From ff0e0bf73eeab48681ecbc058b28a052a83bca77 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Fri, 6 Nov 2020 11:59:09 +0100 Subject: [PATCH] Add picture of platform and sections summary --- figs/SP_assembly.png | Bin 0 -> 108936 bytes figs/centralized_control.eps | Bin 0 -> 16634 bytes figs/centralized_control.pdf | Bin 0 -> 8219 bytes figs/stewart_platform_rotations.eps | Bin 0 -> 233115 bytes figs/stewart_platform_rotations.pdf | Bin 68399 -> 74613 bytes figs/stewart_platform_rotations.png | Bin 91275 -> 130490 bytes figs/stewart_platform_translations.eps | Bin 0 -> 232014 bytes figs/stewart_platform_translations.pdf | Bin 77724 -> 73816 bytes figs/stewart_platform_translations.png | Bin 114954 -> 126488 bytes figs/svd_control.eps | Bin 0 -> 19585 bytes index.html | 838 ++++++++++--------------- index.org | 220 +++---- 12 files changed, 438 insertions(+), 620 deletions(-) create mode 100644 figs/SP_assembly.png create mode 100644 figs/centralized_control.eps create mode 100644 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z@+%1;ud9}Ep{hlqc|8*{>Gd^G$;iBV)7s0Zx-kBs0>nT@BAmAlWs7NPZy!}v!^;tT pIX`(4uRy*T=^qDr;2+80AKiF$-0+Wy97ntROZ^iZ|MTVOe*yNwH4Fd% literal 0 HcmV?d00001 diff --git a/index.html b/index.html index 14664c7..b6daed1 100644 --- a/index.html +++ b/index.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + SVD Control @@ -35,75 +35,56 @@

Table of Contents

-
-

1 Gravimeter - Simscape Model

+
+

1 Gravimeter - Simscape Model

-
-

1.1 Introduction

+
+

1.1 Introduction

-
+

gravimeter_model.png

Figure 1: Model of the gravimeter

@@ -111,8 +92,8 @@
-
-

1.2 Simscape Model - Parameters

+
+

1.2 Simscape Model - Parameters

open('gravimeter.slx')
@@ -143,8 +124,8 @@ g = 0; % Gravity [m/s2]
 
-
-

1.3 System Identification - Without Gravity

+
+

1.3 System Identification - Without Gravity

%% Name of the Simulink File
@@ -166,7 +147,7 @@ G.OutputName = {'Ax1', 
 
-
+
 pole(G)
 ans =
       -0.000473481142385795 +      21.7596190728632i
@@ -191,7 +172,7 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
 
 
 
-
+

open_loop_tf.png

Figure 2: Open Loop Transfer Function from 3 Actuators to 4 Accelerometers

@@ -199,8 +180,8 @@ State-space model with 4 outputs, 3 inputs, and 6 states.
-
-

1.4 System Identification - With Gravity

+
+

1.4 System Identification - With Gravity

g = 9.80665; % Gravity [m/s2]
@@ -217,7 +198,7 @@ Gg.OutputName = {'Ax1', 
 We can now see that the system is unstable due to gravity.
 

-
+
 pole(Gg)
 ans =
           -10.9848275341252 +                     0i
@@ -229,7 +210,7 @@ ans =
 
-
+

open_loop_tf_g.png

Figure 3: Open Loop Transfer Function from 3 Actuators to 4 Accelerometers with an without gravity

@@ -237,12 +218,12 @@ ans =
-
-

1.5 Analytical Model

+
+

1.5 Analytical Model

-
-

1.5.1 Parameters

+
+

1.5.1 Parameters

Bode options. @@ -274,8 +255,8 @@ Frequency vector.

-
-

1.5.2 Generation of the State Space Model

+
+

1.5.2 Generation of the State Space Model

Mass matrix @@ -376,11 +357,11 @@ State-space model with 12 outputs, 6 inputs, and 6 states.

-
-

1.5.3 Comparison with the Simscape Model

+
+

1.5.3 Comparison with the Simscape Model

-
+

gravimeter_analytical_system_open_loop_models.png

Figure 4: Comparison of the analytical and the Simscape models

@@ -388,8 +369,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
-
-

1.5.4 Analysis

+
+

1.5.4 Analysis

% figure
@@ -457,8 +438,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.
 
