Solving positioning problems with minimal data

Abstract

Global and local positioning systems (LPS) make use of nonlinear equations systems to calculate coordinates of unknown points. There exist several methods, such as Sturmfels' resultant, Groebner bases and least squares, for dealing with this kind of equations. We introduce two methods for solving this problem with the aid of symbolic techniques relying on closed-form solutions for the solution set of a system of linear equations. We suppose the receiver just detects or chooses minimal data, i.e., four satellites in global positioning systems (GPS) or three stations in LPS. Both methods proceed by parameterizing the line joining two solution points to later solve a nonlinear univariate equation, either quadratic or with degree smaller than 6. The first one uses the Generalized Cramer Identities, which is a different presentation of the generalized Moore---Penrose inverse, and ends with a degree 6 univariate equation for GPS and a degree 4 univariate equation for LPS. The second one solves the system by dealing with a more geometric way, ending with a quadratic equation. Our approach covers all possible cases with a finite number of solutions, while Bancroft's method cannot be applied when the four satellites, taking the clock bias as fourth coordinate, and the origin lay in the same hyperplane of $${\mathbb{R}}^{4}$$R4, and the method by Grafarend and Shan fails when the four satellites are in the same plane in $${\mathbb{R}}^{3}$$R3. The two proposed methods fail only when the pseudorange 4 point problem has infinite solutions.