Robustness to a change of actuator position
Let say we change the position of the actuators:
-
@@ -822,7 +822,7 @@ The closed-loop system are still stable, and their
-la = l/2*0.7; % Position of Act. [m] - ha = h/2*0.7; % Position of Act. [m] +la = l/2*0.7; % Position of Act. [m] +ha = h/2*0.7; % Position of Act. [m]
+
-
Figure 19: Transmissibility for the initial CL system and when the position of actuators are changed
@@ -830,8 +830,8 @@ The closed-loop system are still stable, and their
-
1.12 Combined / comparison of K and M decoupling
+
+
-
1.12 Combined / comparison of K and M decoupling
If we want to decouple the system at low frequency (determined by the stiffness matrix), we have to compute the Jacobians at a point where the stiffness matrix is diagonal. @@ -852,39 +852,39 @@ To do so, the actuators (springs) should be positioned such that the stiffness m
-
1.12.1 Decoupling of the mass matrix
+
+
1.12.1 Decoupling of the mass matrix
-
+
Figure 20: Choice of {O} such that the Mass Matrix is Diagonal
-
la = l/2; % Position of Act. [m] - ha = h/2; % Position of Act. [m] +la = l/2; % Position of Act. [m] +ha = h/2; % Position of Act. [m]
-
@@ -892,26 +892,26 @@ To do so, the actuators (springs) should be positioned such that the stiffness m
Decoupling at the CoM (Mass decoupled)
%% Name of the Simulink File - mdl = 'gravimeter'; +%% 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; +%% 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 = 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]; +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]; +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'}; +GM = pinv(JMa)*G*pinv(JMt'); +GM.InputName = {'Fx', 'Fy', 'Mz'}; +GM.OutputName = {'Dx', 'Dy', 'Rz'};
+
-
Figure 21: Diagonal and off-diagonal elements of the decoupled plant
@@ -919,11 +919,11 @@ Decoupling at the CoM (Mass decoupled)
-
1.12.2 Decoupling of the stiffness matrix
+
+
1.12.2 Decoupling of the stiffness matrix
-
+
Figure 22: Choice of {O} such that the Stiffness Matrix is Diagonal
@@ -933,14 +933,14 @@ Decoupling at the CoM (Mass decoupled) Decoupling at the point where K is diagonal (x = 0, y = -h/2 from the schematic {O} frame):
-
@@ -948,14 +948,14 @@ Decoupling at the point where K is diagonal (x = 0, y = -h/2 from the schematic
And the plant \(\bm{G}_x\) is computed:
JKa = [1 0 0 - 0 1 -l/2 - 1 0 -h - 0 1 0]; +JKa = [1 0 0 + 0 1 -l/2 + 1 0 -h + 0 1 0]; - JKt = [1 0 0 - 0 1 -la - 0 1 la]; +JKt = [1 0 0 + 0 1 -la + 0 1 la];
-
-GK = pinv(JKa)*G*pinv(JKt'); - GK.InputName = {'Fx', 'Fy', 'Mz'}; - GK.OutputName = {'Dx', 'Dy', 'Rz'}; +GK = pinv(JKa)*G*pinv(JKt'); +GK.InputName = {'Fx', 'Fy', 'Mz'}; +GK.OutputName = {'Dx', 'Dy', 'Rz'};
+
-
Figure 23: Diagonal and off-diagonal elements of the decoupled plant
@@ -963,11 +963,11 @@ And the plant \(\bm{G}_x\) is computed:
-
1.12.3 Combined decoupling of the mass and stiffness matrices
+
+
1.12.3 Combined decoupling of the mass and stiffness matrices
-
+
Figure 24: Ideal location of the actuators such that both the mass and stiffness matrices are diagonal
@@ -978,52 +978,52 @@ To do so, the actuator position should be modified
-
la = l/2; % Position of Act. [m] - ha = 0; % Position of Act. [m] +la = l/2; % Position of Act. [m] +ha = 0; % Position of Act. [m]
-
%% Name of the Simulink File - mdl = 'gravimeter'; +%% 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; +%% 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 = 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]; +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]; +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'}; +GKM = pinv(JMa)*G*pinv(JMt'); +GKM.InputName = {'Fx', 'Fy', 'Mz'}; +GKM.OutputName = {'Dx', 'Dy', 'Rz'};
+
-
Figure 25: Diagonal and off-diagonal elements of the decoupled plant
@@ -1031,8 +1031,8 @@ To do so, the actuator position should be modified
-
1.12.4 Conclusion
+
+
-1.12.4 Conclusion
Ideally, the mechanical system should be designed in order to have a decoupled stiffness matrix at the CoM of the solid. @@ -1046,8 +1046,8 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
-
1.13 SVD decoupling performances
+
+
1.13 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. @@ -1058,30 +1058,30 @@ Let’s do the SVD decoupling on a plant that is mostly real (low damping) a
-Start with small damping, the obtained diagonal and off-diagonal terms are shown in Figure 26. +Start with small damping, the obtained diagonal and off-diagonal terms are shown in Figure 26.
-
- c = 2e1; % Actuator Damping [N/(m/s)]
+c = 2e1; % Actuator Damping [N/(m/s)]
+
Figure 26: Diagonal and off-diagonal term when decoupling with SVD on the gravimeter with small damping
-Now take a larger damping, the obtained diagonal and off-diagonal terms are shown in Figure 27. +Now take a larger damping, the obtained diagonal and off-diagonal terms are shown in Figure 27.
-
- c = 5e2; % Actuator Damping [N/(m/s)]
+c = 5e2; % Actuator Damping [N/(m/s)]
+
-
Figure 27: Diagonal and off-diagonal term when decoupling with SVD on the gravimeter with high damping
@@ -1090,14 +1090,14 @@ Now take a larger damping, the obtained diagonal and off-diagonal terms are show
-
2 Stewart Platform - Simscape Model
+
+
2 Stewart Platform - Simscape Model
-In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure 28. +In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure 28.
@@ -1110,7 +1110,7 @@ Some notes about the system: -
+
-
Figure 28: Stewart Platform CAD View
@@ -1120,26 +1120,26 @@ Some notes about the system: The analysis of the SVD/Jacobian control applied to the Stewart platform is performed in the following sections:-
-
- Section 2.1: The parameters of the Simscape model of the Stewart platform are defined -
- Section 2.2: The plant is identified from the Simscape model and the system coupling is shown -
- Section 2.3: The plant is first decoupled using the Jacobian -
- Section 2.4: The decoupling is performed thanks to the SVD. To do so a real approximation of the plant is computed. -
- Section 2.5: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii -
- Section 2.6: -
- Section 2.7: The dynamics of the decoupled plants are shown -
- Section 2.8: A diagonal controller is defined to control the decoupled plant -
- Section 2.9: Finally, the closed loop system properties are studied +
- Section 2.1: The parameters of the Simscape model of the Stewart platform are defined +
- Section 2.2: The plant is identified from the Simscape model and the system coupling is shown +
- Section 2.3: The plant is first decoupled using the Jacobian +
- Section 2.4: The decoupling is performed thanks to the SVD. To do so a real approximation of the plant is computed. +
- Section 2.5: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii +
- Section 2.6: +
- Section 2.7: The dynamics of the decoupled plants are shown +
- Section 2.8: A diagonal controller is defined to control the decoupled plant +
- Section 2.9: Finally, the closed loop system properties are studied
-
2.1 Simscape Model - Parameters
+
+
2.1 Simscape Model - Parameters
-
@@ -1147,13 +1147,13 @@ The analysis of the SVD/Jacobian control applied to the Stewart platform is perf
Definition of spring parameters:
open('drone_platform.slx');
+open('drone_platform.slx');
-
@@ -1161,7 +1161,7 @@ Definition of spring parameters:
We suppose the sensor is perfectly positioned.
kx = 0.5*1e3/3; % [N/m] - ky = 0.5*1e3/3; - kz = 1e3/3; +kx = 0.5*1e3/3; % [N/m] +ky = 0.5*1e3/3; +kz = 1e3/3; - cx = 0.025; % [Nm/rad] - cy = 0.025; - cz = 0.025; +cx = 0.025; % [Nm/rad] +cy = 0.025; +cz = 0.025;
-
@@ -1169,7 +1169,7 @@ We suppose the sensor is perfectly positioned.
Gravity:
sens_pos_error = zeros(3,1); +sens_pos_error = zeros(3,1);
-
@@ -1177,7 +1177,7 @@ Gravity:
We load the Jacobian (previously computed from the geometry):
g = 0; +g = 0;
-
@@ -1185,21 +1185,21 @@ We load the Jacobian (previously computed from the geometry):
We initialize other parameters:
load('jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J'); +load('jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
-
-U = eye(6); - V = eye(6); - Kc = tf(zeros(6)); +U = eye(6); +V = eye(6); +Kc = tf(zeros(6));
+
-
Figure 29: General view of the Simscape Model
+
-
Figure 30: Simscape model of the Stewart platform
@@ -1207,15 +1207,15 @@ We initialize other parameters:
-
2.2 Identification of the plant
+
+
2.2 Identification of the plant
-The plant shown in Figure 31 is identified from the Simscape model. +The plant shown in Figure 31 is identified from the Simscape model.
@@ -1231,31 +1231,31 @@ The outputs are the 6 accelerations measured by the inertial unit.
-
+
Figure 31: Considered plant \(\bm{G} = \begin{bmatrix}G_d\\G_u\end{bmatrix}\). \(D_w\) is the translation/rotation of the support, \(\tau\) the actuator forces, \(a\) the acceleration/angular acceleration of the top platform
-
@@ -1263,7 +1263,7 @@ The outputs are the 6 accelerations measured by the inertial unit.
There are 24 states (6dof for the bottom platform + 6dof for the top platform).
%% Name of the Simulink File - mdl = 'drone_platform'; +%% Name of the Simulink File +mdl = 'drone_platform'; - %% Input/Output definition - clear io; io_i = 1; - io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1; % Ground Motion - io(io_i) = linio([mdl, '/V-T'], 1, 'openinput'); io_i = io_i + 1; % Actuator Forces - io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Top platform acceleration +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1; % Ground Motion +io(io_i) = linio([mdl, '/V-T'], 1, 'openinput'); io_i = io_i + 1; % Actuator Forces +io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Top platform acceleration - G = linearize(mdl, io); - G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ... - 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; - G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}; +G = linearize(mdl, io); +G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ... + 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; +G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}; - % Plant - Gu = G(:, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); - % Disturbance dynamics - Gd = G(:, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'}); +% Plant +Gu = G(:, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); +% Disturbance dynamics +Gd = G(:, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'});
-
@@ -1273,7 +1273,7 @@ State-space model with 6 outputs, 12 inputs, and 24 states.
size(G) +size(G)
-The elements of the transfer matrix \(\bm{G}\) corresponding to the transfer function from actuator forces \(\tau\) to the measured acceleration \(a\) are shown in Figure 32. +The elements of the transfer matrix \(\bm{G}\) corresponding to the transfer function from actuator forces \(\tau\) to the measured acceleration \(a\) are shown in Figure 32.
@@ -1281,7 +1281,7 @@ One can easily see that the system is strongly coupled.
-
+
-
Figure 32: Magnitude of all 36 elements of the transfer function matrix \(G_u\)
@@ -1289,12 +1289,12 @@ One can easily see that the system is strongly coupled.
-
2.3 Decoupling using the Jacobian
+
+
2.3 Decoupling using the Jacobian
- -Consider the control architecture shown in Figure 33. + +Consider the control architecture shown in Figure 33. The Jacobian matrix is used to transform forces/torques applied on the top platform to the equivalent forces applied by each actuator.
@@ -1376,7 +1376,7 @@ The Jacobian matrix is computed from the geometry of the platform (position and -
+
-
Figure 33: Decoupled plant \(\bm{G}_x\) using the Jacobian matrix \(J\)
@@ -1392,18 +1392,18 @@ We define a new plant:
-
Gx = Gu*inv(J'); - Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; +Gx = Gu*inv(J'); +Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
-
2.4 Decoupling using the SVD
+
+
2.4 Decoupling using the SVD
@@ -1414,9 +1414,9 @@ In order to decouple the plant using the SVD, first a real approximation of the Let’s compute a real approximation of the complex matrix \(H_1\) which corresponds to the the transfer function \(G_u(j\omega_c)\) from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency \(\omega_c\).
-
@@ -1424,8 +1424,8 @@ Let’s compute a real approximation of the complex matrix \(H_1\) which cor
The real approximation is computed as follows:
wc = 2*pi*30; % Decoupling frequency [rad/s] +wc = 2*pi*30; % Decoupling frequency [rad/s] - H1 = evalfr(Gu, j*wc); +H1 = evalfr(Gu, j*wc);
-
@@ -1588,7 +1588,7 @@ Now, the Singular Value Decomposition of \(H_1\) is performed:
D = pinv(real(H1'*H1)); - H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2)))); +D = pinv(real(H1'*H1)); +H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
-
@@ -1739,11 +1739,11 @@ Now, the Singular Value Decomposition of \(H_1\) is performed:
[U,~,V] = svd(H1);
+[U,~,V] = svd(H1);
-The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 34. +The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 34.
-
+
-
Figure 34: Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition
@@ -1755,17 +1755,17 @@ The decoupled plant is then:
-
Gsvd = inv(U)*Gu*inv(V'); +Gsvd = inv(U)*Gu*inv(V');
-
2.5 Verification of the decoupling using the “Gershgorin Radii”
+
+
2.5 Verification of the decoupling using the “Gershgorin Radii”
@@ -1781,7 +1781,7 @@ The “Gershgorin Radii” of a matrix \(S\) is defined by: This is computed over the following frequencies.