-
-

1.5.5 Control Section

+
+

1.5.5 Control Section

system_dec_10Hz = freqresp(system_dec,2*pi*10);
@@ -598,8 +579,8 @@ legend('Control OFF','D
 
-
-

1.5.6 Greshgorin radius

+
+

1.5.6 Greshgorin radius

system_dec_freq = freqresp(system_dec,w);
@@ -645,8 +626,8 @@ ylabel('Greshgorin radius [-]');
 
-
-

1.5.7 Injecting ground motion in the system to have the output

+
+

1.5.7 Injecting ground motion in the system to have the output

Fr = logspace(-2,3,1e3);
@@ -702,15 +683,15 @@ rot = PHI(:,11,11);
 
-
-

2 Gravimeter - Functions

+
+

2 Gravimeter - Functions

-
-

2.1 align

+
+

2.1 align

- +

@@ -739,11 +720,11 @@ This Matlab function is accessible here.

-
-

2.2 pzmap_testCL

+
+

2.2 pzmap_testCL

- +

@@ -792,57 +773,41 @@ This Matlab function is accessible here.

-
-

3 Stewart Platform - Simscape Model

+
+

3 Stewart Platform - Simscape Model

+

+In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure 5. +

+ + +
+

SP_assembly.png +

+

Figure 5: Stewart Platform CAD View

-
-

3.1 Jacobian

+ +

+The analysis of the SVD control applied to the Stewart platform is performed in the following sections: +

+
    +
  • Section 3.1: The parameters of the Simscape model of the Stewart platform are defined
  • +
  • Section 3.2: The plant is identified from the Simscape model and the centralized plant is computed thanks to the Jacobian
  • +
  • Section 3.3: The identified Dynamics is shown
  • +
  • Section 3.4: A real approximation of the plant is computed for further decoupling using the Singular Value Decomposition (SVD)
  • +
  • Section 3.5: The decoupling is performed thanks to the SVD. The effectiveness of the decoupling is verified using the Gershorin Radii
  • +
  • Section 3.6: The dynamics of the decoupled plant is shown
  • +
  • Section 3.7: A diagonal controller is defined to control the decoupled plant
  • +
  • Section 3.8: Finally, the closed loop system properties are studied
  • +
+
+
+

3.1 Simscape Model - Parameters

-First, the position of the “joints” (points of force application) are estimated and the Jacobian computed. +

-
open('drone_platform_jacobian.slx');
-
-
- -
-
sim('drone_platform_jacobian');
-
-
- -
-
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,:)]';
-
-As = (Ab - Aa)./vecnorm(Ab - Aa);
-
-l = vecnorm(Ab - Aa)';
-
-J = [As' , cross(Ab, As)'];
-
-save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
-
-
-
-
- -
-

3.2 Simscape Model

-
-
open('drone_platform.slx');
 
@@ -862,7 +827,15 @@ cz = 0.025;

-We load the Jacobian. +Gravity: +

+
+
g = 0;
+
+
+ +

+We load the Jacobian (previously computed from the geometry).

load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
@@ -871,9 +844,13 @@ We load the Jacobian.
 
-
-

3.3 Identification of the plant

-
+
+

3.2 Identification of the plant

+
+

+ +

+

The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform.

@@ -907,69 +884,64 @@ State-space model with 6 outputs, 12 inputs, and 24 states.
-
-
% G = G*blkdiag(inv(J), eye(6));
-% G.InputName  = {'Dw1', 'Dw2', 'Dw3', 'Dw4', 'Dw5', 'Dw6', ...
-%                 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
-
+

+The “centralized” plant \(\bm{G}_x\) is now computed (Figure 6). +

+ + +
+

centralized_control.png +

+

Figure 6: Centralized control architecture

-Thanks to the Jacobian, we compute the transfer functions in the frame of the legs and in an inertial frame. +Thanks to the Jacobian, we compute the transfer functions in the inertial frame (transfer function from forces and torques applied to the top platform to the absolute acceleration of the top platform).

Gx = G*blkdiag(eye(6), inv(J'));
 Gx.InputName  = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
                  'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
-
-Gl = J*G;
-Gl.OutputName  = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
 
-
-

3.4 Obtained Dynamics

-
+
+

3.3 Obtained Dynamics

+
+

+ +

-
+ +

stewart_platform_translations.png

-

Figure 5: Stewart Platform Plant from forces applied by the legs to the acceleration of the platform

+

Figure 7: Stewart Platform Plant from forces applied by the legs to the acceleration of the platform

-
+

stewart_platform_rotations.png

-

Figure 6: Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform

+

Figure 8: Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform

+
+
- -
-

stewart_platform_legs.png +

+

3.4 Real Approximation of \(G\) at the decoupling frequency

+
+

+

-

Figure 7: Stewart Platform Plant from forces applied by the legs to displacement of the legs

-
- -
-

stewart_platform_transmissibility.png -

-

Figure 8: Transmissibility

-
-
-
- -
-

3.5 Real Approximation of \(G\) at the decoupling frequency

-

Let’s compute a real approximation of the complex matrix \(H_1\) which corresponds to the the transfer function \(G_c(j\omega_c)\) from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency \(\omega_c\).

