Circular/hyperbolic/elliptic localization via Euclidean norm elimination

Localization of a target of interest based on measurements using multiple spatially separated sensors has been one of the central problems in numerous application areas. Many measurement types in the three-dimensional space are convertible to distance information which can be classified as range, range-difference and range-sum, which correspond to solving a set of circular, hyperbolic and elliptic equations, respectively. In this work, we develop methods for circular/hyperbolic/elliptic localization. The key idea is to remove the Euclidean norm in the resultant optimization formulations via the introduction of auxiliary vector magnitude and phase variables. Moreover, the vector magnitude and phase angle parameters are decoupled, leading to separable and computationally efficient updating rules. The optimality of the developed methods is demonstrated by comparing with the Cramér-Rao lower bound via computer simulations.