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

Some comments:

1) You have already checked some steps against exact answers. I suggest that you create toy problems with known amounts of random noise added to the observations. Since you know the right answer in this case you can see what happens with error propagation. If your method works well here but badly on real data you might want to think about horrid behaviour in real life, such as one or a few of the distances being seriously wrong.

2) I don't know why your solution is only up to scale, as the underlying data are properly scaled - if I went out there with ropes cut to length and tied them to the fixed points there would be no ambiguity. When you use SVD to solve the equations (7) are you doing something like www.cse.unr.edu/~bebis/MathMethods/SVD/lecture.pdf to get out a least squares solution? That should give you x, y, and z without ambiguity.

3) I'm not at all sure about how observational errors work through (7). I don't like all the divisions, for one thing. It might be worth writing down an equation for the sum of the squares of the differences between measured distances and computed distances given x,y,z for the unknown position, and then minimising this for x,y,z. The Wikipedia article discards this approach due to its cost, but it might give you a more accurate answer, and computing and comparing this answer might tell you something even if you can't use this method in practice.