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

I followed @zerm's code shown above and it worked fairly well (yellow is calculated point from 3 towers). Results are shown in the Folium snippet. Multilateration using linalg.SVD

However, when I followed the same algorithm with the changes that @mcdowella suggested (#2) using the Least Squares of an MxN System solution, results are much better. Multilateration using Least Squares MxN

Here is the amended code:

A = []
b = []
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]]
    b += [[Dm]]
# Solve using Least Squares of an MxN System
# A*x = b --> x = (ATA)_inv.AT.b = A+.b
A = np.array(A)
b = np.array(b)
AT = A.T
ATA = np.matmul(AT,A)
ATA_inv = np.linalg.inv(ATA)
Aplus = np.matmul(ATA_inv,AT)
x = np.matmul(Aplus,b)
# convert back to lat/long from ECEF
# convert to degrees
lat = math.degrees(math.asin(x[2] / R))
lon = math.degrees(math.atan2(x[1],x[0]))

I'm still exploring other multilateration methods but this post really allowed me to understand the basics of N-points MLAT. Thanks!