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Copy pathan.intergrated.INS.GPS.kalman.m
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an.intergrated.INS.GPS.kalman.m
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% SANS Kalman Filter in
% "An integrated INS/GPS navigation system for small AUVs using an asynchronous Kalman filter"
% By Glenn Hernandez
% 8 May 98
% edited from pdf by nicholasferguson@wingarch.com 2019
% Number of minutes for simulated run
minutes = 1;
% Define the resolution of the simulation here
delta_t = .01; % For 100 Hz resolution
% Number of samples
samples = 1/delta_t * 2 * minutes; % Gives 60000 samples at delta_t = .01 seconds
% Time Constants
tau_1 = 60; % seconds for velocity
tau_2 = 60; % seconds for GPS
tau_3 = 3600; % seconds for ocean current
% Process Noise Vector
wl = randn(1, samples)
w2 = randn(1, samples)
w3 = randn(1, samples);
w4 = randn(1, samples);
w5 = 600 * randn(1, samples); % Gives a GPS standard deviation of 3 m
w6 = 600 * randn(1,samples); % Gives a GPS standard deviation of 3 m
w7 = zeros(1,samples); % No white noise input for x7
w8 = zeros(1,samples); % No white noise input for x8
w = [wl;w2;w3;w4;w5;w6;w7;w8];
% Measurement Noise Vector
vl = randn(1,samples);
v2 = randn(1,samples) ;
v3 = randn(1,samples) ;
v4 = randn(1,samples) ;
v_0 = [vl;v2]; % Noise vector without GPS input
v_l = [vl;v2;v3;v4] ; % Noise vector with GPS input
% Generate GPS Sampling
gps_flag = zeros(samples,1);
for g = 1:1/delta_t: samples % Need at least .01 seconds between GPS fixes
j = rand; % Normalized random generator between and 1
if j < .5
gps_flag(g) = 0; % No GPS Signal Available
else
gps_flag(g) = 1; % GPS Signal Available
end
end
'gps flags generated'
% System Matrix
A = [1/tau_1 0 0 0 0 0 0 0;
0 1/tau_1 0 0 0 0 0 0;
0 0 1/tau_2 0 0 0 0 0;
0 0 0 1/tau_2 0 0 0 0;
0 0 0 0 1/tau_3 0 0 0;
0 0 0 0 0 1/tau_3 0 0;
1/tau_1 0 1/tau_2 0 0 0 0 0;
0 1/tau_1 0 1/tau_2 0 0 0 0];
% Input Noise Matrix
B = [wl;w2 ;w3;w4;w5;w6;w7;w8 ];
% Output Matrix
C = eye(8);
D = zeros(8,samples);
% State Transition Matrix
phi = [exp(-delta_t/tau_1) 0 0 0 0 0 0 0;
0 exp(-delta_t/tau_1) 0 0 0 0 0 0;
0 0 exp(-delta_t/tau_2) 0 0 0 0 0;
0 0 0 exp(-delta_t/tau_2) 0 0 0 0;
0 0 0 0 exp(-delta_t/tau_3) 0 0 0;
0 0 0 0 0 exp(-delta_t/tau_3) 0 0;
tau_1*(1-exp(-delta_t/tau_1)) 0 tau_2*(1-exp(-delta_t/tau_2)) 0 0 0 0 0;
0 tau_1*(1-exp(-delta_t/tau_1)) 0 tau_2*(1-exp(-delta_t/tau_2)) 0 0 0 0];
Q = [((1/(2*tau_1) )*(1-exp((-2*delta_t)/tau_1))) 0 0 0 0 0 0 0;
0 ((1/(2*tau_1) )*(1-exp((-2*delta_t)/tau_1))) 0 0 0 0 0 0;
0 0 ((1/(2*tau_2))*(1-exp((-2*delta_t)/tau_2))) 0 0 0 0 0;
0 0 0 ((1/(2*tau_2))*(1-exp((-2*delta_t)/tau_2))) 0 0 0 0;
0 0 0 0 ((1/(2*tau_3))*(1-exp((-2*delta_t)/tau_3))) 0 0 0;
0 0 0 0 0 ((1/(2*tau_3))*(1-exp((-2*delta_t)/tau_3))) 0 0;
0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0];
% Generate Process Noise Vectors
process_noise = sqrt(Q)*w;
% Error Covariance Matrix
R_0 = diag([.5 .5]); % Without GPS signal
R_l = diag([.5 .5 0 0]); % With GPS signal
% Generate Measurement Noise Vectors
sensor_noise_0 = sqrt(R_0) * v_0; % Without GPS signal
sensor_noise_1 = sqrt(R_l) * v_l; % With GPS signal
% Initial x_hat_minus
x_hat_minus (8,samples) = zeros;
% Initial x_hat_plus
x_hat_plus (8,samples) = zeros;