-
+
-
Figure 35: Gershgorin Radii of the Coupled and Decoupled plants
@@ -1789,11 +1789,11 @@ This is computed over the following frequencies.
-
2.6 Verification of the decoupling using the “Relative Gain Array”
+
+
2.6 Verification of the decoupling using the “Relative Gain Array”
@@ -1807,11 +1807,11 @@ where \(\times\) denotes an element by element multiplication and \(G(s)\) is an
-The obtained RGA elements are shown in Figure 36. +The obtained RGA elements are shown in Figure 36.
-
+
-
Figure 36: Obtained norm of RGA elements for the SVD decoupled plant and the Jacobian decoupled plant
@@ -1819,30 +1819,30 @@ The obtained RGA elements are shown in Figure 36.
-
2.7 Obtained Decoupled Plants
+
+
2.7 Obtained Decoupled Plants
-The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 37. +The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 37.
-
+
Figure 37: Decoupled Plant using SVD
-Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure 38. +Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure 38.
-
+
-
Figure 38: Stewart Platform Plant from forces (resp. torques) applied by the legs to the acceleration (resp. angular acceleration) of the platform as well as all the coupling terms between the two (non-diagonal terms of the transfer function matrix)
@@ -1850,12 +1850,12 @@ Similarly, the bode plots of the diagonal elements and off-diagonal elements of
-
2.8 Diagonal Controller
+
+
2.8 Diagonal Controller
- -The control diagram for the centralized control is shown in Figure 39. + +The control diagram for the centralized control is shown in Figure 39.
@@ -1864,19 +1864,19 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied
-
+
Figure 39: Control Diagram for the Centralized control
-The SVD control architecture is shown in Figure 40. +The SVD control architecture is shown in Figure 40. The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).
-
+
Figure 40: Control Diagram for the SVD control
@@ -1893,31 +1893,31 @@ We choose the controller to be a low pass filter:
-
wc = 2*pi*80; % Crossover Frequency [rad/s] - w0 = 2*pi*0.1; % Controller Pole [rad/s] +wc = 2*pi*80; % Crossover Frequency [rad/s] +w0 = 2*pi*0.1; % Controller Pole [rad/s]
-
K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0); - L_cen = K_cen*Gx; - G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]); +K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0); +L_cen = K_cen*Gx; +G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);
-
K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0); - L_svd = K_svd*Gsvd; - G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]); +K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0); +L_svd = K_svd*Gsvd; +G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]);
-The obtained diagonal elements of the loop gains are shown in Figure 41. +The obtained diagonal elements of the loop gains are shown in Figure 41.
-
+
Figure 41: Comparison of the diagonal elements of the loop gains for the SVD control architecture and the Jacobian one
@@ -1925,18 +1925,18 @@ The obtained diagonal elements of the loop gains are shown in Figure -2.9 Closed-Loop system Performances
+
+
2.9 Closed-Loop system Performances
Let’s first verify the stability of the closed-loop systems:
-
@@ -1948,7 +1948,7 @@ ans =
isstable(G_cen) +isstable(G_cen)
-
@@ -1960,11 +1960,11 @@ ans =
isstable(G_svd) +isstable(G_svd)
-The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 42. +The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 42.
-
+
>
+<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
- <>
+<>
#+end_src
#+begin_src matlab :tangle no
- addpath('gravimeter');
+addpath('gravimeter');
#+end_src
#+begin_src matlab
- freqs = logspace(-1, 2, 1000);
+freqs = logspace(-1, 2, 1000);
#+end_src
** Gravimeter Model - Parameters
<>
#+begin_src matlab :exports none
- open('gravimeter.slx')
+open('gravimeter.slx')
#+end_src
The model of the gravimeter is schematically shown in Figure [[fig:gravimeter_model]].
@@ -111,43 +111,43 @@ The model of the gravimeter is schematically shown in Figure [[fig:gravimeter_mo
The parameters used for the simulation are the following:
#+begin_src matlab
- l = 1.0; % Length of the mass [m]
- h = 1.7; % Height of the mass [m]
+l = 1.0; % Length of the mass [m]
+h = 1.7; % Height of the mass [m]
- la = l/2; % Position of Act. [m]
- ha = h/2; % Position of Act. [m]
+la = l/2; % Position of Act. [m]
+ha = h/2; % Position of Act. [m]
- m = 400; % Mass [kg]
- I = 115; % Inertia [kg m^2]
+m = 400; % Mass [kg]
+I = 115; % Inertia [kg m^2]
- k = 15e3; % Actuator Stiffness [N/m]
- c = 2e1; % Actuator Damping [N/(m/s)]
+k = 15e3; % Actuator Stiffness [N/m]
+c = 2e1; % Actuator Damping [N/(m/s)]
- deq = 0.2; % Length of the actuators [m]
+deq = 0.2; % Length of the actuators [m]
- g = 0; % Gravity [m/s2]
+g = 0; % Gravity [m/s2]
#+end_src
** System Identification
<>
#+begin_src matlab
- %% Name of the Simulink File
- mdl = 'gravimeter';
+%% 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;
+%% 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 = linearize(mdl, io);
+G.InputName = {'F1', 'F2', 'F3'};
+G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
#+end_src
The inputs and outputs of the plant are shown in Figure [[fig:gravimeter_plant_schematic]].
@@ -162,13 +162,13 @@ And 4 outputs (the two 2-DoF accelerometers):
\end{equation}
#+begin_src latex :file gravimeter_plant_schematic.pdf :tangle no :exports results
- \begin{tikzpicture}
- \node[block] (G) {$\bm{G}$};
+\begin{tikzpicture}
+ \node[block] (G) {$\bm{G}$};
- % 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]{$\bm{a} = \begin{bmatrix} a_{1x} \\ a_{1y} \\ a_{2x} \\ a_{2y} \end{bmatrix}$};
- \end{tikzpicture}
+ % 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]{$\bm{a} = \begin{bmatrix} a_{1x} \\ a_{1y} \\ a_{2x} \\ a_{2y} \end{bmatrix}$};
+\end{tikzpicture}
#+end_src
#+name: fig:gravimeter_plant_schematic
@@ -178,7 +178,7 @@ And 4 outputs (the two 2-DoF accelerometers):
We can check the poles of the plant:
#+begin_src matlab :results value replace :exports results
- pole(G)
+pole(G)
#+end_src
#+RESULTS:
@@ -191,7 +191,7 @@ We can check the poles of the plant:
As expected, the plant as 6 states (2 translations + 1 rotation)
#+begin_src matlab :results output replace
- size(G)
+size(G)
#+end_src
#+RESULTS:
@@ -200,33 +200,33 @@ As expected, the plant as 6 states (2 translations + 1 rotation)
The bode plot of all elements of the plant are shown in Figure [[fig:open_loop_tf]].
#+begin_src matlab :exports none
- figure;
- tiledlayout(4, 3, 'TileSpacing', 'None', 'Padding', 'None');
+figure;
+tiledlayout(4, 3, 'TileSpacing', 'None', 'Padding', 'None');
- for out_i = 1:4
- for in_i = 1:3
- nexttile;
- plot(freqs, abs(squeeze(freqresp(G(out_i,in_i), freqs, 'Hz'))), '-');
- set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
- xlim([1e-1, 2e1]); ylim([1e-4, 1e0]);
+for out_i = 1:4
+ for in_i = 1:3
+ nexttile;
+ plot(freqs, abs(squeeze(freqresp(G(out_i,in_i), freqs, 'Hz'))), '-');
+ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ xlim([1e-1, 2e1]); ylim([1e-4, 1e0]);
- if in_i == 1
- ylabel('Amplitude [m/N]')
- else
- set(gca, 'YTickLabel',[]);
- end
+ if in_i == 1
+ ylabel('Amplitude [m/N]')
+ else
+ set(gca, 'YTickLabel',[]);
+ end
- if out_i == 4
- xlabel('Frequency [Hz]')
- else
- set(gca, 'XTickLabel',[]);
- end
- end
- end
+ if out_i == 4
+ xlabel('Frequency [Hz]')
+ else
+ set(gca, 'XTickLabel',[]);
+ end
+ end
+end
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
- exportFig('figs/open_loop_tf.pdf', 'width', 'full', 'height', 'full');
+exportFig('figs/open_loop_tf.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:open_loop_tf
@@ -256,22 +256,22 @@ We thus define a new plant as defined in Figure [[fig:gravimeter_decouple_jacobi
$\bm{G}_x(s)$ correspond to the $3 \times 3$transfer function matrix from forces and torques applied to the gravimeter at its center of mass to the absolute acceleration of the gravimeter's center of mass (Figure [[fig:gravimeter_decouple_jacobian]]).
#+begin_src latex :file gravimeter_decouple_jacobian.pdf :tangle no :exports results
- \begin{tikzpicture}
- \node[block] (G) {$\bm{G}$};
- \node[block, left=0.6 of G] (Jt) {$J_{\tau}^{-T}$};
- \node[block, right=0.6 of G] (Ja) {$J_{a}^{-1}$};
+\begin{tikzpicture}
+ \node[block] (G) {$\bm{G}$};
+ \node[block, left=0.6 of G] (Jt) {$J_{\tau}^{-T}$};
+ \node[block, right=0.6 of G] (Ja) {$J_{a}^{-1}$};
- % Connections and labels
- \draw[<-] (Jt.west) -- ++(-2.5, 0) node[above right]{$\bm{\mathcal{F}} = \begin{bmatrix}F_x \\ F_y \\ M_z \end{bmatrix}$};
- \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]{$\bm{\mathcal{A}} = \begin{bmatrix}a_x \\ a_y \\ a_{R_z} \end{bmatrix}$};
+ % Connections and labels
+ \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]{$\bm{\mathcal{A}} = \begin{bmatrix}a_x \\ a_y \\ a_{R_z} \end{bmatrix}$};
- \begin{scope}[on background layer]
- \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gx) {};
- \node[below right] at (Gx.north west) {$\bm{G}_x$};
- \end{scope}
- \end{tikzpicture}
+ \begin{scope}[on background layer]
+ \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gx) {};
+ \node[below right] at (Gx.north west) {$\bm{G}_x$};
+ \end{scope}
+\end{tikzpicture}
#+end_src
#+name: fig:gravimeter_decouple_jacobian
@@ -281,25 +281,25 @@ $\bm{G}_x(s)$ correspond to the $3 \times 3$transfer function matrix from forces
The Jacobian corresponding to the sensors and actuators are defined below:
#+begin_src matlab
- Ja = [1 0 -h/2
- 0 1 l/2
- 1 0 h/2
- 0 1 0];
+Ja = [1 0 -h/2
+ 0 1 l/2
+ 1 0 h/2
+ 0 1 0];
- Jt = [1 0 -ha
- 0 1 la
- 0 1 -la];
+Jt = [1 0 -ha
+ 0 1 la
+ 0 1 -la];
#+end_src
And the plant $\bm{G}_x$ is computed:
#+begin_src matlab
- Gx = pinv(Ja)*G*pinv(Jt');
- Gx.InputName = {'Fx', 'Fy', 'Mz'};
- Gx.OutputName = {'Dx', 'Dy', 'Rz'};
+Gx = pinv(Ja)*G*pinv(Jt');
+Gx.InputName = {'Fx', 'Fy', 'Mz'};
+Gx.OutputName = {'Dx', 'Dy', 'Rz'};
#+end_src
#+begin_src matlab :results output replace :exports results
- size(Gx)
+size(Gx)
#+end_src
#+RESULTS:
@@ -309,31 +309,31 @@ And the plant $\bm{G}_x$ is computed:
The diagonal and off-diagonal elements of $G_x$ are shown in Figure [[fig:gravimeter_jacobian_plant]].
#+begin_src matlab :exports none
- figure;
+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(Gx(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
- 'HandleVisibility', 'off');
- end
- end
- plot(freqs, abs(squeeze(freqresp(Gx(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
+% Magnitude
+hold on;
+for i_in = 1:3
+ for i_out = [1:i_in-1, i_in+1:3]
+ plot(freqs, abs(squeeze(freqresp(Gx(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+ 'HandleVisibility', 'off');
+ end
+end
+plot(freqs, abs(squeeze(freqresp(Gx(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(Gx(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
+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 :tangle no :exports results :results file replace
- exportFig('figs/gravimeter_jacobian_plant.pdf', 'width', 'wide', 'height', 'normal');
+exportFig('figs/gravimeter_jacobian_plant.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:gravimeter_jacobian_plant
@@ -348,19 +348,19 @@ In order to decouple the plant using the SVD, first a real approximation of the
Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G(j\omega_c)$ 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*10; % Decoupling frequency [rad/s]
+wc = 2*pi*10; % Decoupling frequency [rad/s]
- H1 = evalfr(G, j*wc);
+H1 = evalfr(G, j*wc);
#+end_src
The real approximation is computed as follows:
#+begin_src matlab
- D = pinv(real(H1'*H1));
- H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
+D = pinv(real(H1'*H1));
+H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no
- data2orgtable(H1, {}, {}, ' %.2g ');
+data2orgtable(H1, {}, {}, ' %.2g ');
#+end_src
#+caption: Real approximate of $G$ at the decoupling frequency $\omega_c$
@@ -375,11 +375,11 @@ Now, the Singular Value Decomposition of $H_1$ is performed:
\[ H_1 = U \Sigma V^H \]
#+begin_src matlab
- [U,S,V] = svd(H1);
+[U,S,V] = svd(H1);
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no
- data2orgtable(U, {}, {}, ' %.2f ');
+data2orgtable(U, {}, {}, ' %.2f ');
#+end_src
#+caption: $U$ matrix
@@ -390,7 +390,7 @@ Now, the Singular Value Decomposition of $H_1$ is performed:
| 0.03 | 0.73 | 0.06 | 0.68 |
#+begin_src matlab :exports results :results value table replace :tangle no
- data2orgtable(V, {}, {}, ' %.2f ');
+data2orgtable(V, {}, {}, ' %.2f ');
#+end_src
#+caption: $V$ matrix
@@ -402,23 +402,23 @@ Now, the Singular Value Decomposition of $H_1$ is performed:
The obtained matrices $U$ and $V$ are used to decouple the system as shown in Figure [[fig:gravimeter_decouple_svd]].
