Uniqueness and hyperconic geometry of positioning with biased distance measurements

Abstract

AbstractPositioning an object with biased distance measurements is exactly solvable if exact knowledge of the reference locations and noise-free range measurements are assumed. By examining the positioning algebra, this paper obtains explicit necessary and sufficient conditions for the positioning problem to have a unique or twin solutions. The intersection of negative hyperconic halves is shown to be an exact geometric interpretation of the positioning solutions. The placement of non-coplanar references largely ensures but does not guarantee unique positioning. Given a set of references, object region for non-unique positioning is identifiable by using the conditions derived in this work. The placement of five references to form an asymmetric hexahedron is postulated to be sufficient for unique positioning in a three-dimensional environment. Illustrative examples explain these findings.