A Convex Geometry Approach to Dynamic GNSS Satellite Selection for a Multi-Constellation System

In the near future GPS, GLONASS, Galileo and Compass will constitute four GNSS constellations providing up to more than 40 satellites in view at a given point and time at the Earth's surface. The problem of choosing the best set of satellites for that super-constellation will become intractable unless we adopt an efficient selection method. In a previous work the authors presented a low computational-cost method for selecting a subset of the satellites in view that provides the smallest Geometric Dilution of Precision (GDOP). The method allows to reduce up to one fourth the number of satellites to be considered for the position computation. This reduction is based on the fact that the set of satellite positions defining the maximum volume polytope (polygon in 2D, polyhedron in 3D) that can be constructed are the ones that yield the smallest GDOP. The maximum volume of a set of points is contained in the minimal convex set that encompasses all the points. Thus, the solution is given by the set of satellites defining the convex hull or smallest convex envelope of the set of satellite positions. A static scenario was previously assumed by the authors while a time-varying environment is considered now, where the dynamics of both user and satellites constantly change the convex hull. An optimal solution for convex hull computation with lower bound computational burden is proposed for both scenarios. GDOP values attained for this solution are presented. A solution for further reducing the set of satellites from the 2D solution is proposed. Finally, a comparison in terms of computational complexity against current approaches is provided.