-
wc = 2*pi*20; % Decoupling frequency [rad/s]
+
wc = 2*pi*30; % Decoupling frequency [rad/s]
 
 Gc = G({'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}, ...
        {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); % Transfer function to find a real approximation
@@ -986,12 +958,169 @@ The real approximation is computed as follows:
 H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
 
+ + + + +++ ++ ++ ++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Table 1: Real approximate of \(G\) at the decoupling frequency \(\omega_c\)
4.4-2.1-2.14.4-2.4-2.4
-0.2-3.93.90.2-3.83.8
3.43.43.43.43.43.4
-367.1-323.8323.8367.143.3-43.3
-162.0-237.0-237.0-162.0398.9398.9
220.6-220.6220.6-220.6220.6-220.6
+ + +

+Please not that the plant \(G\) at \(\omega_c\) is already an almost real matrix. +This can be seen on the Bode plots where the phase is close to 1. +This can be verified below where only the real value of \(G(\omega_c)\) is shown +

+ + + + +++ ++ ++ ++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
4.4-2.1-2.14.4-2.4-2.4
-0.2-3.93.90.2-3.83.8
3.43.43.43.43.43.4
-367.1-323.8323.8367.143.3-43.3
-162.0-237.0-237.0-162.0398.9398.9
220.6-220.6220.6-220.6220.6-220.6
-
-

3.6 Verification of the decoupling using the “Gershgorin Radii”

-
+
+

3.5 Verification of the decoupling using the “Gershgorin Radii”

+
+

+ +

+

First, the Singular Value Decomposition of \(H_1\) is performed: \[ H_1 = U \Sigma V^H \] @@ -1058,7 +1187,7 @@ H = abs(squeeze(freqresp(Gj, freqs, 'Hz')));

-
+

simscape_model_gershgorin_radii.png

Figure 9: Gershgorin Radii of the Coupled and Decoupled plants

@@ -1066,23 +1195,27 @@ H = abs(squeeze(freqresp(Gj, freqs, 'Hz')));
-
-

3.7 Decoupled Plant

-
+
+

3.6 Decoupled Plant

+
+

+ +

+

Let’s see the bode plot of the decoupled plant \(G_d(s)\). \[ G_d(s) = U^T G_c(s) V \]

-
+

simscape_model_decoupled_plant_svd.png

Figure 10: Decoupled Plant using SVD

-
+

simscape_model_decoupled_plant_jacobian.png

Figure 11: Decoupled Plant using the Jacobian

@@ -1090,9 +1223,13 @@ Let’s see the bode plot of the decoupled plant \(G_d(s)\).
-
-

3.8 Diagonal Controller

-
+
+

3.7 Diagonal Controller

+
+

+ +

+

The controller \(K\) is a diagonal controller consisting a low pass filters with a crossover frequency \(\omega_c\) and a DC gain \(C_g\).

@@ -1104,14 +1241,9 @@ C_g = 50; % DC Gain K = eye(6)*C_g/(s+wc);
-
-
-
-

3.9 Centralized Control

-

-The control diagram for the centralized control is shown below. +The control diagram for the centralized control is shown in Figure 6.

@@ -1120,34 +1252,33 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied

-
+

centralized_control.png

+

Figure 12: Control Diagram for the Centralized control

+

+The feedback system is computed as shown below. +

G_cen = feedback(G, inv(J')*K, [7:12], [1:6]);
 
-
-
-
-

3.10 SVD Control

-

-The SVD control architecture is shown below. +The SVD control architecture is shown in Figure 13. The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).

- -
+

svd_control.png

+

Figure 13: Control Diagram for the SVD control

-SVD Control +The feedback system is computed as shown below.

G_svd = feedback(G, pinv(V')*K*pinv(U), [7:12], [1:6]);
@@ -1156,9 +1287,13 @@ SVD Control
 
-
-

3.11 Results

-
+
+

3.8 Closed-Loop system Performances

+
+

+ +

+

Let’s first verify the stability of the closed-loop systems:

@@ -1187,11 +1322,11 @@ ans =

-The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 14. +The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 14.