#+begin_src latex :file gravimeter_decouple_svd.pdf :tangle no :exports results
- \begin{tikzpicture}
- \node[block] (G) {$\bm{G}$};
+\begin{tikzpicture}
+ \node[block] (G) {$\bm{G}$};
- \node[block, left=0.6 of G.west] (V) {$V^{-T}$};
- \node[block, right=0.6 of G.east] (U) {$U^{-1}$};
+ \node[block, left=0.6 of G.west] (V) {$V^{-T}$};
+ \node[block, right=0.6 of G.east] (U) {$U^{-1}$};
- % Connections and labels
- \draw[<-] (V.west) -- ++(-1.0, 0) node[above right]{$u$};
- \draw[->] (V.east) -- (G.west) node[above left]{$\tau$};
- \draw[->] (G.east) -- (U.west) node[above left]{$a$};
- \draw[->] (U.east) -- ++( 1.0, 0) node[above left]{$y$};
+ % Connections and labels
+ \draw[<-] (V.west) -- ++(-1.0, 0) node[above right]{$u$};
+ \draw[->] (V.east) -- (G.west) node[above left]{$\tau$};
+ \draw[->] (G.east) -- (U.west) node[above left]{$a$};
+ \draw[->] (U.east) -- ++( 1.0, 0) node[above left]{$y$};
- \begin{scope}[on background layer]
- \node[fit={(V.south west) (G.north-|U.east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gsvd) {};
- \node[below right] at (Gsvd.north west) {$\bm{G}_{SVD}$};
- \end{scope}
- \end{tikzpicture}
+ \begin{scope}[on background layer]
+ \node[fit={(V.south west) (G.north-|U.east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gsvd) {};
+ \node[below right] at (Gsvd.north west) {$\bm{G}_{SVD}$};
+ \end{scope}
+\end{tikzpicture}
#+end_src
#+name: fig:gravimeter_decouple_svd
@@ -430,11 +430,11 @@ The decoupled plant is then:
\[ \bm{G}_{SVD}(s) = U^{-1} \bm{G}(s) V^{-H} \]
#+begin_src matlab
- Gsvd = inv(U)*G*inv(V');
+Gsvd = inv(U)*G*inv(V');
#+end_src
#+begin_src matlab :results output replace :exports results
- size(Gsvd)
+size(Gsvd)
#+end_src
#+RESULTS:
@@ -443,36 +443,36 @@ The decoupled plant is then:
The 4th output (corresponding to the null singular value) is discarded, and we only keep the $3 \times 3$ plant:
#+begin_src matlab
- Gsvd = Gsvd(1:3, 1:3);
+Gsvd = Gsvd(1:3, 1:3);
#+end_src
The diagonal and off-diagonal elements of the "SVD" plant are shown in Figure [[fig:gravimeter_svd_plant]].
#+begin_src matlab :exports none
- figure;
+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
+% 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', 'southwest', 'FontSize', 8);
- ylim([1e-8, 1e0]);
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Magnitude');
+legend('location', 'southwest', 'FontSize', 8);
+ylim([1e-8, 1e0]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
- exportFig('figs/gravimeter_svd_plant.pdf', 'width', 'wide', 'height', 'normal');
+exportFig('figs/gravimeter_svd_plant.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:gravimeter_svd_plant
@@ -489,51 +489,51 @@ The "Gershgorin Radii" of a matrix $S$ is defined by:
\[ \zeta_i(j\omega) = \frac{\sum\limits_{j\neq i}|S_{ij}(j\omega)|}{|S_{ii}(j\omega)|} \]
#+begin_src matlab :exports none
- % Gershgorin Radii for the coupled plant:
- Gr_coupled = zeros(length(freqs), size(G,2));
- H = abs(squeeze(freqresp(G, freqs, 'Hz')));
- for out_i = 1:size(G,2)
- Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
- end
+% Gershgorin Radii for the coupled plant:
+Gr_coupled = zeros(length(freqs), size(G,2));
+H = abs(squeeze(freqresp(G, freqs, 'Hz')));
+for out_i = 1:size(G,2)
+ Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
+end
- % Gershgorin Radii for the decoupled plant using SVD:
- Gr_decoupled = zeros(length(freqs), size(Gsvd,2));
- H = abs(squeeze(freqresp(Gsvd, freqs, 'Hz')));
- for out_i = 1:size(Gsvd,2)
- Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
- end
+% Gershgorin Radii for the decoupled plant using SVD:
+Gr_decoupled = zeros(length(freqs), size(Gsvd,2));
+H = abs(squeeze(freqresp(Gsvd, freqs, 'Hz')));
+for out_i = 1:size(Gsvd,2)
+ Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
+end
- % Gershgorin Radii for the decoupled plant using the Jacobian:
- Gr_jacobian = zeros(length(freqs), size(Gx,2));
- H = abs(squeeze(freqresp(Gx, freqs, 'Hz')));
- for out_i = 1:size(Gx,2)
- Gr_jacobian(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
- end
+% Gershgorin Radii for the decoupled plant using the Jacobian:
+Gr_jacobian = zeros(length(freqs), size(Gx,2));
+H = abs(squeeze(freqresp(Gx, freqs, 'Hz')));
+for out_i = 1:size(Gx,2)
+ Gr_jacobian(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
+end
#+end_src
#+begin_src matlab :exports results
- figure;
- hold on;
- plot(freqs, Gr_coupled(:,1), 'DisplayName', 'Coupled');
- plot(freqs, Gr_decoupled(:,1), 'DisplayName', 'SVD');
- plot(freqs, Gr_jacobian(:,1), 'DisplayName', 'Jacobian');
- for in_i = 2:3
- set(gca,'ColorOrderIndex',1)
- plot(freqs, Gr_coupled(:,in_i), 'HandleVisibility', 'off');
- set(gca,'ColorOrderIndex',2)
- plot(freqs, Gr_decoupled(:,in_i), 'HandleVisibility', 'off');
- set(gca,'ColorOrderIndex',3)
- plot(freqs, Gr_jacobian(:,in_i), 'HandleVisibility', 'off');
- end
- set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
- hold off;
- xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
- legend('location', 'southwest');
- ylim([1e-4, 1e2]);
+figure;
+hold on;
+plot(freqs, Gr_coupled(:,1), 'DisplayName', 'Coupled');
+plot(freqs, Gr_decoupled(:,1), 'DisplayName', 'SVD');
+plot(freqs, Gr_jacobian(:,1), 'DisplayName', 'Jacobian');
+for in_i = 2:3
+ set(gca,'ColorOrderIndex',1)
+ plot(freqs, Gr_coupled(:,in_i), 'HandleVisibility', 'off');
+ set(gca,'ColorOrderIndex',2)
+ plot(freqs, Gr_decoupled(:,in_i), 'HandleVisibility', 'off');
+ set(gca,'ColorOrderIndex',3)
+ plot(freqs, Gr_jacobian(:,in_i), 'HandleVisibility', 'off');
+end
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+hold off;
+xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
+legend('location', 'southwest');
+ylim([1e-4, 1e2]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
- exportFig('figs/gravimeter_gershgorin_radii.pdf', 'eps', true, 'width', 'wide', 'height', 'normal');
+exportFig('figs/gravimeter_gershgorin_radii.pdf', 'eps', true, 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:gravimeter_gershgorin_radii
@@ -553,75 +553,75 @@ where $\times$ denotes an element by element multiplication and $G(s)$ is an $n
The obtained RGA elements are shown in Figure [[fig:gravimeter_rga]].
#+begin_src matlab :exports none
- % Relative Gain Array for the decoupled plant using SVD:
- RGA_svd = zeros(length(freqs), size(Gsvd,1), size(Gsvd,2));
- Gsvd_inv = inv(Gsvd);
- for f_i = 1:length(freqs)
+% Relative Gain Array for the decoupled plant using SVD:
+RGA_svd = zeros(length(freqs), size(Gsvd,1), size(Gsvd,2));
+Gsvd_inv = inv(Gsvd);
+for f_i = 1:length(freqs)
RGA_svd(f_i, :, :) = abs(evalfr(Gsvd, j*2*pi*freqs(f_i)).*evalfr(Gsvd_inv, j*2*pi*freqs(f_i))');
- end
+end
- % Relative Gain Array for the decoupled plant using the Jacobian:
- RGA_x = zeros(length(freqs), size(Gx,1), size(Gx,2));
- Gx_inv = inv(Gx);
- for f_i = 1:length(freqs)
+% Relative Gain Array for the decoupled plant using the Jacobian:
+RGA_x = zeros(length(freqs), size(Gx,1), size(Gx,2));
+Gx_inv = inv(Gx);
+for f_i = 1:length(freqs)
RGA_x(f_i, :, :) = abs(evalfr(Gx, j*2*pi*freqs(f_i)).*evalfr(Gx_inv, j*2*pi*freqs(f_i))');
- end
+end
#+end_src
#+begin_src matlab :exports none
- figure;
- tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
+figure;
+tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
- ax1 = nexttile;
- hold on;
- for i_in = 1:3
- for i_out = [1:i_in-1, i_in+1:3]
- plot(freqs, RGA_svd(:, i_out, i_in), '--', 'color', [0 0 0 0.2], ...
- 'HandleVisibility', 'off');
- end
- end
- plot(freqs, RGA_svd(:, 1, 2), '--', 'color', [0 0 0 0.2], ...
- 'DisplayName', '$RGA_{SVD}(i,j),\ i \neq j$');
+ax1 = nexttile;
+hold on;
+for i_in = 1:3
+ for i_out = [1:i_in-1, i_in+1:3]
+ plot(freqs, RGA_svd(:, i_out, i_in), '--', 'color', [0 0 0 0.2], ...
+ 'HandleVisibility', 'off');
+ end
+end
+plot(freqs, RGA_svd(:, 1, 2), '--', 'color', [0 0 0 0.2], ...
+ 'DisplayName', '$RGA_{SVD}(i,j),\ i \neq j$');
- plot(freqs, RGA_svd(:, 1, 1), 'k-', ...
- 'DisplayName', '$RGA_{SVD}(i,i)$');
- for ch_i = 1:3
+plot(freqs, RGA_svd(:, 1, 1), 'k-', ...
+ 'DisplayName', '$RGA_{SVD}(i,i)$');
+for ch_i = 1:3
plot(freqs, RGA_svd(:, ch_i, ch_i), 'k-', ...
'HandleVisibility', 'off');
- end
- hold off;
- set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
- ylabel('Magnitude'); xlabel('Frequency [Hz]');
- legend('location', 'southwest');
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Magnitude'); xlabel('Frequency [Hz]');
+legend('location', 'southwest');
- ax2 = nexttile;
- hold on;
- for i_in = 1:3
- for i_out = [1:i_in-1, i_in+1:3]
- plot(freqs, RGA_x(:, i_out, i_in), '--', 'color', [0 0 0 0.2], ...
- 'HandleVisibility', 'off');
- end
- end
- plot(freqs, RGA_x(:, 1, 2), '--', 'color', [0 0 0 0.2], ...
- 'DisplayName', '$RGA_{X}(i,j),\ i \neq j$');
+ax2 = nexttile;
+hold on;
+for i_in = 1:3
+ for i_out = [1:i_in-1, i_in+1:3]
+ plot(freqs, RGA_x(:, i_out, i_in), '--', 'color', [0 0 0 0.2], ...
+ 'HandleVisibility', 'off');
+ end
+end
+plot(freqs, RGA_x(:, 1, 2), '--', 'color', [0 0 0 0.2], ...
+ 'DisplayName', '$RGA_{X}(i,j),\ i \neq j$');
- plot(freqs, RGA_x(:, 1, 1), 'k-', ...
- 'DisplayName', '$RGA_{X}(i,i)$');
- for ch_i = 1:3
+plot(freqs, RGA_x(:, 1, 1), 'k-', ...
+ 'DisplayName', '$RGA_{X}(i,i)$');
+for ch_i = 1:3
plot(freqs, RGA_x(:, ch_i, ch_i), 'k-', ...
