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Copy pathINS.GPS.kalman.q
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INS.GPS.kalman.q
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/ for documentation see my directory kalman.filter.studies
/ Read [MS]An integrated INS GPS navigation system for small - kalman(1998)
/ And octave file: an.intergrated.INS.GPS.kalman.v2
/ code.analysis.an.intergrated.INS.GPS.kalman.v2.docx
/------ helper functions
make_diagA:{[x]`float$x*{x=/:x}til count x};
make_diag:{[x]
:make_diagA (x)#1f;
};
make_diagY:{[x;y]
:make_diagA (x)#y;
};
zeroM1:{[x] (x,x)#0f,x#0f}; / Returns an x by x matrix of 0f
zeroM2:{[x;y] (x;y)#0.0f }; / Returns an x by y matrix of 0f
/ from stat.q. similar to octave/matlab randn
pi:acos -1
nor:{$[x=2*n:x div 2;raze sqrt[-2*log n?1f]*/:(sin;cos)@\:(2*pi)*n?1f;-1_.z.s 1+x]}
/--Sample Size--
minutes:6;
delta_t:.01;
/ Sample size
samples:(2 * minutes * 60 % delta_t);
s:`long$samples;
/------------ DB
xV:zeroM2[8;s];
xV1:zeroM2[8;s];
x:zeroM2[8;s];
/ Time Constants
tau_1:60; / seconds for velocity
tau_2:60; / seconds for GPS
tau_3:3600; / seconds for ocean current
/ size of square matrices
n:8;
/ Process Noise Vector
wA:nor s;
w2:nor s;
w3:nor s;
w4:nor s;
w5:600.0f * nor s; / Gives a GPS standard deviation of 3 m
w6:600.0f * nor s; / Gives a GPS standard deviation of 3 m
w7:s#0.0f; / No white noise input for x7
w8:s#0.0f; / No white noise input for x8
w:zeroM2[n;s];
w[0;]:wA;w[1;]:w2;w[2;]:w3;w[3;]:w4;
w[4;]:w5;w[5;]:w6;w[6;]:w7;w[7;]:w8;
/ Measurement Noise Vector
vA:nor s;
vB:nor s;
vC:nor s;
vD:nor s;
v_1:zeroM2[4;s];
v_0:zeroM2[2;s];
v_1[0;]:vA;v_1[1;]:vB;v_1[2;]:vC;v_1[3;]:vD; / Noise vector with GPS input
v_0[0;]:vA;v_0[1;]:vB; / Noise vector without GPS input
/ Generate GPS Sampling
/ gps_flag:nor s;
/ State Transition Matrix
F:zeroM2[8;8];
F[0;0]:exp[-1*delta_t%tau_1];
F[1;1]:exp[-1*delta_t%tau_1];
F[2;2]:exp[-1*delta_t%tau_2];
F[3;3]:exp[-1*delta_t%tau_2];
F[4;4]:exp[-1*delta_t%tau_3];
F[5;5]:exp[-1*delta_t%tau_3];
F[6;0]:tau_1*(1-exp[-1*delta_t%tau_1]);
F[6;2]:tau_2*(1-exp[-1*delta_t%tau_2]);
F[7;1]:tau_1*(1-exp[-1*delta_t%tau_1]);
F[7;3]:tau_2*(1-exp[-1*delta_t%tau_2]);
/ Q is the mean of Process Noise Vector 'w'
/ Q = E[w wT]
Q:zeroM2[8;8];
Q[0;0]:(1%(2*tau_1))*(1-exp[-2*delta_t]%tau_1);
Q[1;1]:(1%(2*tau_1))*(1-exp[-2*delta_t]%tau_1);
Q[2;2]:(1%(2*tau_2))*(1-exp[-2*delta_t]%tau_2);
Q[3;3]:(1%(2*tau_2))*(1-exp[-2*delta_t]%tau_2);
Q[4;4]:(1%(2*tau_3))*(1-exp[-2*delta_t]%tau_3);
Q[5;5]:(1%(2*tau_3))*(1-exp[-2*delta_t]%tau_3);
/ Generate Process Noise Vectors
process_noise:mmu[xexp[Q;.5];w];
/ Identify Matrix
EY:make_diag[8];
/ Error Covariance Matrix
R_0:make_diag[2];
R_0[0;0]:0.5;R_0[1;1]:0.5;