-
+

stewart_platform_simscape_cl_transmissibility.png

Figure 14: Obtained Transmissibility

@@ -1199,309 +1334,10 @@ The obtained transmissibility in Open-loop, for the centralized control as well
- -
-

4 Stewart Platform - Analytical Model

-
-
-
-

4.1 Characteristics

-
-
-
L  = 0.055; % Leg length [m]
-Zc = 0;     % ?
-m  = 0.2;   % Top platform mass [m]
-k  = 1e3;   % Total vertical stiffness [N/m]
-c  = 2*0.1*sqrt(k*m); % Damping ? [N/(m/s)]
-
-Rx = 0.04; % ?
-Rz = 0.04; % ?
-Ix = m*Rx^2; % ?
-Iy = m*Rx^2; % ?
-Iz = m*Rz^2; % ?
-
-
-
-
- -
-

4.2 Mass Matrix

-
-
-
M = m*[1   0 0  0         Zc        0;
-       0   1 0 -Zc        0         0;
-       0   0 1  0         0         0;
-       0 -Zc 0  Rx^2+Zc^2 0         0;
-       Zc  0 0  0         Rx^2+Zc^2 0;
-       0   0 0  0         0         Rz^2];
-
-
-
-
- -
-

4.3 Jacobian Matrix

-
-
-
Bj=1/sqrt(6)*[ 1             1          -2          1         1        -2;
-               sqrt(3)      -sqrt(3)     0          sqrt(3)  -sqrt(3)   0;
-               sqrt(2)       sqrt(2)     sqrt(2)    sqrt(2)   sqrt(2)   sqrt(2);
-               0             0           L          L        -L         -L;
-               -L*2/sqrt(3) -L*2/sqrt(3) L/sqrt(3)  L/sqrt(3) L/sqrt(3)  L/sqrt(3);
-               L*sqrt(2)    -L*sqrt(2)   L*sqrt(2) -L*sqrt(2) L*sqrt(2) -L*sqrt(2)];
-
-
-
-
- -
-

4.4 Stifnness and Damping matrices

-
-
-
kv = k/3;     % Vertical Stiffness of the springs [N/m]
-kh = 0.5*k/3; % Horizontal Stiffness of the springs [N/m]
-
-K = diag([3*kh, 3*kh, 3*kv, 3*kv*Rx^2/2, 3*kv*Rx^2/2, 3*kh*Rx^2]); % Stiffness Matrix
-C = c*K/100000; % Damping Matrix
-
-
-
-
- -
-

4.5 State Space System

-
-
-
A  = [ zeros(6) eye(6); ...
-      -M\K     -M\C];
-Bw = [zeros(6); -eye(6)];
-Bu = [zeros(6); M\Bj];
-
-Co = [-M\K -M\C];
-
-D  = [zeros(6) M\Bj];
-
-ST = ss(A,[Bw Bu],Co,D);
-
-
- -
    -
  • OUT 1-6: 6 dof
  • -
  • IN 1-6 : ground displacement in the directions of the legs
  • -
  • IN 7-12: forces in the actuators.
  • -
-
-
ST.StateName = {'x';'y';'z';'theta_x';'theta_y';'theta_z';...
-                'dx';'dy';'dz';'dtheta_x';'dtheta_y';'dtheta_z'};
-
-ST.InputName = {'w1';'w2';'w3';'w4';'w5';'w6';...
-                'u1';'u2';'u3';'u4';'u5';'u6'};
-
-ST.OutputName = {'ax';'ay';'az';'atheta_x';'atheta_y';'atheta_z'};
-
-
-
-
- -
-

4.6 Transmissibility

-
-
-
TR=ST*[eye(6); zeros(6)];
-
-
- -
-
figure
-subplot(231)
-bodemag(TR(1,1));
-subplot(232)
-bodemag(TR(2,2));
-subplot(233)
-bodemag(TR(3,3));
-subplot(234)
-bodemag(TR(4,4));
-subplot(235)
-bodemag(TR(5,5));
-subplot(236)
-bodemag(TR(6,6));
-
-
- - -
-

stewart_platform_analytical_transmissibility.png -

-

Figure 15: Transmissibility

-
-
-
- -
-

4.7 Real approximation of \(G(j\omega)\) at decoupling frequency

-
-
-
sys1 = ST*[zeros(6); eye(6)]; % take only the forces inputs
-
-dec_fr = 20;
-H1 = evalfr(sys1,j*2*pi*dec_fr);
-H2 = H1;
-D = pinv(real(H2'*H2));
-H1 = inv(D*real(H2'*diag(exp(j*angle(diag(H2*D*H2.'))/2)))) ;
-[U,S,V] = svd(H1);
-
-wf = logspace(-1,2,1000);
-for i  = 1:length(wf)
-    H = abs(evalfr(sys1,j*2*pi*wf(i)));
-    H_dec = abs(evalfr(U'*sys1*V,j*2*pi*wf(i)));
-    for j = 1:size(H,2)
-        g_r1(i,j) =  (sum(H(j,:))-H(j,j))/H(j,j);
-        g_r2(i,j) =  (sum(H_dec(j,:))-H_dec(j,j))/H_dec(j,j);
-        %     keyboard
-    end
-    g_lim(i) = 0.5;
-end
-
-
-
-
- -
-

