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I need to compute the position of a static object based on the distance to multiple fixed stations (it the same thing we do to calculate the GPS receiver position based on the satellites position).

I found this article with a nice algorithm withe an analytical solution based only on four fixed stations: Exact Solution of a Three Dimensional Hyperbolic Positioning System.

What I need to do is generalize this to multiple fixed stations (can be 8, 9, 13, or $n$). So I imagine it should be an inversion problem, but I couldn't formulate a matrix for it.

It comes to solve a set of equation of the form

$$ \begin{align} (x-x_1)^2+(y-y_1)^2+(z-z_1)^2-D_1^2 &= 0 \\ (x-x_2)^2+(y-y_2)^2+(z-z_2)^2-D_2^2 &= 0 \\ \vdots \\ (x-x_n)^2+(y-y_n)^2+(z-z_n)^2-D_n^2 &= 0 \\ \end{align} $$

where $x_i$, $y_i$, $z_i$, and $D_i$ are known, and we need to find $x$, $y$ and $z$.

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migrated from Jun 6 '13 at 14:56

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It seems the following. You can simply take any four non-coplanar stations and they should determine the position of the object. Or, if you wish fairly use your data, you can take the objects coordinates averaged over all non-coplanar fours of stations. If you wish to obtain more precise objects coordinates from an experimental data, for instance, minimizing an average square deviation (ASD), you can take one of the previous object coordinates as a starting point of iterative calculation minimizing the ASD. Of course, such method have a problem of convergence proof to the global minimum and yield no analytical solution, but I hope this is not essential for your applications.

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