'HandleVisibility', 'off');
- end
- hold off;
- set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
- xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
- legend('location', 'southwest');
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
+legend('location', 'southwest');
- linkaxes([ax1,ax2],'y');
- ylim([1e-5, 1e1]);
+linkaxes([ax1,ax2],'y');
+ylim([1e-5, 1e1]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
- exportFig('figs/gravimeter_rga.pdf', 'width', 'wide', 'height', 'tall');
+exportFig('figs/gravimeter_rga.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:gravimeter_rga
@@ -635,35 +635,35 @@ The RGA-number is also a measure of diagonal dominance:
\end{equation}
#+begin_src matlab :exports none
- % 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)
+% 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
+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)
+% 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
+end
#+end_src
#+begin_src matlab :exports none
- RGA_num_svd = squeeze(sum(sum(RGA_svd - eye(3))));
- RGA_num_x = squeeze(sum(sum(RGA_x - eye(3))));
+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');
+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');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
- exportFig('figs/gravimeter_rga_num.pdf', 'width', 'wide', 'height', 'normal');
+exportFig('figs/gravimeter_rga_num.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:gravimeter_rga_num
@@ -677,48 +677,48 @@ The RGA-number is also a measure of diagonal dominance:
The bode plot of the diagonal and off-diagonal elements of $G_{SVD}$ are shown in Figure [[fig:gravimeter_decoupled_plant_svd]].
#+begin_src matlab :exports none
- figure;
- tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
- % Magnitude
- ax1 = nexttile([2, 1]);
- 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(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
- 'DisplayName', '$G_{SVD}(i,j),\ i \neq j$');
- set(gca,'ColorOrderIndex',1)
- for ch_i = 1:3
+% Magnitude
+ax1 = nexttile([2, 1]);
+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(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
+ 'DisplayName', '$G_{SVD}(i,j),\ i \neq j$');
+set(gca,'ColorOrderIndex',1)
+for ch_i = 1:3
plot(freqs, abs(squeeze(freqresp(Gsvd(ch_i, ch_i), freqs, 'Hz'))), ...
'DisplayName', sprintf('$G_{SVD}(%i,%i)$', ch_i, ch_i));
- end
- hold off;
- set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
- ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
- legend('location', 'southwest');
- ylim([1e-8, 1e0])
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
+legend('location', 'southwest');
+ylim([1e-8, 1e0])
- % Phase
- ax2 = nexttile;
- hold on;
- for ch_i = 1:3
+% Phase
+ax2 = nexttile;
+hold on;
+for ch_i = 1:3
plot(freqs, 180/pi*angle(squeeze(freqresp(Gsvd(ch_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([-180:90:360]);
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-180, 180]);
+yticks([-180:90:360]);
- linkaxes([ax1,ax2],'x');
+linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
- exportFig('figs/gravimeter_decoupled_plant_svd.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
+exportFig('figs/gravimeter_decoupled_plant_svd.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:gravimeter_decoupled_plant_svd
@@ -729,47 +729,47 @@ The bode plot of the diagonal and off-diagonal elements of $G_{SVD}$ are shown i
Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant $G_x(s)$ using the Jacobian are shown in Figure [[fig:gravimeter_decoupled_plant_jacobian]].
#+begin_src matlab :exports none
- figure;
- tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
- % Magnitude
- ax1 = nexttile([2, 1]);
- hold on;
- for i_in = 1:3
- for i_out = [1:i_in-1, i_in+1:3]
- plot(freqs, abs(squeeze(freqresp(Gx(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
- 'HandleVisibility', 'off');
- end
- end
- plot(freqs, abs(squeeze(freqresp(Gx(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
- 'DisplayName', '$G_x(i,j),\ i \neq j$');
- 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', '$G_x(3,3) = R_z/M_z$');
- hold off;
- set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
- ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
- legend('location', 'southwest');
- ylim([1e-8, 1e0])
+% Magnitude
+ax1 = nexttile([2, 1]);
+hold on;
+for i_in = 1:3
+ for i_out = [1:i_in-1, i_in+1:3]
+ plot(freqs, abs(squeeze(freqresp(Gx(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+ 'HandleVisibility', 'off');
+ end
+end
+plot(freqs, abs(squeeze(freqresp(Gx(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
+ 'DisplayName', '$G_x(i,j),\ i \neq j$');
+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', '$G_x(3,3) = R_z/M_z$');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
+legend('location', 'southwest');
+ylim([1e-8, 1e0])
- % Phase
- ax2 = nexttile;
- hold on;
- plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(1, 1), freqs, 'Hz'))));
- plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(2, 2), freqs, 'Hz'))));
- plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(3, 3), freqs, 'Hz'))));
- hold off;
- set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
- ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
- ylim([-180, 180]);
- yticks([0:45:360]);
+% Phase
+ax2 = nexttile;
+hold on;
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(1, 1), freqs, 'Hz'))));
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(2, 2), freqs, 'Hz'))));
+plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(3, 3), freqs, 'Hz'))));
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-180, 180]);
+yticks([0:45:360]);
- linkaxes([ax1,ax2],'x');
+linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
- exportFig('figs/gravimeter_decoupled_plant_jacobian.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
+exportFig('figs/gravimeter_decoupled_plant_jacobian.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:gravimeter_decoupled_plant_jacobian
@@ -785,24 +785,24 @@ 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_gravimeter.pdf :tangle no :exports results
- \begin{tikzpicture}
- \node[block] (G) {$\bm{G}$};
- \node[block, left=0.6 of G] (Jt) {$J_{\tau}^{-T}$};
- \node[block, right=0.6 of G] (Ja) {$J_{a}^{-1}$};
- \node[block, left=1.2 of Jt] (K) {$K_c$};
+\begin{tikzpicture}
+ \node[block] (G) {$\bm{G}$};
+ \node[block, left=0.6 of G] (Jt) {$J_{\tau}^{-T}$};
+ \node[block, right=0.6 of G] (Ja) {$J_{a}^{-1}$};
+ \node[block, left=1.2 of Jt] (K) {$K_c$};
- % Connections and labels
- \draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$};
- \draw[->] (G.east) -- (Ja.west) node[above left]{$\bm{a}$};
- \draw[->] (Ja.east) -- ++(1.4, 0);
- \draw[->] ($(Ja.east) + (0.8, 0)$) node[branch]{} node[above]{$\bm{\mathcal{A}}$} -- ++(0, -1.2) -| ($(K.west) + (-0.6, 0)$) -- (K.west);
- \draw[->] (K.east) -- (Jt.west) node[above left]{$\bm{\mathcal{F}}$};
+ % Connections and labels
+ \draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$};
+ \draw[->] (G.east) -- (Ja.west) node[above left]{$\bm{a}$};
+ \draw[->] (Ja.east) -- ++(1.4, 0);
+ \draw[->] ($(Ja.east) + (0.8, 0)$) node[branch]{} node[above]{$\bm{\mathcal{A}}$} -- ++(0, -1.2) -| ($(K.west) + (-0.6, 0)$) -- (K.west);
+ \draw[->] (K.east) -- (Jt.west) node[above left]{$\bm{\mathcal{F}}$};
- \begin{scope}[on background layer]
- \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gx) {};
- \node[below right] at (Gx.north west) {$\bm{G}_x$};
- \end{scope}
- \end{tikzpicture}
+ \begin{scope}[on background layer]
+ \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gx) {};
+ \node[below right] at (Gx.north west) {$\bm{G}_x$};
+ \end{scope}
+\end{tikzpicture}
#+end_src
#+name: fig:centralized_control_gravimeter
@@ -814,25 +814,25 @@ The SVD control architecture is shown in Figure [[fig:svd_control_gravimeter]].
The matrices $U$ and $V$ are used to decoupled the plant $G$.
#+begin_src latex :file svd_control_gravimeter.pdf :tangle no :exports results
- \begin{tikzpicture}
- \node[block] (G) {$\bm{G}$};
+\begin{tikzpicture}
+ \node[block] (G) {$\bm{G}$};
- \node[block, left=0.6 of G.west] (V) {$V^{-T}$};
- \node[block, right=0.6 of G.east] (U) {$U^{-1}$};
- \node[block, left=1.2 of V] (K) {$K_c$};
+ \node[block, left=0.6 of G.west] (V) {$V^{-T}$};
+ \node[block, right=0.6 of G.east] (U) {$U^{-1}$};
+ \node[block, left=1.2 of V] (K) {$K_c$};
- % Connections and labels
- \draw[->] (V.east) -- (G.west) node[above left]{$\tau$};
- \draw[->] (G.east) -- (U.west) node[above left]{$a$};
- \draw[->] (U.east) -- ++( 1.4, 0);
- \draw[->] ($(U.east) + (0.8, 0)$) node[branch]{} node[above]{$y$} -- ++(0, -1.2) -| ($(K.west) + (-0.6, 0)$) -- (K.west);
- \draw[->] (K.east) -- (V.west) node[above left]{$u$};
+ % Connections and labels
+ \draw[->] (V.east) -- (G.west) node[above left]{$\tau$};
+ \draw[->] (G.east) -- (U.west) node[above left]{$a$};
+ \draw[->] (U.east) -- ++( 1.4, 0);
+ \draw[->] ($(U.east) + (0.8, 0)$) node[branch]{} node[above]{$y$} -- ++(0, -1.2) -| ($(K.west) + (-0.6, 0)$) -- (K.west);
+ \draw[->] (K.east) -- (V.west) node[above left]{$u$};
- \begin{scope}[on background layer]
- \node[fit={(V.south west) (G.north-|U.east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gsvd) {};
- \node[below right] at (Gsvd.north west) {$\bm{G}_{SVD}$};
- \end{scope}
- \end{tikzpicture}
+ \begin{scope}[on background layer]
+ \node[fit={(V.south west) (G.north-|U.east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gsvd) {};
+ \node[below right] at (Gsvd.north west) {$\bm{G}_{SVD}$};
+ \end{scope}
+\end{tikzpicture}
#+end_src
#+name: fig:svd_control_gravimeter
@@ -847,74 +847,74 @@ We choose the controller to be a low pass filter:
$G_0$ is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to $\omega_c$
#+begin_src matlab
- wc = 2*pi*10; % Crossover Frequency [rad/s]
- w0 = 2*pi*0.1; % Controller Pole [rad/s]
+wc = 2*pi*10; % Crossover Frequency [rad/s]
+w0 = 2*pi*0.1; % Controller Pole [rad/s]
#+end_src
#+begin_src matlab
- K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
- L_cen = K_cen*Gx;
- G_cen = feedback(G, pinv(Jt')*K_cen*pinv(Ja));
+K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
+L_cen = K_cen*Gx;
+G_cen = feedback(G, pinv(Jt')*K_cen*pinv(Ja));
#+end_src
#+begin_src matlab
- K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
- L_svd = K_svd*Gsvd;
- U_inv = inv(U);
- G_svd = feedback(G, inv(V')*K_svd*U_inv(1:3, :));
+K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
+L_svd = K_svd*Gsvd;
+U_inv = inv(U);
+G_svd = feedback(G, inv(V')*K_svd*U_inv(1:3, :));
#+end_src
The obtained diagonal elements of the loop gains are shown in Figure [[fig:gravimeter_comp_loop_gain_diagonal]].
#+begin_src matlab :exports none
- figure;
- tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
- % Magnitude
- ax1 = nexttile([2, 1]);
- hold on;
- plot(freqs, abs(squeeze(freqresp(L_svd(1, 1), freqs, 'Hz'))), 'DisplayName', '$L_{SVD}(i,i)$');
- for i_in_out = 2:3
+% Magnitude
+ax1 = nexttile([2, 1]);
+hold on;
+plot(freqs, abs(squeeze(freqresp(L_svd(1, 1), freqs, 'Hz'))), 'DisplayName', '$L_{SVD}(i,i)$');
+for i_in_out = 2:3
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off');
- end
+end
- set(gca,'ColorOrderIndex',2)
- plot(freqs, abs(squeeze(freqresp(L_cen(1, 1), freqs, 'Hz'))), ...
- 'DisplayName', '$L_{J}(i,i)$');
- for i_in_out = 2:3
+set(gca,'ColorOrderIndex',2)
+plot(freqs, abs(squeeze(freqresp(L_cen(1, 1), freqs, 'Hz'))), ...
+ 'DisplayName', '$L_{J}(i,i)$');
+for i_in_out = 2:3
set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off');
- end
- hold off;
- set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
- ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
- legend('location', 'northwest');
- ylim([5e-2, 2e3])
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
+legend('location', 'northwest');
+ylim([5e-2, 2e3])
- % Phase
- ax2 = nexttile;
- hold on;
- for i_in_out = 1:3
+% Phase
+ax2 = nexttile;
+hold on;
+for i_in_out = 1:3
set(gca,'ColorOrderIndex',1)
plot(freqs, 180/pi*angle(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))));
- end
- set(gca,'ColorOrderIndex',2)
- for i_in_out = 1:3
+end
+set(gca,'ColorOrderIndex',2)
+for i_in_out = 1:3
set(gca,'ColorOrderIndex',2)
plot(freqs, 180/pi*angle(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))));
- end
- hold off;
- set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
- ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
- ylim([-180, 180]);
- yticks([-180:90:360]);
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-180, 180]);
+yticks([-180:90:360]);
- linkaxes([ax1,ax2],'x');
+linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
- exportFig('figs/gravimeter_comp_loop_gain_diagonal.pdf', 'width', 'wide', 'height', 'tall');
+exportFig('figs/gravimeter_comp_loop_gain_diagonal.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:gravimeter_comp_loop_gain_diagonal
@@ -927,7 +927,7 @@ The obtained diagonal elements of the loop gains are shown in Figure [[fig:gravi
Let's first verify the stability of the closed-loop systems:
#+begin_src matlab :results output replace text
- isstable(G_cen)
+isstable(G_cen)
#+end_src
#+RESULTS:
@@ -936,7 +936,7 @@ Let's first verify the stability of the closed-loop systems:
: 1
#+begin_src matlab :results output replace text
- isstable(G_svd)
+isstable(G_svd)
#+end_src
#+RESULTS:
@@ -947,49 +947,49 @@ Let's first verify the stability of the closed-loop systems:
The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure [[fig:gravimeter_platform_simscape_cl_transmissibility]].
