Multilateration of GPS Coordinates

Question

I have N GPS coordinates with N distances given to an unknown position which I wish to determine.

My first approach was to use just three p

Answer 1

Eventually, I figured it out myself - or at least improve the accuracy significantly.

The approach described at wikipedia (Eq.7) is apparently not very suited for this application, but in this case it is already a lot easier.

Considering Eq. 6 from wikipedia, we can simplify it a lot: R_0 can be guessed as the earth radius, as the origin of ECEF coordinates lies in the center of earth. Therefore, there is no need to shift everything to make one Point the origin and we can use all N equations.

In python, with P an array of ECEF coordinates and dists the distances to these points, it all boils down to

R = 6378137 # Earth radius in meters
A = []
for m in range(0,len(P)):
    x = P[m][0]
    y = P[m][1]
    z = P[m][2]
    Am = -2*x
    Bm = -2*y
    Cm = -2*z
    Dm = R*R + (pow(x,2)+pow(y,2)+pow(z,2)) - pow(dists[m],2)
    A += [[Am,Bm,Cm,Dm]]
# Solve using SVD
A = numpy.array(A)
(_,_,v) = numpy.linalg.svd(A)
# Get the minimizer
w = v[3,:]
w /= w[3] # Resulting position in ECEF

With this approach, what I described as Step 4 is no longer necessary. In fact, it even makes the solution worse.

Now, accuracy ranges between 2km and 275m -- in most cases better than the "optimal" trilateration with an error of 464m.