4.8 Coupled and Decoupled Plant “Gershgorin Radii”

-
-
-
figure;
-title('Coupled plant')
-loglog(wf,g_r1(:,1),wf,g_r1(:,2),wf,g_r1(:,3),wf,g_r1(:,4),wf,g_r1(:,5),wf,g_r1(:,6),wf,g_lim,'--');
-legend('$a_x$','$a_y$','$a_z$','$\theta_x$','$\theta_y$','$\theta_z$','Limit');
-xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
-
-
- - -
-

gershorin_raddii_coupled_analytical.png -

-

Figure 16: Gershorin Raddi for the coupled plant

-
- -
-
figure;
-title('Decoupled plant (10 Hz)')
-loglog(wf,g_r2(:,1),wf,g_r2(:,2),wf,g_r2(:,3),wf,g_r2(:,4),wf,g_r2(:,5),wf,g_r2(:,6),wf,g_lim,'--');
-legend('$S_1$','$S_2$','$S_3$','$S_4$','$S_5$','$S_6$','Limit');
-xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
-
-
- - -
-

gershorin_raddii_decoupled_analytical.png -

-

Figure 17: Gershorin Raddi for the decoupled plant

-
-
-
- -
-

4.9 Decoupled Plant

-
-
-
figure;
-bodemag(U'*sys1*V,opts)
-
-
- - -
-

stewart_platform_analytical_decoupled_plant.png -

-

Figure 18: Decoupled Plant

-
-
-
- -
-

4.10 Controller

-
-
-
fc = 2*pi*0.1; % Crossover Frequency [rad/s]
-c_gain = 50; %
-
-cont = eye(6)*c_gain/(s+fc);
-
-
-
-
- -
-

4.11 Closed Loop System

-
-
-
FEEDIN  = [7:12]; % Input of controller
-FEEDOUT = [1:6]; % Output of controller
-
-
- -

-Centralized Control -

-
-
STcen = feedback(ST, inv(Bj)*cont, FEEDIN, FEEDOUT);
-TRcen = STcen*[eye(6); zeros(6)];
-
-
- -

-SVD Control -

-
-
STsvd = feedback(ST, pinv(V')*cont*pinv(U), FEEDIN, FEEDOUT);
-TRsvd = STsvd*[eye(6); zeros(6)];
-
-
-
-
- -
-

4.12 Results

-
-
-
figure
-subplot(231)
-bodemag(TR(1,1),TRcen(1,1),TRsvd(1,1),opts)
-legend('OL','Centralized','SVD')
-subplot(232)
-bodemag(TR(2,2),TRcen(2,2),TRsvd(2,2),opts)
-legend('OL','Centralized','SVD')
-subplot(233)
-bodemag(TR(3,3),TRcen(3,3),TRsvd(3,3),opts)
-legend('OL','Centralized','SVD')
-subplot(234)
-bodemag(TR(4,4),TRcen(4,4),TRsvd(4,4),opts)
-legend('OL','Centralized','SVD')
-subplot(235)
-bodemag(TR(5,5),TRcen(5,5),TRsvd(5,5),opts)
-legend('OL','Centralized','SVD')
-subplot(236)
-bodemag(TR(6,6),TRcen(6,6),TRsvd(6,6),opts)
-legend('OL','Centralized','SVD')
-
-
- - -
-

stewart_platform_analytical_svd_cen_comp.png -

-

Figure 19: Comparison of the obtained transmissibility for the centralized control and the SVD control