#+begin_src matlab :exports results
- freqs = logspace(-2, 2, 1000);
+freqs = logspace(-2, 2, 1000);
- figure;
- tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None');
+figure;
+tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None');
- ax1 = nexttile;
- hold on;
- plot(freqs, abs(squeeze(freqresp(G( 1,1)/s^2, freqs, 'Hz'))), 'DisplayName', 'Open-Loop');
- plot(freqs, abs(squeeze(freqresp(G_cen(1,1)/s^2, freqs, 'Hz'))), 'DisplayName', 'Centralized');
- plot(freqs, abs(squeeze(freqresp(G_svd(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');
+ax1 = nexttile;
+hold on;
+plot(freqs, abs(squeeze(freqresp(G( 1,1)/s^2, freqs, 'Hz'))), 'DisplayName', 'Open-Loop');
+plot(freqs, abs(squeeze(freqresp(G_cen(1,1)/s^2, freqs, 'Hz'))), 'DisplayName', 'Centralized');
+plot(freqs, abs(squeeze(freqresp(G_svd(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( 2,2)/s^2, freqs, 'Hz'))));
- plot(freqs, abs(squeeze(freqresp(G_cen(2,2)/s^2, freqs, 'Hz'))));
- 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('$D_y/D_{w,y}$');
+ax2 = nexttile;
+hold on;
+plot(freqs, abs(squeeze(freqresp(G( 2,2)/s^2, freqs, 'Hz'))));
+plot(freqs, abs(squeeze(freqresp(G_cen(2,2)/s^2, freqs, 'Hz'))));
+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('$D_y/D_{w,y}$');
- ax3 = nexttile;
- hold on;
- plot(freqs, abs(squeeze(freqresp(G( 3,3)/s^2, freqs, 'Hz'))));
- plot(freqs, abs(squeeze(freqresp(G_cen(3,3)/s^2, freqs, 'Hz'))));
- 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('$R_z/R_{w,z}$');
+ax3 = nexttile;
+hold on;
+plot(freqs, abs(squeeze(freqresp(G( 3,3)/s^2, freqs, 'Hz'))));
+plot(freqs, abs(squeeze(freqresp(G_cen(3,3)/s^2, freqs, 'Hz'))));
+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('$R_z/R_{w,z}$');
- linkaxes([ax1,ax2,ax3],'xy');
- xlim([freqs(1), freqs(end)]);
- xlim([1e-2, 5e1]); ylim([1e-7, 1e-2]);
+linkaxes([ax1,ax2,ax3],'xy');
+xlim([freqs(1), freqs(end)]);
+xlim([1e-2, 5e1]); ylim([1e-7, 1e-2]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
- exportFig('figs/gravimeter_platform_simscape_cl_transmissibility.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
+exportFig('figs/gravimeter_platform_simscape_cl_transmissibility.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:gravimeter_platform_simscape_cl_transmissibility
@@ -998,26 +998,26 @@ The obtained transmissibility in Open-loop, for the centralized control as well
[[file:figs/gravimeter_platform_simscape_cl_transmissibility.png]]
#+begin_src matlab :exports results
- freqs = logspace(-2, 2, 1000);
+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]');
+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]');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
- exportFig('figs/gravimeter_cl_transmissibility_coupling.pdf', 'width', 'wide', 'height', 'normal');
+exportFig('figs/gravimeter_cl_transmissibility_coupling.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:gravimeter_cl_transmissibility_coupling
@@ -1030,32 +1030,32 @@ The obtained transmissibility in Open-loop, for the centralized control as well
Let say we change the position of the actuators:
#+begin_src matlab
- la = l/2*0.7; % Position of Act. [m]
- ha = h/2*0.7; % Position of Act. [m]
+la = l/2*0.7; % Position of Act. [m]
+ha = h/2*0.7; % Position of Act. [m]
#+end_src
#+begin_src matlab :exports none
- %% Name of the Simulink File
- mdl = 'gravimeter';
+%% 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;
+%% 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 = linearize(mdl, io);
+G.InputName = {'F1', 'F2', 'F3'};
+G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
#+end_src
#+begin_src matlab :exports none
- G_cen_b = feedback(G, pinv(Jt')*K_cen*pinv(Ja));
- G_svd_b = feedback(G, inv(V')*K_svd*U_inv(1:3, :));
+G_cen_b = feedback(G, pinv(Jt')*K_cen*pinv(Ja));
+G_svd_b = feedback(G, inv(V')*K_svd*U_inv(1:3, :));
#+end_src
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.
@@ -1063,49 +1063,49 @@ The new plant is computed, and the centralized and SVD control architectures are
The closed-loop system are still stable, and their
#+begin_src matlab :exports results
- freqs = logspace(-2, 2, 1000);
+freqs = logspace(-2, 2, 1000);
- figure;
- tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None');
+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');
+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('$D_y/D_{w,y}$');
+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('$R_z/R_{w,z}$');
+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]);
+linkaxes([ax1,ax2,ax3],'xy');
+xlim([freqs(1), freqs(end)]);
+xlim([1e-2, 5e1]); ylim([1e-7, 3e-4]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
- exportFig('figs/gravimeter_transmissibility_offset_act.pdf', 'width', 'wide', 'height', 'normal');
+exportFig('figs/gravimeter_transmissibility_offset_act.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:gravimeter_transmissibility_offset_act
@@ -1135,73 +1135,73 @@ To do so, the actuators (springs) should be positioned such that the stiffness m
[[file:figs/gravimeter_model_M.png]]
#+begin_src matlab
- la = l/2; % Position of Act. [m]
- ha = h/2; % Position of Act. [m]
+la = l/2; % Position of Act. [m]
+ha = h/2; % Position of Act. [m]
#+end_src
#+begin_src matlab
- %% Name of the Simulink File
- mdl = 'gravimeter';
+%% 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;
+%% 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 = linearize(mdl, io);
+G.InputName = {'F1', 'F2', 'F3'};
+G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
#+end_src
Decoupling at the CoM (Mass decoupled)
#+begin_src matlab
- JMa = [1 0 -h/2
- 0 1 l/2
- 1 0 h/2
- 0 1 0];
+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];
+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'};
+GM = pinv(JMa)*G*pinv(JMt');
+GM.InputName = {'Fx', 'Fy', 'Mz'};
+GM.OutputName = {'Dx', 'Dy', 'Rz'};
#+end_src
#+begin_src matlab :exports none
- figure;
+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
+% 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
+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 :tangle no :exports results :results file replace
- exportFig('figs/jac_decoupling_M.pdf', 'width', 'wide', 'height', 'normal');
+exportFig('figs/jac_decoupling_M.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:jac_decoupling_M
@@ -1217,49 +1217,49 @@ Decoupling at the CoM (Mass decoupled)
Decoupling at the point where K is diagonal (x = 0, y = -h/2 from the schematic {O} frame):
#+begin_src matlab
- JKa = [1 0 0
- 0 1 -l/2
- 1 0 -h
- 0 1 0];
+JKa = [1 0 0
+ 0 1 -l/2
+ 1 0 -h
+ 0 1 0];
- JKt = [1 0 0
- 0 1 -la
- 0 1 la];
+JKt = [1 0 0
+ 0 1 -la
+ 0 1 la];
#+end_src
And the plant $\bm{G}_x$ is computed:
#+begin_src matlab
- GK = pinv(JKa)*G*pinv(JKt');
- GK.InputName = {'Fx', 'Fy', 'Mz'};
- GK.OutputName = {'Dx', 'Dy', 'Rz'};
+GK = pinv(JKa)*G*pinv(JKt');
+GK.InputName = {'Fx', 'Fy', 'Mz'};
+GK.OutputName = {'Dx', 'Dy', 'Rz'};
#+end_src
#+begin_src matlab :exports none
- figure;
+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
+% 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]);
+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 :tangle no :exports results :results file replace
- exportFig('figs/jac_decoupling_K.pdf', 'width', 'wide', 'height', 'normal');
+exportFig('figs/jac_decoupling_K.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:jac_decoupling_K
@@ -1276,72 +1276,72 @@ And the plant $\bm{G}_x$ is computed:
To do so, the actuator position should be modified
#+begin_src matlab
- la = l/2; % Position of Act. [m]
- ha = 0; % Position of Act. [m]
+la = l/2; % Position of Act. [m]
+ha = 0; % Position of Act. [m]
#+end_src
#+begin_src matlab
- %% Name of the Simulink File
- mdl = 'gravimeter';
+%% 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;
+%% 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 = linearize(mdl, io);
+G.InputName = {'F1', 'F2', 'F3'};
+G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
#+end_src
#+begin_src matlab
- JMa = [1 0 -h/2
- 0 1 l/2
- 1 0 h/2
- 0 1 0];
+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];
+JMt = [1 0 -ha
+ 0 1 la
+ 0 1 -la];
#+end_src
#+begin_src matlab
- GKM = pinv(JMa)*G*pinv(JMt');
- GKM.InputName = {'Fx', 'Fy', 'Mz'};
- GKM.OutputName = {'Dx', 'Dy', 'Rz'};
+GKM = pinv(JMa)*G*pinv(JMt');
+GKM.InputName = {'Fx', 'Fy', 'Mz'};
+GKM.OutputName = {'Dx', 'Dy', 'Rz'};
#+end_src
#+begin_src matlab :exports none
- figure;
+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
+% 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]);
+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 :tangle no :exports results :results file replace
- exportFig('figs/jac_decoupling_KM.pdf', 'width', 'wide', 'height', 'normal');
+exportFig('figs/jac_decoupling_KM.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:jac_decoupling_KM
@@ -1363,61 +1363,61 @@ Let's do the SVD decoupling on a plant that is mostly real (low damping) and one
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)]
+c = 2e1; % Actuator Damping [N/(m/s)]
#+end_src
#+begin_src matlab :exports none
- %% Name of the Simulink File
- mdl = 'gravimeter';
+%% 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;
+%% 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 = 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');
+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');
#+end_src
#+begin_src matlab :exports none
- figure;
+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
+% 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
+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');
+exportFig('figs/gravimeter_svd_low_damping.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:gravimeter_svd_low_damping
@@ -1427,61 +1427,61 @@ Start with small damping, the obtained diagonal and off-diagonal terms are shown
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)]
+c = 5e2; % Actuator Damping [N/(m/s)]
#+end_src
#+begin_src matlab :exports none
- %% Name of the Simulink File
- mdl = 'gravimeter';
+%% 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;
+%% 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 = 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');
+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');
#+end_src
#+begin_src matlab :exports none
- figure;
+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', '$G_{svd}(i,j)\ i \neq j$');
- set(gca,'ColorOrderIndex',1)
- for i_in_out = 1:3
+% 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', '$G_{svd}(i,j)\ i \neq j$');
+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('$G_{svd}(%d,%d)$', 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
+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_high_damping.pdf', 'width', 'wide', 'height', 'normal');
+exportFig('figs/gravimeter_svd_high_damping.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:gravimeter_svd_high_damping
@@ -1521,97 +1521,97 @@ The analysis of the SVD/Jacobian control applied to the Stewart platform is perf
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
- <>
+<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
- <>
+<>
#+end_src
#+begin_src matlab :tangle no
- addpath('stewart_platform');
- addpath('stewart_platform/STEP');
+addpath('stewart_platform');
+addpath('stewart_platform/STEP');
#+end_src
#+begin_src matlab :eval no
- addpath('STEP');
+addpath('STEP');
#+end_src
#+begin_src matlab
- freqs = logspace(-1, 2, 1000);
+freqs = logspace(-1, 2, 1000);
#+end_src
** Jacobian :noexport:
First, the position of the "joints" (points of force application) are estimated and the Jacobian computed.
#+begin_src matlab :tangle no
- open('drone_platform_jacobian.slx');
+open('drone_platform_jacobian.slx');
#+end_src
#+begin_src matlab :tangle no
- sim('drone_platform_jacobian');
+sim('drone_platform_jacobian');
#+end_src
#+begin_src matlab :tangle no
- Aa = [a1.Data(1,:);
- a2.Data(1,:);
- a3.Data(1,:);
- a4.Data(1,:);
- a5.Data(1,:);
- a6.Data(1,:)]';
+Aa = [a1.Data(1,:);
+ a2.Data(1,:);
+ a3.Data(1,:);
+ a4.Data(1,:);
+ a5.Data(1,:);
+ a6.Data(1,:)]';
- Ab = [b1.Data(1,:);
- b2.Data(1,:);
- b3.Data(1,:);
- b4.Data(1,:);
- b5.Data(1,:);
- b6.Data(1,:)]';
+Ab = [b1.Data(1,:);
+ b2.Data(1,:);
+ b3.Data(1,:);
+ b4.Data(1,:);
+ b5.Data(1,:);
+ b6.Data(1,:)]';
- As = (Ab - Aa)./vecnorm(Ab - Aa);
+As = (Ab - Aa)./vecnorm(Ab - Aa);
- l = vecnorm(Ab - Aa)';
+l = vecnorm(Ab - Aa)';
- J = [As' , cross(Ab, As)'];
+J = [As' , cross(Ab, As)'];
- save('stewart_platform/jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
+save('stewart_platform/jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
#+end_src
** Simscape Model - Parameters
<>
#+begin_src matlab
- open('drone_platform.slx');
+open('drone_platform.slx');
#+end_src
Definition of spring parameters:
#+begin_src matlab
- kx = 0.5*1e3/3; % [N/m]
- ky = 0.5*1e3/3;
- kz = 1e3/3;
+kx = 0.5*1e3/3; % [N/m]
+ky = 0.5*1e3/3;
+kz = 1e3/3;
- cx = 0.025; % [Nm/rad]
- cy = 0.025;
- cz = 0.025;
+cx = 0.025; % [Nm/rad]
+cy = 0.025;
+cz = 0.025;
#+end_src
We suppose the sensor is perfectly positioned.
