Complex-Valued Function Modeling of Bilateration and its Applications

Bilateration is one of the localization methods that provides the location information of the target node using two anchors' location information and two distances between the target node and each anchor. This paper shows that bilateration with different configurations in a three-anchor positioning system gives better precision than trilateration and the same precision as the nonlinear least square (NLS) method when the target is outside the convex hull of the anchors, and also that the combinatorial use of bilaterations gives the same level of precision as the NLS method or tilateration even when the target is inside the convex hull of the anchors. For the rigorous proof of this claim, we first develop a theory that the bilateration method can be naturally modeled and understood by adopting a pair of distances as a complex variable and the mapping in the bilateration method as a complex-valued sine function with a complex variable, respectively. We explain why the proposed complexification is more intuitive and applicable to real-world problems than the real-valued bilateration method. Since the complex sine function has many nice properties and one-to-one correspondence, it can be regarded as a fundamental transformation from the measurement space to position space in a two-dimensional positioning system. Next, based on the complex-valued function theory for bilateration, we describe the error propagation and show that a nonlinear least square method can be replaced with the closed formula in the bilateration localization system. The error propagation of the bilateration has a unique property: the error in position space propagates only in the horizontal direction with anchors and is unchanged in the other direction regardless of the target's position.