-
-
-
-

Author: Dehaeze Thomas

-

Created: 2020-10-13 mar. 14:53

+

Created: 2020-11-06 ven. 11:59

diff --git a/index.org b/index.org index c314118..3396f7e 100644 --- a/index.org +++ b/index.org @@ -689,6 +689,24 @@ This Matlab function is accessible [[file:gravimeter/pzmap_testCL.m][here]]. :PROPERTIES: :header-args:matlab+: :tangle stewart_platform/simscape_model.m :END: +** Introduction :ignore: + +In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure [[fig:SP_assembly]]. + +#+name: fig:SP_assembly +#+caption: Stewart Platform CAD View +[[file:figs/SP_assembly.png]] + +The analysis of the SVD control applied to the Stewart platform is performed in the following sections: +- Section [[sec:stewart_simscape]]: The parameters of the Simscape model of the Stewart platform are defined +- Section [[sec:stewart_identification]]: The plant is identified from the Simscape model and the centralized plant is computed thanks to the Jacobian +- Section [[sec:stewart_dynamics]]: The identified Dynamics is shown +- Section [[sec:stewart_real_approx]]: A real approximation of the plant is computed for further decoupling using the Singular Value Decomposition (SVD) +- Section [[sec:stewart_svd_decoupling]]: The decoupling is performed thanks to the SVD. The effectiveness of the decoupling is verified using the Gershorin Radii +- Section [[sec:stewart_decoupled_plant]]: The dynamics of the decoupled plant is shown +- Section [[sec:stewart_diagonal_control]]: A diagonal controller is defined to control the decoupled plant +- Section [[sec:stewart_closed_loop_results]]: Finally, the closed loop system properties are studied + ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> @@ -707,7 +725,7 @@ This Matlab function is accessible [[file:gravimeter/pzmap_testCL.m][here]]. addpath('STEP'); #+end_src -** Jacobian +** Jacobian :noexport: First, the position of the "joints" (points of force application) are estimated and the Jacobian computed. #+begin_src matlab open('drone_platform_jacobian.slx'); @@ -741,7 +759,8 @@ First, the position of the "joints" (points of force application) are estimated save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J'); #+end_src -** Simscape Model +** Simscape Model - Parameters +<> #+begin_src matlab open('drone_platform.slx'); #+end_src @@ -757,16 +776,19 @@ Definition of spring parameters cz = 0.025; #+end_src +Gravity: #+begin_src matlab g = 0; #+end_src -We load the Jacobian. +We load the Jacobian (previously computed from the geometry). #+begin_src matlab load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J'); #+end_src ** Identification of the plant +<> + The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform. #+begin_src matlab %% Name of the Simulink File @@ -792,39 +814,41 @@ There are 24 states (6dof for the bottom platform + 6dof for the top platform). #+RESULTS: : State-space model with 6 outputs, 12 inputs, and 24 states. -#+begin_src matlab - % G = G*blkdiag(inv(J), eye(6)); - % G.InputName = {'Dw1', 'Dw2', 'Dw3', 'Dw4', 'Dw5', 'Dw6', ... - % 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; -#+end_src +The "centralized" plant $\bm{G}_x$ is now computed (Figure [[fig:centralized_control]]). -Thanks to the Jacobian, we compute the transfer functions in the frame of the legs and in an inertial frame. +#+name: fig:centralized_control +#+caption: Centralized control architecture +[[file:figs/centralized_control.png]] + +Thanks to the Jacobian, we compute the transfer functions in the inertial frame (transfer function from forces and torques applied to the top platform to the absolute acceleration of the top platform). #+begin_src matlab Gx = G*blkdiag(eye(6), inv(J')); Gx.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ... 'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; - - % Gl = J*G; - % Gl.OutputName = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'}; #+end_src ** Obtained Dynamics +<> + #+begin_src matlab :exports none freqs = logspace(-1, 2, 1000); figure; + tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); - ax1 = subplot(2, 1, 1); + % Magnitude + ax1 = nexttile([2, 1]); hold on; plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))), 'DisplayName', '$A_x/F_x$'); plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))), 'DisplayName', '$A_y/F_y$'); plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$A_z/F_z$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]); legend('location', 'southeast'); - ax2 = subplot(2, 1, 2); + % 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')))); @@ -839,7 +863,7 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le #+end_src #+begin_src matlab :tangle no :exports results :results file replace - exportFig('figs/stewart_platform_translations.pdf', 'width', 'full', 'height', 'full'); + exportFig('figs/stewart_platform_translations.pdf', 'eps', true, 