Need method for trilateration of 3 GPS satellites

[tribaljunkie]

Can someone please shed some light on the exact method for finding the two points at which the spheres created by the GPS signals from 3 different satellites intersect? I have been working on a uni project for 4 days now and still can't get this relatively simple part of my MATLAB code to work. I realize the 4th satellite is for a time error, but that's not part of my report, i just have the position of that to select the intersection point with the range closest to the pseudorange from the 4th satellite.

the data i have is 4 satellite positions and their corresponding pseudoranges.

if someone could give me the mathematical method, or if you're able even the MATLAB process required as I am a bit rusty with it.

any help much appreciated, cheers.

 

[tribaljunkie]

after a bit more research iv found i need to use the non linear least squares method in MATLAB, but still having trouble coding it.

if the ranges from the satellites were perfect i could use the simultaneous equations

0 = (ri_ia(1,1)-ri_is1(1,1))^2+(ri_ia(2,1)-ri_is1(2,1))^2+(ri_ia(3,1)-ri_is1(3,1))^2 - rho_1^2;
0 = (ri_ia(1,1)-ri_is2(1,1))^2+(ri_ia(2,1)-ri_is2(2,1))^2+(ri_ia(3,1)-ri_is2(3,1))^2 - rho_2^2;
0 = (ri_ia(1,1)-ri_is3(1,1))^2+(ri_ia(2,1)-ri_is3(2,1))^2+(ri_ia(3,1)-ri_is3(3,1))^2 - rho_3^2;
0 = (ri_ia(1,1)-ri_is4(1,1))^2+(ri_ia(2,1)-ri_is4(2,1))^2+(ri_ia(3,1)-ri_is4(3,1))^2 - rho_4^2;

where ri_ia is the receiver position (1x3 vector), ri_is is the satellite position (1x3 vector) and rho is the corresponding range from receiver to satellite (scalar). Also rho includes unavoidable errors.

as the measurements are not exact i must use the least squares approach minimising the error dri_ia.

dri_ia1 = (ri_ia(1,1)-ri_is1(1,1))^2+(ri_ia(2,1)-ri_is1(2,1))^2+(ri_ia(3,1)-ri_is1(3,1))^2 - rho_1^2;
dri_ia2 = (ri_ia(1,1)-ri_is2(1,1))^2+(ri_ia(2,1)-ri_is2(2,1))^2+(ri_ia(3,1)-ri_is2(3,1))^2 - rho_2^2;
dri_ia3 = (ri_ia(1,1)-ri_is3(1,1))^2+(ri_ia(2,1)-ri_is3(2,1))^2+(ri_ia(3,1)-ri_is3(3,1))^2 - rho_3^2;
dri_ia4 = (ri_ia(1,1)-ri_is4(1,1))^2+(ri_ia(2,1)-ri_is4(2,1))^2+(ri_ia(3,1)-ri_is4(3,1))^2 - rho_4^2;

thing is i have no idea how to get MATLAB to do this for me, could anyone lend a hand. i know the function is LSQNONLIN but i am not experienced enough with MATLAB to understand the help file and implement it.any help would be great, thanks.