#+begin_src matlab
- sens_pos_error = zeros(3,1);
+sens_pos_error = zeros(3,1);
#+end_src
Gravity:
#+begin_src matlab
- g = 0;
+g = 0;
#+end_src
We load the Jacobian (previously computed from the geometry):
#+begin_src matlab
- load('jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
+load('jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
#+end_src
We initialize other parameters:
#+begin_src matlab
- U = eye(6);
- V = eye(6);
- Kc = tf(zeros(6));
+U = eye(6);
+V = eye(6);
+Kc = tf(zeros(6));
#+end_src
#+name: fig:stewart_simscape
@@ -1636,19 +1636,19 @@ The inputs are:
The outputs are the 6 accelerations measured by the inertial unit.
#+begin_src latex :file stewart_platform_plant.pdf :tangle no :exports results
- \begin{tikzpicture}
- \node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G_u\end{bmatrix}$};
- \node[above] at (G.north) {$\bm{G}$};
+\begin{tikzpicture}
+ \node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G_u\end{bmatrix}$};
+ \node[above] at (G.north) {$\bm{G}$};
- % 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
+ % 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[<-] (inputu) -- ++(-0.8, 0) node[above right]{$\tau$};
- \draw[->] (G.east) -- ++(0.8, 0) node[above left]{$a$};
- \end{tikzpicture}
+ \draw[<-] (inputd) -- ++(-0.8, 0) node[above right]{$D_w$};
+ \draw[<-] (inputu) -- ++(-0.8, 0) node[above right]{$\tau$};
+ \draw[->] (G.east) -- ++(0.8, 0) node[above left]{$a$};
+\end{tikzpicture}
#+end_src
#+name: fig:stewart_platform_plant
@@ -1657,29 +1657,29 @@ The outputs are the 6 accelerations measured by the inertial unit.
[[file:figs/stewart_platform_plant.png]]
#+begin_src matlab
- %% Name of the Simulink File
- mdl = 'drone_platform';
+%% Name of the Simulink File
+mdl = 'drone_platform';
- %% Input/Output definition
- clear io; io_i = 1;
- io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1; % Ground Motion
- io(io_i) = linio([mdl, '/V-T'], 1, 'openinput'); io_i = io_i + 1; % Actuator Forces
- io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Top platform acceleration
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1; % Ground Motion
+io(io_i) = linio([mdl, '/V-T'], 1, 'openinput'); io_i = io_i + 1; % Actuator Forces
+io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Top platform acceleration
- G = linearize(mdl, io);
- G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
- 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
- G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
+G = linearize(mdl, io);
+G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
+ 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
- % Plant
- Gu = G(:, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'});
- % Disturbance dynamics
- Gd = G(:, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'});
+% Plant
+Gu = G(:, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'});
+% Disturbance dynamics
+Gd = G(:, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'});
#+end_src
There are 24 states (6dof for the bottom platform + 6dof for the top platform).
#+begin_src matlab :results output replace
- size(G)
+size(G)
#+end_src
#+RESULTS:
@@ -1690,31 +1690,31 @@ The elements of the transfer matrix $\bm{G}$ corresponding to the transfer funct
One can easily see that the system is strongly coupled.
#+begin_src matlab :exports none
- figure;
+figure;
- % Magnitude
- hold on;
- for i_in = 1:6
- for i_out = [1:i_in-1, i_in+1:6]
- plot(freqs, abs(squeeze(freqresp(Gu(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
- 'HandleVisibility', 'off');
- end
- end
- plot(freqs, abs(squeeze(freqresp(Gu(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
- 'DisplayName', '$G_u(i,j)\ i \neq j$');
- set(gca,'ColorOrderIndex',1)
- for i_in_out = 1:6
+% Magnitude
+hold on;
+for i_in = 1:6
+ for i_out = [1:i_in-1, i_in+1:6]
+ plot(freqs, abs(squeeze(freqresp(Gu(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+ 'HandleVisibility', 'off');
+ end
+end
+plot(freqs, abs(squeeze(freqresp(Gu(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+ 'DisplayName', '$G_u(i,j)\ i \neq j$');
+set(gca,'ColorOrderIndex',1)
+for i_in_out = 1:6
plot(freqs, abs(squeeze(freqresp(Gu(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_u(%d,%d)$', i_in_out, i_in_out));
- end
- hold off;
- set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
- xlabel('Frequency [Hz]'); ylabel('Magnitude');
- ylim([1e-2, 1e5]);
- legend('location', 'northwest');
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Magnitude');
+ylim([1e-2, 1e5]);
+legend('location', 'northwest');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
- exportFig('figs/stewart_platform_coupled_plant.pdf', 'eps', true, 'width', 'wide', 'height', 'normal');
+exportFig('figs/stewart_platform_coupled_plant.pdf', 'eps', true, 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:stewart_platform_coupled_plant
@@ -1730,7 +1730,7 @@ The Jacobian matrix is used to transform forces/torques applied on the top platf
The Jacobian matrix is computed from the geometry of the platform (position and orientation of the actuators).
#+begin_src matlab :exports results :results value table replace :tangle no
- data2orgtable(J, {}, {}, ' %.3f ');
+data2orgtable(J, {}, {}, ' %.3f ');
#+end_src
#+caption: Computed Jacobian Matrix
@@ -1743,20 +1743,20 @@ The Jacobian matrix is computed from the geometry of the platform (position and
| -0.406 | 0.703 | 0.584 | -0.002 | 0.019 | -0.025 |
#+begin_src latex :file plant_decouple_jacobian.pdf :tangle no :exports results
- \begin{tikzpicture}
- \node[block] (G) {$G_u$};
- \node[block, left=0.6 of G] (J) {$J^{-T}$};
+\begin{tikzpicture}
+ \node[block] (G) {$G_u$};
+ \node[block, left=0.6 of G] (J) {$J^{-T}$};
- % Connections and labels
- \draw[<-] (J.west) -- ++(-1.0, 0) node[above right]{$\mathcal{F}$};
- \draw[->] (J.east) -- (G.west) node[above left]{$\tau$};
- \draw[->] (G.east) -- ++( 1.0, 0) node[above left]{$a$};
+ % Connections and labels
+ \draw[<-] (J.west) -- ++(-1.0, 0) node[above right]{$\mathcal{F}$};
+ \draw[->] (J.east) -- (G.west) node[above left]{$\tau$};
+ \draw[->] (G.east) -- ++( 1.0, 0) node[above left]{$a$};
- \begin{scope}[on background layer]
- \node[fit={(J.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gx) {};
- \node[below right] at (Gx.north west) {$\bm{G}_x$};
- \end{scope}
- \end{tikzpicture}
+ \begin{scope}[on background layer]
+ \node[fit={(J.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gx) {};
+ \node[below right] at (Gx.north west) {$\bm{G}_x$};
+ \end{scope}
+\end{tikzpicture}
#+end_src
#+name: fig:plant_decouple_jacobian
@@ -1770,8 +1770,8 @@ We define a new plant:
$G_x(s)$ correspond to the transfer function from forces and torques applied to the top platform to the absolute acceleration of the top platform.
#+begin_src matlab
- Gx = Gu*inv(J');
- Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
+Gx = Gu*inv(J');
+Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
#+end_src
** Decoupling using the SVD
@@ -1781,19 +1781,19 @@ In order to decouple the plant using the SVD, first a real approximation of the
Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_u(j\omega_c)$ 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*30; % Decoupling frequency [rad/s]
+wc = 2*pi*30; % Decoupling frequency [rad/s]
- H1 = evalfr(Gu, j*wc);
+H1 = evalfr(Gu, 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))));
+D = pinv(real(H1'*H1));
+H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no
- data2orgtable(H1, {}, {}, ' %.1f ');
+data2orgtable(H1, {}, {}, ' %.1f ');
#+end_src
#+caption: Real approximate of $G$ at the decoupling frequency $\omega_c$
@@ -1811,7 +1811,7 @@ 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 $G_u(\omega_c)$ is shown
#+begin_src matlab :exports results :results value table replace :tangle no
- data2orgtable(real(evalfr(Gu, j*wc)), {}, {}, ' %.1f ');
+data2orgtable(real(evalfr(Gu, j*wc)), {}, {}, ' %.1f ');
#+end_src
#+caption: Real part of $G$ at the decoupling frequency $\omega_c$
@@ -1827,11 +1827,11 @@ Now, the Singular Value Decomposition of $H_1$ is performed:
\[ H_1 = U \Sigma V^H \]
#+begin_src matlab
- [U,~,V] = svd(H1);
+[U,~,V] = svd(H1);
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no
- data2orgtable(U, {}, {}, ' %.1g ');
+data2orgtable(U, {}, {}, ' %.1g ');
#+end_src
#+caption: Obtained matrix $U$
@@ -1844,7 +1844,7 @@ Now, the Singular Value Decomposition of $H_1$ is performed:
| -4e-09 | 5e-06 | -1 | 6e-11 | -2e-09 | -1e-08 |
#+begin_src matlab :exports results :results value table replace :tangle no
- data2orgtable(V, {}, {}, ' %.1g ');
+data2orgtable(V, {}, {}, ' %.1g ');
#+end_src
#+caption: Obtained matrix $V$
@@ -1859,23 +1859,23 @@ Now, the Singular Value Decomposition of $H_1$ is performed:
The obtained matrices $U$ and $V$ are used to decouple the system as shown in Figure [[fig:plant_decouple_svd]].
#+begin_src latex :file plant_decouple_svd.pdf :tangle no :exports results
- \begin{tikzpicture}
- \node[block] (G) {$G_u$};
+\begin{tikzpicture}
+ \node[block] (G) {$G_u$};
- \node[block, left=0.6 of G.west] (V) {$V^{-T}$};
- \node[block, right=0.6 of G.east] (U) {$U^{-1}$};
+ \node[block, left=0.6 of G.west] (V) {$V^{-T}$};
+ \node[block, right=0.6 of G.east] (U) {$U^{-1}$};
- % Connections and labels
- \draw[<-] (V.west) -- ++(-1.0, 0) node[above right]{$u$};
- \draw[->] (V.east) -- (G.west) node[above left]{$\tau$};
- \draw[->] (G.east) -- (U.west) node[above left]{$a$};
- \draw[->] (U.east) -- ++( 1.0, 0) node[above left]{$y$};
+ % Connections and labels
+ \draw[<-] (V.west) -- ++(-1.0, 0) node[above right]{$u$};
+ \draw[->] (V.east) -- (G.west) node[above left]{$\tau$};
+ \draw[->] (G.east) -- (U.west) node[above left]{$a$};
+ \draw[->] (U.east) -- ++( 1.0, 0) node[above left]{$y$};
- \begin{scope}[on background layer]
- \node[fit={(V.south west) (G.north-|U.east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gsvd) {};
- \node[below right] at (Gsvd.north west) {$\bm{G}_{SVD}$};
- \end{scope}
- \end{tikzpicture}
+ \begin{scope}[on background layer]
+ \node[fit={(V.south west) (G.north-|U.east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gsvd) {};
+ \node[below right] at (Gsvd.north west) {$\bm{G}_{SVD}$};
+ \end{scope}
+\end{tikzpicture}
#+end_src
#+name: fig:plant_decouple_svd
@@ -1887,7 +1887,7 @@ The decoupled plant is then:
\[ G_{SVD}(s) = U^{-1} G_u(s) V^{-H} \]
#+begin_src matlab
- Gsvd = inv(U)*Gu*inv(V');
+Gsvd = inv(U)*Gu*inv(V');
#+end_src
** Verification of the decoupling using the "Gershgorin Radii"
@@ -1900,51 +1900,51 @@ The "Gershgorin Radii" of a matrix $S$ is defined by:
This is computed over the following frequencies.