% Initial State Vector
x(8,samples) = zeros;
% Initial z as globals for graphing
z_gps = zeros(1,samples);
z_gps1 = zeros(1,samples);
z_gps2 = zeros(1,samples);
z_gps3 = zeros(1,samples);
z_gps4 = zeros(1,samples);
z_gps_time = zeros(1,samples);
% Initial Error Covariance Matrix
P_minus = [0.5 0 0 0 0 0 0 0;
0 0.5 0 0 0 0 0 0;
0 0 1 0 0 0 0 0;
0 0 0 1 0 0 0 0;
0 0 0 0 3 0 0 0;
0 0 0 0 0 3 0 0;
0 0 0 0 0 0 5 0;
0 0 0 0 0 0 0 5];
time_index_a =1; % Initial Index forMeasurement Vector without GPS
time_index_b =1; % Initial Index forMeasurement Vector with GPS
'Beginning Kalman Loops'
% Begin Simulation
for k = 2:samples
% Generate State Vectors and Measurement Vectors
x(:,k) = phi * x(:,k-1) + process_noise( :,k-1);
% Kalman loop with out GPS signal
if gps_flag(k) == 0
H = [1 0 0 0 0 0 0 0;
0 1 0 0 0 0 0 0];
R = diag([.5 .5]);
sensor_noise_0( :,k) = sqrt(R) * v_0(:,k);
z_vell(time_index_a) = H(1,:) * x(:,k) + sensor_noise_0(1,k)
z_vel2(time_index_a) = H(2,:) * x(:,k) + sensor_noise_0(2,k)
z_vel_time(time_index_a) = k * delta_t;
z_vel = [z_vell ; z_vel2];
% Compute Ka1man Gain
K = P_minus * H' * inv(H * P_minus * H' + R);
% Update Estimate
x_hat_p1us(:,k) = x_hat_minus(:,k-1)+K*(z_vel(:,time_index_a) - H *x_hat_minus(:,k-1));
% Compute Error Covariance for Updated Estimate
P_p1us = (eye(8)- K * H )*P_minus;
P_p1us = (P_p1us+ P_p1us')/2;
time_index_a = time_index_a +1; % Increase the measurement vector index by 1
else
H= [1 0 0 0 0 0 0 0;
0 1 0 0 0 0 0 0;
0 0 0 0 1 0 1 0;
0 0 0 0 0 1 0 1];
R = diag([.5 .5 0 0]);
z_gps1(time_index_b) = H(1,:)* x(:,k)+ sensor_noise_1(1,k);
z_gps2(time_index_b) = H(2,:)* x(:,k)+ sensor_noise_1(2,k);
z_gps3(time_index_b) = H(3,:)* x(:,k)+ sensor_noise_1(3,k);
z_gps4(time_index_b) = H(4,:)* x(:,k)+ sensor_noise_1(4,k);
z_gps = [z_gps1 ; z_gps2 ; z_gps3 ; z_gps4];
z_gps_time(time_index_b) = k * delta_t;
% Compute Ka1man Gain
K = P_minus * H' * inv(H * P_minus * H' + R);
% Update Estimate
x_hat_p1us(:,k) = x_hat_minus(:,k-1)+ K *(z_gps(:,time_index_b) - H *x_hat_minus (:,k-1));
% Compute Error Covariance for Updated Estimate
P_p1us = ( eye(8) - K * H ) * P_minus;
P_p1us = ( P_p1us + P_p1us') / 2;
time_index_b = time_index_b +1; % Increase the measurement vector index by 1
end
pmin( :,:,k) = P_p1us; % Save Matrix for future ana1ysis
% Project Ahead
x_hat_minus(:,k) = phi * x_hat_p1us( :,k);
P_minus = phi * P_p1us * phi' + Q;
P_minus = ( P_minus + P_minus') / 2;
time(k) = k * delta_t; % Memorize time index
% what
end
figure(1)
subplot(3,1,1)
plot(time, x(7,:),'b-',time, x_hat_minus(7,:),'r-.',z_gps_time,z_gps3, 'g^')
xlabel('time (seconds )')
ylabel('position (meters)')
title('North Position vs Time')
axis([0 max(time) -1.5*max(abs(z_gps3)) 1.5*max(abs(z_gps3))])
subplot(3,1,2)
plot(time, x(8,:),'b-',time, x_hat_minus(8,:), 'r-.',z_gps_time,z_gps4, 'g^')
xlabel('time (seconds)')
ylabel('position (meters)')
title('East Position vs Time')
axis([0 max(time) -1.5*max(abs(z_gps4) ) 1.5*max(abs(z_gps4))])
subplot(3,1,3)
plot(x_hat_minus(7,:), x_hat_minus(8,:), 'b-')
xlabel('North Position')
ylabel('East Position')
title('North-East Position plot')
axis([-1.5*max(abs(x_hat_minus(7,:))) 1.5*max(abs(x_hat_minus(7,:)))...
-1.5*max(abs(x_hat_minus(8,:))) 1.5*max(abs(x_hat_minus(8,:)))])
orient tall