Not being a statistician I apologize in advance. With multilateration I mean a 2D geometric grid of distances measured with a laser-meter (affected by a random error), such as the one below:

I currently estimate the real distances numerically (using scipy Python package optimize.minimize function), by minimizing the objective function

$$\sum_{i=1}^{n} (De_i-Dm_i)^2$$

where $Dm_i$ are the $n$ measured distances and $De_i$ are the estimated real distances expressed in terms of vertices coordinates $(x,y)$

$$De = \sqrt{(x_b-x_a)^2+(y_b-y_a)^2}$$

which are the actual ultimate unknowns of the problem (at least with this approach).

I wonder the above approach is correct, and in the affirmative case if there is a way to find the solution directly (non numerically).



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