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:stewart_platform_translations @@ -851,18 +875,21 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le freqs = logspace(-1, 2, 1000); figure; + tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); - ax1 = subplot(2, 1, 1); + % Magnitude + ax1 = nexttile([2, 1]); hold on; plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))), 'DisplayName', '$A_{R_x}/M_x$'); plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))), 'DisplayName', '$A_{R_y}/M_y$'); plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$A_{R_z}/M_z$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude [rad/(Nm)]'); set(gca, 'XTickLabel',[]); + ylabel('Magnitude [rad/(Nm)]'); set(gca, 'XTickLabel',[]); legend('location', 'southeast'); - ax2 = subplot(2, 1, 2); + % Phase + ax2 = nexttile; hold on; plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz')))); plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz')))); @@ -877,7 +904,7 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le #+end_src #+begin_src matlab :tangle no :exports results :results file replace - exportFig('figs/stewart_platform_rotations.pdf', 'width', 'full', 'height', 'full'); + exportFig('figs/stewart_platform_rotations.pdf', 'eps', true, 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:stewart_platform_rotations @@ -885,94 +912,9 @@ Thanks to the Jacobian, we compute the transfer functions in the frame of the le #+RESULTS: [[file:figs/stewart_platform_rotations.png]] -#+begin_src matlab :exports none - freqs = logspace(-1, 2, 1000); - - figure; - - ax1 = subplot(2, 1, 1); - hold on; - for out_i = 1:5 - for in_i = i+1:6 - plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', out_i), sprintf('F%i', in_i)), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]); - end - end - for ch_i = 1:6 - plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', ch_i), sprintf('F%i', ch_i)), freqs, 'Hz')))); - end - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); - - ax2 = subplot(2, 1, 2); - hold on; - for ch_i = 1:6 - plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(sprintf('A%i', ch_i), sprintf('F%i', ch_i)), freqs, 'Hz')))); - end - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); - ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); - ylim([-180, 180]); - yticks([-360:90:360]); - - linkaxes([ax1,ax2],'x'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace - exportFig('figs/stewart_platform_legs.pdf', 'width', 'full', 'height', 'full'); -#+end_src - -#+name: fig:stewart_platform_legs -#+caption: Stewart Platform Plant from forces applied by the legs to displacement of the legs -#+RESULTS: -[[file:figs/stewart_platform_legs.png]] - -#+begin_src matlab :exports none - freqs = logspace(-1, 2, 1000); - - figure; - - ax1 = subplot(2, 1, 1); - hold on; - plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_x/D_{w,x}$'); - plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_y/D_{w,y}$'); - plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Dwz')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_z/D_{w,z}$'); - % set(gca,'ColorOrderIndex',1) - % plot(freqs, abs(squeeze(freqresp(TR(1,1), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$'); - % plot(freqs, abs(squeeze(freqresp(TR(2,2), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$'); - % plot(freqs, abs(squeeze(freqresp(TR(3,3), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$'); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Transmissibility - Translations'); xlabel('Frequency [Hz]'); - legend('location', 'northeast'); - - ax2 = subplot(2, 1, 2); - hold on; - plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Rwx')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_x/R_{w,x}$'); - plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'Rwy')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_y/R_{w,y}$'); - plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Rwz')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_z/R_{w,z}$'); - % set(gca,'ColorOrderIndex',1) - % plot(freqs, abs(squeeze(freqresp(TR(4,4), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$'); - % plot(freqs, abs(squeeze(freqresp(TR(5,5), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$'); - % plot(freqs, abs(squeeze(freqresp(TR(6,6), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$'); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Transmissibility - Rotations'); xlabel('Frequency [Hz]'); - legend('location', 'northeast'); - - linkaxes([ax1,ax2],'x'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace - exportFig('figs/stewart_platform_transmissibility.pdf', 'width', 'full', 'height', 'full'); -#+end_src - -#+name: fig:stewart_platform_transmissibility -#+caption: Transmissibility -#+RESULTS: -[[file:figs/stewart_platform_transmissibility.png]] - ** Real Approximation of $G$ at the