R_1:make_diag[4];
R_1[0;0]:0.5;R_1[1;1]:0.5;R_1[2;2]:0.0;R_1[3;3]:0.0;
/ Generate Measurement Noise Vectors
sensor_noise_0:mmu[sqrt[R_0];v_0]; / Without GPS signal
sensor_noise_1:mmu[sqrt[R_1];v_1]; / With GPS signal
/ Initial z as globals for graphing
z_gps:zeroM2[1;s];
z_gpsA:zeroM2[1;s];
z_gps2:zeroM2[1;s];
z_gps3:zeroM2[1;s];
z_gps4:zeroM2[1;s];
z_gps_time:zeroM2[1;s];
/ System Error Covariance
/ P is initial Error Covariance Matrix
P:make_diag[8];
P[0;0]:0.5;P[1;1]:0.5;P[4;4]:3.0;P[5;5]:3.0;
P[6;6]:5.0;P[7;7]:5.0;
P1:zeroM2[8;8];
/ Index for Measurement Vector
tia:0;tia:`long$tia; /Initial Index for Measurement Vector without GPS
tib:0;tib:`long$tib; /Initial Index for Measurement Vector with GPS
/ Vector with measurements
z_vel:zeroM2[2;s];
z_vel1:s#0f;
z_vel2:s#0f;
/ Vector with measurements
z_gps:zeroM2[4;s];
z_gps1:zeroM2[1;s]
z_gps2:zeroM2[1;s]
z_gps3:zeroM2[1;s]
z_gps4:zeroM2[1;s]
/ noiseless connection between measurement and state vector
H0:2 8#0.0; / without GPS
H0[0;0]:1.0;H0[1;1]:1.0;
H1:4 8#0f; / with GPS
H1[0;0]:1f;H1[1;1]:1f;H1[2;4]:1f;H1[2;6]:1f;H1[3;5]:1f;H1[3;7]:1f;
/ Measurement error covariance.
R_0:2 2#0.0;R_0[0;0]:0.5;R_0[1;1]:0.5; / Without GPS
R_1:4 4#0.0;R_1[0;0]:0.5;R_1[1;1]:0.5;R_1[2;2]:0.0;R_1[3;3]:0.0; / with GPS
myfunction:{[it]
x[;it]::process_noise[;it-1]+ mmu[F;x[;it-1]];
j:first nor 1;
$[j < 0.5;gps_flag::0f;gps_flag::1f];
/ show "gps_flag";show gps_flag;
if[gps_flag=0f; / loop w/o GPS Signal
[
sensor_noise_0[;it]: mmu[xexp[R_0;0.5];v_0[;it]];
z_vel1[tia]::mmu[H0[0;];x[;it]]+sensor_noise_0[0;it];
z_vel2[tia]::mmu[H0[1;];x[;it]]+sensor_noise_0[1;it];
z_vel::(z_vel1;z_vel2);
/ Compute Ka1man Gain
K:mmu[mmu[P;flip H0];inv[mmu[mmu[H0;P];flip H0]+R_0]];
/ Update Estimate
xV1[;it]::xV[;it-1]+ mmu[K;(z_vel[;tia]-mmu[H0;xV[;it-1]])];
/ Compute Error Covariance for Updated Estimate
P1::mmu[(EY- mmu[K;H0]);P];
P1::mmu[P1;flip P1]%2;
tia+:1;
]];
if[gps_flag=1f; / Loop with GPS Signal
[
z_gps1:mmu[H1[0;];x[;it]]+sensor_noise_1[0;it];
z_gps2:mmu[H1[1;];x[;it]]+sensor_noise_1[1;it];
z_gps3:mmu[H1[2;];x[;it]]+sensor_noise_1[2;it];
z_gps4:mmu[H1[3;];x[;it]]+sensor_noise_1[3;it];
z_gps[0;it]:first z_gps1;
z_gps[1;it]:first z_gps2;
z_gps[2;it]:first z_gps3;
z_gps[3;it]:first z_gps4;
/ Compute Ka1man Gain
K:mmu[mmu[P;flip H1];inv[mmu[H1;mmu[P;flip H1]]+R_1]];
/ Update Estimate
xV1[;it]::xV[;it-1]+ mmu[K;(z_gps[;tib]-mmu[H1;xV[;it-1]])];
/ Compute Error Covariance for Updated Estimate
P1::mmu[(EY- mmu[K;H1]);P];
P1::mmu[P1;flip P1]%2;
tib+:1;
]];
xV[;it]::process_noise[;it-1] + mmu[F;xV1[;it]];
P::Q + mmu[F;mmu[P1;flip F]];
P::mmu[P;flip P]%2;
x[;it]::xV[;it];
}
simulation:{[]
it:1;
while[it< s;
myfunction[it];
it+:1;
]
}
show "sample size";
show s;
simulation[];