#+begin_src matlab :exports none
- % Gershgorin Radii for the coupled plant:
- Gr_coupled = zeros(length(freqs), size(Gu,2));
- H = abs(squeeze(freqresp(Gu, freqs, 'Hz')));
- for out_i = 1:size(Gu,2)
- Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
- end
+% Gershgorin Radii for the coupled plant:
+Gr_coupled = zeros(length(freqs), size(Gu,2));
+H = abs(squeeze(freqresp(Gu, freqs, 'Hz')));
+for out_i = 1:size(Gu,2)
+ Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
+end
- % Gershgorin Radii for the decoupled plant using SVD:
- Gr_decoupled = zeros(length(freqs), size(Gsvd,2));
- H = abs(squeeze(freqresp(Gsvd, freqs, 'Hz')));
- for out_i = 1:size(Gsvd,2)
- Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
- end
+% Gershgorin Radii for the decoupled plant using SVD:
+Gr_decoupled = zeros(length(freqs), size(Gsvd,2));
+H = abs(squeeze(freqresp(Gsvd, freqs, 'Hz')));
+for out_i = 1:size(Gsvd,2)
+ Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
+end
- % Gershgorin Radii for the decoupled plant using the Jacobian:
- Gr_jacobian = zeros(length(freqs), size(Gx,2));
- H = abs(squeeze(freqresp(Gx, freqs, 'Hz')));
- for out_i = 1:size(Gx,2)
- Gr_jacobian(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
- end
+% Gershgorin Radii for the decoupled plant using the Jacobian:
+Gr_jacobian = zeros(length(freqs), size(Gx,2));
+H = abs(squeeze(freqresp(Gx, freqs, 'Hz')));
+for out_i = 1:size(Gx,2)
+ Gr_jacobian(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
+end
#+end_src
#+begin_src matlab :exports results
- figure;
- hold on;
- plot(freqs, Gr_coupled(:,1), 'DisplayName', 'Coupled');
- plot(freqs, Gr_decoupled(:,1), 'DisplayName', 'SVD');
- plot(freqs, Gr_jacobian(:,1), 'DisplayName', 'Jacobian');
- for in_i = 2:6
- set(gca,'ColorOrderIndex',1)
- plot(freqs, Gr_coupled(:,in_i), 'HandleVisibility', 'off');
- set(gca,'ColorOrderIndex',2)
- plot(freqs, Gr_decoupled(:,in_i), 'HandleVisibility', 'off');
- set(gca,'ColorOrderIndex',3)
- plot(freqs, Gr_jacobian(:,in_i), 'HandleVisibility', 'off');
- end
- set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
- hold off;
- xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
- legend('location', 'northwest');
- ylim([1e-3, 1e3]);
+figure;
+hold on;
+plot(freqs, Gr_coupled(:,1), 'DisplayName', 'Coupled');
+plot(freqs, Gr_decoupled(:,1), 'DisplayName', 'SVD');
+plot(freqs, Gr_jacobian(:,1), 'DisplayName', 'Jacobian');
+for in_i = 2:6
+ set(gca,'ColorOrderIndex',1)
+ plot(freqs, Gr_coupled(:,in_i), 'HandleVisibility', 'off');
+ set(gca,'ColorOrderIndex',2)
+ plot(freqs, Gr_decoupled(:,in_i), 'HandleVisibility', 'off');
+ set(gca,'ColorOrderIndex',3)
+ plot(freqs, Gr_jacobian(:,in_i), 'HandleVisibility', 'off');
+end
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+hold off;
+xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
+legend('location', 'northwest');
+ylim([1e-3, 1e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
- exportFig('figs/simscape_model_gershgorin_radii.pdf', 'eps', true, 'width', 'wide', 'height', 'normal');
+exportFig('figs/simscape_model_gershgorin_radii.pdf', 'eps', true, 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:simscape_model_gershgorin_radii
@@ -1964,82 +1964,82 @@ where $\times$ denotes an element by element multiplication and $G(s)$ is an $n
The obtained RGA elements are shown in Figure [[fig:simscape_model_rga]].
#+begin_src matlab :exports none
- % Relative Gain Array for the coupled plant:
- RGA_coupled = zeros(length(freqs), size(Gu,1), size(Gu,2));
- Gu_inv = inv(Gu);
- for f_i = 1:length(freqs)
+% Relative Gain Array for the coupled plant:
+RGA_coupled = zeros(length(freqs), size(Gu,1), size(Gu,2));
+Gu_inv = inv(Gu);
+for f_i = 1:length(freqs)
RGA_coupled(f_i, :, :) = abs(evalfr(Gu, j*2*pi*freqs(f_i)).*evalfr(Gu_inv, j*2*pi*freqs(f_i))');
- end
+end
- % Relative Gain Array for the decoupled plant using SVD:
- RGA_svd = zeros(length(freqs), size(Gsvd,1), size(Gsvd,2));
- Gsvd_inv = inv(Gsvd);
- for f_i = 1:length(freqs)
+% Relative Gain Array for the decoupled plant using SVD:
+RGA_svd = zeros(length(freqs), size(Gsvd,1), size(Gsvd,2));
+Gsvd_inv = inv(Gsvd);
+for f_i = 1:length(freqs)
RGA_svd(f_i, :, :) = abs(evalfr(Gsvd, j*2*pi*freqs(f_i)).*evalfr(Gsvd_inv, j*2*pi*freqs(f_i))');
- end
+end
- % Relative Gain Array for the decoupled plant using the Jacobian:
- RGA_x = zeros(length(freqs), size(Gx,1), size(Gx,2));
- Gx_inv = inv(Gx);
- for f_i = 1:length(freqs)
+% Relative Gain Array for the decoupled plant using the Jacobian:
+RGA_x = zeros(length(freqs), size(Gx,1), size(Gx,2));
+Gx_inv = inv(Gx);
+for f_i = 1:length(freqs)
RGA_x(f_i, :, :) = abs(evalfr(Gx, j*2*pi*freqs(f_i)).*evalfr(Gx_inv, j*2*pi*freqs(f_i))');
- end
+end
#+end_src
#+begin_src matlab :exports none
- figure;
- tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
+figure;
+tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
- ax1 = nexttile;
- hold on;
- for i_in = 1:6
- for i_out = [1:i_in-1, i_in+1:6]
- plot(freqs, RGA_svd(:, i_out, i_in), '--', 'color', [0 0 0 0.2], ...
- 'HandleVisibility', 'off');
- end
- end
- plot(freqs, RGA_svd(:, 1, 2), '--', 'color', [0 0 0 0.2], ...
- 'DisplayName', '$RGA_{SVD}(i,j),\ i \neq j$');
+ax1 = nexttile;
+hold on;
+for i_in = 1:6
+ for i_out = [1:i_in-1, i_in+1:6]
+ plot(freqs, RGA_svd(:, i_out, i_in), '--', 'color', [0 0 0 0.2], ...
+ 'HandleVisibility', 'off');
+ end
+end
+plot(freqs, RGA_svd(:, 1, 2), '--', 'color', [0 0 0 0.2], ...
+ 'DisplayName', '$RGA_{SVD}(i,j),\ i \neq j$');
- plot(freqs, RGA_svd(:, 1, 1), 'k-', ...
- 'DisplayName', '$RGA_{SVD}(i,i)$');
- for ch_i = 1:6
+plot(freqs, RGA_svd(:, 1, 1), 'k-', ...
+ 'DisplayName', '$RGA_{SVD}(i,i)$');
+for ch_i = 1:6
plot(freqs, RGA_svd(:, ch_i, ch_i), 'k-', ...
'HandleVisibility', 'off');
- end
- hold off;
- set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
- ylabel('Magnitude'); xlabel('Frequency [Hz]');
- legend('location', 'southwest');
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Magnitude'); xlabel('Frequency [Hz]');
+legend('location', 'southwest');
- ax2 = nexttile;
- hold on;
- for i_in = 1:6
- for i_out = [1:i_in-1, i_in+1:6]
- plot(freqs, RGA_x(:, i_out, i_in), '--', 'color', [0 0 0 0.2], ...
- 'HandleVisibility', 'off');
- end
- end
- plot(freqs, RGA_x(:, 1, 2), '--', 'color', [0 0 0 0.2], ...
- 'DisplayName', '$RGA_{X}(i,j),\ i \neq j$');
+ax2 = nexttile;
+hold on;
+for i_in = 1:6
+ for i_out = [1:i_in-1, i_in+1:6]
+ plot(freqs, RGA_x(:, i_out, i_in), '--', 'color', [0 0 0 0.2], ...
+ 'HandleVisibility', 'off');
+ end
+end
+plot(freqs, RGA_x(:, 1, 2), '--', 'color', [0 0 0 0.2], ...
+ 'DisplayName', '$RGA_{X}(i,j),\ i \neq j$');
- plot(freqs, RGA_x(:, 1, 1), 'k-', ...
- 'DisplayName', '$RGA_{X}(i,i)$');
- for ch_i = 1:6
+plot(freqs, RGA_x(:, 1, 1), 'k-', ...
+ 'DisplayName', '$RGA_{X}(i,i)$');
+for ch_i = 1:6
plot(freqs, RGA_x(:, ch_i, ch_i), 'k-', ...
'HandleVisibility', 'off');
- end
- hold off;
- set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
- xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
- legend('location', 'southwest');
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
+legend('location', 'southwest');
- linkaxes([ax1,ax2],'y');
- ylim([1e-5, 1e1]);
+linkaxes([ax1,ax2],'y');
+ylim([1e-5, 1e1]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
- exportFig('figs/simscape_model_rga.pdf', 'width', 'wide', 'height', 'tall');
+exportFig('figs/simscape_model_rga.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:simscape_model_rga
@@ -2053,48 +2053,48 @@ The obtained RGA elements are shown in Figure [[fig:simscape_model_rga]].
The bode plot of the diagonal and off-diagonal elements of $G_{SVD}$ are shown in Figure [[fig:simscape_model_decoupled_plant_svd]].
#+begin_src matlab :exports none
- figure;
- tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
- % Magnitude
- ax1 = nexttile([2, 1]);
- hold on;
- for i_in = 1:6
- for i_out = [1:i_in-1, i_in+1:6]
- 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(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
- 'DisplayName', '$G_{SVD}(i,j),\ i \neq j$');
- set(gca,'ColorOrderIndex',1)
- for ch_i = 1:6
+% Magnitude
+ax1 = nexttile([2, 1]);
+hold on;
+for i_in = 1:6
+ for i_out = [1:i_in-1, i_in+1:6]
+ 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(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
+ 'DisplayName', '$G_{SVD}(i,j),\ i \neq j$');
+set(gca,'ColorOrderIndex',1)
+for ch_i = 1:6
plot(freqs, abs(squeeze(freqresp(Gsvd(ch_i, ch_i), freqs, 'Hz'))), ...
'DisplayName', sprintf('$G_{SVD}(%i,%i)$', ch_i, ch_i));
- end
- hold off;
- set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
- ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
- legend('location', 'northwest');
- ylim([1e-1, 1e5])
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
+legend('location', 'northwest');
+ylim([1e-1, 1e5])
- % Phase
- ax2 = nexttile;
- hold on;
- for ch_i = 1:6
+% Phase
+ax2 = nexttile;
+hold on;
+for ch_i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gsvd(ch_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([-180:90:360]);
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-180, 180]);
+yticks([-180:90:360]);
- linkaxes([ax1,ax2],'x');
+linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
- exportFig('figs/simscape_model_decoupled_plant_svd.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
+exportFig('figs/simscape_model_decoupled_plant_svd.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:simscape_model_decoupled_plant_svd
@@ -2105,53 +2105,53 @@ The bode plot of the diagonal and off-diagonal elements of $G_{SVD}$ are shown i
Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant $G_x(s)$ using the Jacobian are shown in Figure [[fig:simscape_model_decoupled_plant_jacobian]].
#+begin_src matlab :exports none
- figure;
- tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
- % Magnitude
- ax1 = nexttile([2, 1]);
- hold on;
- for i_in = 1:6
- for i_out = [1:i_in-1, i_in+1:6]
- plot(freqs, abs(squeeze(freqresp(Gx(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
- 'HandleVisibility', 'off');
- end
- end
- plot(freqs, abs(squeeze(freqresp(Gx(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
- 'DisplayName', '$G_x(i,j),\ i \neq j$');
- set(gca,'ColorOrderIndex',1)
- plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))), 'DisplayName', '$G_x(1,1) = A_x/F_x$');
- plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))), 'DisplayName', '$G_x(2,2) = A_y/F_y$');
- plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$G_x(3,3) = A_z/F_z$');
- plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))), 'DisplayName', '$G_x(4,4) = A_{R_x}/M_x$');
- plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))), 'DisplayName', '$G_x(5,5) = A_{R_y}/M_y$');
- plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$G_x(6,6) = A_{R_z}/M_z$');
- hold off;
- set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
- ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
- legend('location', 'northwest');
- ylim([1e-2, 2e6])
+% Magnitude
+ax1 = nexttile([2, 1]);
+hold on;
+for i_in = 1:6
+ for i_out = [1:i_in-1, i_in+1:6]
+ plot(freqs, abs(squeeze(freqresp(Gx(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
+ 'HandleVisibility', 'off');
+ end
+end
+plot(freqs, abs(squeeze(freqresp(Gx(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
+ 'DisplayName', '$G_x(i,j),\ i \neq j$');
+set(gca,'ColorOrderIndex',1)
+plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))), 'DisplayName', '$G_x(1,1) = A_x/F_x$');
+plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))), 'DisplayName', '$G_x(2,2) = A_y/F_y$');
+plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$G_x(3,3) = A_z/F_z$');
+plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))), 'DisplayName', '$G_x(4,4) = A_{R_x}/M_x$');
+plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))), 'DisplayName', '$G_x(5,5) = A_{R_y}/M_y$');
+plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$G_x(6,6) = A_{R_z}/M_z$');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
+legend('location', 'northwest');
+ylim([1e-2, 2e6])
- % Phase
- ax2 = nexttile;
- hold on;
- 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'))));
- 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([0, 180]);
- yticks([0:45:360]);
+% Phase
+ax2 = nexttile;
+hold on;
+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'))));
+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([0, 180]);
+yticks([0:45:360]);
- linkaxes([ax1,ax2],'x');
+linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
- exportFig('figs/simscape_model_decoupled_plant_jacobian.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
+exportFig('figs/simscape_model_decoupled_plant_jacobian.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:simscape_model_decoupled_plant_jacobian
@@ -2167,23 +2167,23 @@ 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 :tangle no :exports results
- \begin{tikzpicture}
- \node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G_u\end{bmatrix}$};
- \node[above] at (G.north) {$\bm{G}$};
- \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) {$J^{-T}$};
+\begin{tikzpicture}
+ \node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G_u\end{bmatrix}$};
+ \node[above] at (G.north) {$\bm{G}$};
+ \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 ($(G.south west)!0.25!(G.north west)$);
+ % 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}
+ % 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
#+name: fig:centralized_control
@@ -2195,25 +2195,25 @@ The SVD control architecture is shown in Figure [[fig:svd_control]].