decoupling frequency +<> + Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_c(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$. #+begin_src matlab wc = 2*pi*30; % Decoupling frequency [rad/s] @@ -989,7 +931,39 @@ The real approximation is computed as follows: H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2)))); #+end_src +#+begin_src matlab :exports results :results value table replace :tangle no + data2orgtable(H1, {}, {}, ' %.1f '); +#+end_src + +#+caption: Real approximate of $G$ at the decoupling frequency $\omega_c$ +#+RESULTS: +| 4.4 | -2.1 | -2.1 | 4.4 | -2.4 | -2.4 | +| -0.2 | -3.9 | 3.9 | 0.2 | -3.8 | 3.8 | +| 3.4 | 3.4 | 3.4 | 3.4 | 3.4 | 3.4 | +| -367.1 | -323.8 | 323.8 | 367.1 | 43.3 | -43.3 | +| -162.0 | -237.0 | -237.0 | -162.0 | 398.9 | 398.9 | +| 220.6 | -220.6 | 220.6 | -220.6 | 220.6 | -220.6 | + + +Please not that the plant $G$ at $\omega_c$ is already an almost real matrix. +This can be seen on the Bode plots where the phase is close to 1. +This can be verified below where only the real value of $G(\omega_c)$ is shown + +#+begin_src matlab :exports results :results value table replace :tangle no + data2orgtable(real(evalfr(Gc, j*wc)), {}, {}, ' %.1f '); +#+end_src + +#+RESULTS: +| 4.4 | -2.1 | -2.1 | 4.4 | -2.4 | -2.4 | +| -0.2 | -3.9 | 3.9 | 0.2 | -3.8 | 3.8 | +| 3.4 | 3.4 | 3.4 | 3.4 | 3.4 | 3.4 | +| -367.1 | -323.8 | 323.8 | 367.1 | 43.3 | -43.3 | +| -162.0 | -237.0 | -237.0 | -162.0 | 398.9 | 398.9 | +| 220.6 | -220.6 | 220.6 | -220.6 | 220.6 | -220.6 | + ** Verification of the decoupling using the "Gershgorin Radii" +<> + First, the Singular Value Decomposition of $H_1$ is performed: \[ H_1 = U \Sigma V^H \] @@ -1069,6 +1043,8 @@ Gershgorin Radii for the decoupled plant using the Jacobian: [[file:figs/simscape_model_gershgorin_radii.png]] ** Decoupled Plant +<> + Let's see the bode plot of the decoupled plant $G_d(s)$. \[ G_d(s) = U^T G_c(s) V \] @@ -1089,7 +1065,7 @@ Let's see the bode plot of the decoupled plant $G_d(s)$. end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude'); xlabel('Frequency [Hz]'); + ylabel('Magnitude'); xlabel('Frequency [Hz]'); legend('location', 'southeast'); #+end_src @@ -1119,7 +1095,7 @@ Let's see the bode plot of the decoupled plant $G_d(s)$. end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude'); xlabel('Frequency [Hz]'); + ylabel('Magnitude'); xlabel('Frequency [Hz]'); legend('location', 'southeast'); #+end_src @@ -1133,6 +1109,8 @@ Let's see the bode plot of the decoupled plant $G_d(s)$. [[file:figs/simscape_model_decoupled_plant_jacobian.png]] ** Diagonal Controller +<> + The controller $K$ is a diagonal controller consisting a low pass filters with a crossover frequency $\omega_c$ and a DC gain $C_g$. #+begin_src matlab @@ -1142,13 +1120,12 @@ The controller $K$ is a diagonal controller consisting a low pass filters with a K = eye(6)*C_g/(s+wc); #+end_src -** Centralized Control -The control diagram for the centralized control is shown below. +The control diagram for the centralized control is shown in Figure [[fig:centralized_control]]. 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 +#+begin_src latex :file centralized_control.pdf :tangle no :exports results \begin{tikzpicture} \node[block={2cm}{1.5cm}] (G) {$G$}; \node[block, below right=0.6 and -0.5 of G] (K) {$K_c$}; @@ -1167,18 +1144,19 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied \end{tikzpicture} #+end_src +#+name: fig:centralized_control +#+caption: Control Diagram for the Centralized control #+RESULTS: [[file:figs/centralized_control.png]] +The feedback system is computed as shown below. #+begin_src matlab G_cen = feedback(G, inv(J')*K, [7:12], [1:6]); #+end_src -** SVD Control -The SVD control architecture is shown below. +The 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 +#+begin_src latex :file svd_control.pdf :tangle no :exports results \begin{tikzpicture} \node[block={2cm}{1.5cm}] (G) {$G$}; \node[block, below right=0.6 and 0 of G] (U) {$U^{-1}$}; @@ -1199,15 +1177,19 @@ The matrices $U$ and $V$ are used to decoupled the plant $G$. \end{tikzpicture} #+end_src +#+name: fig:svd_control +#+caption: Control Diagram for the SVD control #+RESULTS: [[file:figs/svd_control.png]] -SVD Control +The feedback system is computed as shown below. #+begin_src matlab G_svd = feedback(G, pinv(V')*K*pinv(U), [7:12], [1:6]); #+end_src -** Results +** Closed-Loop system Performances +<> + Let's first verify the stability of the closed-loop systems: #+begin_src matlab :results output replace text isstable(G_cen) @@ -1302,7 +1284,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well #+RESULTS: [[file:figs/stewart_platform_simscape_cl_transmissibility.png]] -* Stewart Platform - Analytical Model +* Stewart Platform - Analytical Model :noexport: :PROPERTIES: :header-args:matlab+: :tangle stewart_platform/analytical_model.m :END: