Global positioning: The uniqueness question and a new solution method

Abstract

We provide a new algebraic solution procedure for the global positioning problem in n dimensions using m satellites. We also give a geometric characterization of the situations in which the problem does not have a unique solution. This characterization shows that such cases can happen in any dimension and with any number of satellites, leading to counterexamples to some open conjectures. We fill a gap in the literature by giving a proof for the long-held belief that when $m\ge n+2$, the solution is unique for almost all user positions. Even better, when $m\ge 2n+2$, almost all satellite configurations will guarantee a unique solution for all user positions. Our uniqueness results provide a basis for predicting the behavior of numerical solutions, as ill-conditioning is expected near the threshold between areas of nonuniqueness and uniqueness. Some of our results are obtained using tools from algebraic geometry.