The matrices $U$ and $V$ are used to decoupled the plant $G$.
#+begin_src latex :file svd_control.pdf :tangle no :exports results
- \begin{tikzpicture}
- \node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G_u\end{bmatrix}$};
- \node[above] at (G.north) {$\bm{G}$};
- \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) {$V^{-T}$};
+\begin{tikzpicture}
+ \node[block={2cm}{1.5cm}] (G) {$\begin{bmatrix}G_d\\G_u\end{bmatrix}$};
+ \node[above] at (G.north) {$\bm{G}$};
+ \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 ($(G.south west)!0.25!(G.north west)$);
+ % 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}
+ % 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
#+name: fig:svd_control
@@ -2228,73 +2228,73 @@ We choose the controller to be a low pass filter:
$G_0$ is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to $\omega_c$
#+begin_src matlab
- wc = 2*pi*80; % Crossover Frequency [rad/s]
- w0 = 2*pi*0.1; % Controller Pole [rad/s]
+wc = 2*pi*80; % Crossover Frequency [rad/s]
+w0 = 2*pi*0.1; % Controller Pole [rad/s]
#+end_src
#+begin_src matlab
- K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
- L_cen = K_cen*Gx;
- G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);
+K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
+L_cen = K_cen*Gx;
+G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);
#+end_src
#+begin_src matlab
- K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
- L_svd = K_svd*Gsvd;
- G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]);
+K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
+L_svd = K_svd*Gsvd;
+G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]);
#+end_src
The obtained diagonal elements of the loop gains are shown in Figure [[fig:stewart_comp_loop_gain_diagonal]].
#+begin_src matlab :exports none
- figure;
- tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
- % Magnitude
- ax1 = nexttile([2, 1]);
- hold on;
- plot(freqs, abs(squeeze(freqresp(L_svd(1, 1), freqs, 'Hz'))), 'DisplayName', '$L_{SVD}(i,i)$');
- for i_in_out = 2:6
+% Magnitude
+ax1 = nexttile([2, 1]);
+hold on;
+plot(freqs, abs(squeeze(freqresp(L_svd(1, 1), freqs, 'Hz'))), 'DisplayName', '$L_{SVD}(i,i)$');
+for i_in_out = 2:6
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off');
- end
+end
- set(gca,'ColorOrderIndex',2)
- plot(freqs, abs(squeeze(freqresp(L_cen(1, 1), freqs, 'Hz'))), ...
- 'DisplayName', '$L_{J}(i,i)$');
- for i_in_out = 2:6
+set(gca,'ColorOrderIndex',2)
+plot(freqs, abs(squeeze(freqresp(L_cen(1, 1), freqs, 'Hz'))), ...
+ 'DisplayName', '$L_{J}(i,i)$');
+for i_in_out = 2:6
set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off');
- end
- hold off;
- set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
- ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
- legend('location', 'northwest');
- ylim([5e-2, 2e3])
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
+legend('location', 'northwest');
+ylim([5e-2, 2e3])
- % Phase
- ax2 = nexttile;
- hold on;
- for i_in_out = 1:6
+% Phase
+ax2 = nexttile;
+hold on;
+for i_in_out = 1:6
set(gca,'ColorOrderIndex',1)
plot(freqs, 180/pi*angle(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))));
- end
- set(gca,'ColorOrderIndex',2)
- for i_in_out = 1:6
+end
+set(gca,'ColorOrderIndex',2)
+for i_in_out = 1:6
set(gca,'ColorOrderIndex',2)
plot(freqs, 180/pi*angle(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))));
- end
- hold off;
- set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
- ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
- ylim([-180, 180]);
- yticks([-180:90:360]);
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
+ylim([-180, 180]);
+yticks([-180:90:360]);
- linkaxes([ax1,ax2],'x');
+linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
- exportFig('figs/stewart_comp_loop_gain_diagonal.pdf', 'width', 'wide', 'height', 'tall');
+exportFig('figs/stewart_comp_loop_gain_diagonal.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:stewart_comp_loop_gain_diagonal
@@ -2307,7 +2307,7 @@ The obtained diagonal elements of the loop gains are shown in Figure [[fig:stewa
Let's first verify the stability of the closed-loop systems:
#+begin_src matlab :results output replace text
- isstable(G_cen)
+isstable(G_cen)
#+end_src
#+RESULTS:
@@ -2316,7 +2316,7 @@ Let's first verify the stability of the closed-loop systems:
: 1
#+begin_src matlab :results output replace text
- isstable(G_svd)
+isstable(G_svd)
#+end_src
#+RESULTS:
@@ -2327,61 +2327,61 @@ Let's first verify the stability of the closed-loop systems:
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
- figure;
- tiledlayout(2, 2, 'TileSpacing', 'None', 'Padding', 'None');
+figure;
+tiledlayout(2, 2, 'TileSpacing', 'None', 'Padding', 'None');
- ax1 = nexttile;
- 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');
- set(gca,'ColorOrderIndex',1)
- plot(freqs, abs(squeeze(freqresp(G( 'Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'HandleVisibility', 'off');
- plot(freqs, abs(squeeze(freqresp(G_cen('Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'HandleVisibility', 'off');
- plot(freqs, abs(squeeze(freqresp(G_svd('Ay', 'Dwy')/s^2, freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
- hold off;
- set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
- ylabel('$D_x/D_{w,x}$, $D_y/D_{w, y}$'); set(gca, 'XTickLabel',[]);
- legend('location', 'southwest');
+ax1 = nexttile;
+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');
+set(gca,'ColorOrderIndex',1)
+plot(freqs, abs(squeeze(freqresp(G( 'Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'HandleVisibility', 'off');
+plot(freqs, abs(squeeze(freqresp(G_cen('Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'HandleVisibility', 'off');
+plot(freqs, abs(squeeze(freqresp(G_svd('Ay', 'Dwy')/s^2, freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('$D_x/D_{w,x}$, $D_y/D_{w, y}$'); set(gca, 'XTickLabel',[]);
+legend('location', 'southwest');
- ax2 = nexttile;
- 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('$D_z/D_{w,z}$'); set(gca, 'XTickLabel',[]);
+ax2 = nexttile;
+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('$D_z/D_{w,z}$'); set(gca, 'XTickLabel',[]);
- ax3 = nexttile;
- 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'))), '--');
- set(gca,'ColorOrderIndex',1)
- 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('$R_x/R_{w,x}$, $R_y/R_{w,y}$'); xlabel('Frequency [Hz]');
+ax3 = nexttile;
+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'))), '--');
+set(gca,'ColorOrderIndex',1)
+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('$R_x/R_{w,x}$, $R_y/R_{w,y}$'); xlabel('Frequency [Hz]');
- ax4 = nexttile;
- 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('$R_z/R_{w,z}$'); xlabel('Frequency [Hz]');
+ax4 = nexttile;
+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('$R_z/R_{w,z}$'); xlabel('Frequency [Hz]');
- linkaxes([ax1,ax2,ax3,ax4],'xy');
- xlim([freqs(1), freqs(end)]);
- ylim([1e-3, 1e2]);
+linkaxes([ax1,ax2,ax3,ax4],'xy');
+xlim([freqs(1), freqs(end)]);
+ylim([1e-3, 1e2]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
- exportFig('figs/stewart_platform_simscape_cl_transmissibility.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
+exportFig('figs/stewart_platform_simscape_cl_transmissibility.pdf', 'eps', true, 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:stewart_platform_simscape_cl_transmissibility
diff --git a/stewart_platform/script.m b/stewart_platform/script.m
index 90a8603..4555444 100644
--- a/stewart_platform/script.m
+++ b/stewart_platform/script.m
@@ -108,7 +108,7 @@ plot(freqs, abs(squeeze(freqresp(Gu(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,
'DisplayName', '$G_u(i,j)\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for i_in_out = 1:6
- plot(freqs, abs(squeeze(freqresp(Gu(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_u(%d,%d)$', i_in_out, i_in_out));
+ plot(freqs, abs(squeeze(freqresp(Gu(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_u(%d,%d)$', i_in_out, i_in_out));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
@@ -246,21 +246,21 @@ ylim([1e-3, 1e3]);
RGA_coupled = zeros(length(freqs), size(Gu,1), size(Gu,2));
Gu_inv = inv(Gu);
for f_i = 1:length(freqs)
- RGA_coupled(f_i, :, :) = abs(evalfr(Gu, j*2*pi*freqs(f_i)).*evalfr(Gu_inv, j*2*pi*freqs(f_i))');
+ RGA_coupled(f_i, :, :) = abs(evalfr(Gu, j*2*pi*freqs(f_i)).*evalfr(Gu_inv, j*2*pi*freqs(f_i))');
end
% Relative Gain Array for the decoupled plant using SVD:
RGA_svd = zeros(length(freqs), size(Gsvd,1), size(Gsvd,2));
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))');
+ 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(length(freqs), size(Gx,1), size(Gx,2));
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))');
+ RGA_x(f_i, :, :) = abs(evalfr(Gx, j*2*pi*freqs(f_i)).*evalfr(Gx_inv, j*2*pi*freqs(f_i))');
end
figure;
@@ -280,8 +280,8 @@ plot(freqs, RGA_svd(:, 1, 2), '--', 'color', [0 0 0 0.2], ...
plot(freqs, RGA_svd(:, 1, 1), 'k-', ...
'DisplayName', '$RGA_{SVD}(i,i)$');
for ch_i = 1:6
- plot(freqs, RGA_svd(:, ch_i, ch_i), 'k-', ...
- 'HandleVisibility', 'off');
+ plot(freqs, RGA_svd(:, ch_i, ch_i), 'k-', ...
+ 'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
@@ -302,8 +302,8 @@ plot(freqs, RGA_x(:, 1, 2), '--', 'color', [0 0 0 0.2], ...
plot(freqs, RGA_x(:, 1, 1), 'k-', ...
'DisplayName', '$RGA_{X}(i,i)$');
for ch_i = 1:6
- plot(freqs, RGA_x(:, ch_i, ch_i), 'k-', ...
- 'HandleVisibility', 'off');
+ plot(freqs, RGA_x(:, ch_i, ch_i), 'k-', ...
+ 'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
@@ -335,8 +335,8 @@ plot(freqs, abs(squeeze(freqresp(Gsvd(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5
'DisplayName', '$G_{SVD}(i,j),\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for ch_i = 1:6
- plot(freqs, abs(squeeze(freqresp(Gsvd(ch_i, ch_i), freqs, 'Hz'))), ...
- 'DisplayName', sprintf('$G_{SVD}(%i,%i)$', ch_i, ch_i));
+ plot(freqs, abs(squeeze(freqresp(Gsvd(ch_i, ch_i), freqs, 'Hz'))), ...
+ 'DisplayName', sprintf('$G_{SVD}(%i,%i)$', ch_i, ch_i));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
@@ -348,7 +348,7 @@ ylim([1e-1, 1e5])
ax2 = nexttile;
hold on;
for ch_i = 1:6
- plot(freqs, 180/pi*angle(squeeze(freqresp(Gsvd(ch_i, ch_i), freqs, 'Hz'))));
+ plot(freqs, 180/pi*angle(squeeze(freqresp(Gsvd(ch_i, ch_i), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
@@ -450,16 +450,16 @@ ax1 = nexttile([2, 1]);
hold on;
plot(freqs, abs(squeeze(freqresp(L_svd(1, 1), freqs, 'Hz'))), 'DisplayName', '$L_{SVD}(i,i)$');
for i_in_out = 2:6
- set(gca,'ColorOrderIndex',1)
- plot(freqs, abs(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off');
+ set(gca,'ColorOrderIndex',1)
+ plot(freqs, abs(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off');
end
set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(L_cen(1, 1), freqs, 'Hz'))), ...
'DisplayName', '$L_{J}(i,i)$');
for i_in_out = 2:6
- set(gca,'ColorOrderIndex',2)
- plot(freqs, abs(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off');
+ set(gca,'ColorOrderIndex',2)
+ plot(freqs, abs(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
@@ -471,13 +471,13 @@ ylim([5e-2, 2e3])
ax2 = nexttile;
hold on;
for i_in_out = 1:6
- set(gca,'ColorOrderIndex',1)
- plot(freqs, 180/pi*angle(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))));
+ set(gca,'ColorOrderIndex',1)
+ plot(freqs, 180/pi*angle(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))));
end
set(gca,'ColorOrderIndex',2)
for i_in_out = 1:6
- set(gca,'ColorOrderIndex',2)
- plot(freqs, 180/pi*angle(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))));
+ set(gca,'ColorOrderIndex',2)
+ plot(freqs, 180/pi*angle(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
Figure 42: Obtained Transmissibility
@@ -1975,7 +1975,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well
-
diff --git a/index.org b/index.org
index 7754166..58826e6 100644
--- a/index.org
+++ b/index.org
@@ -77,26 +77,26 @@ In this part, diagonal control using both the SVD and the Jacobian matrices are
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
- <Created: 2021-01-08 ven. 13:57
+Created: 2021-01-11 lun. 09:09