Cramer-Rao Lower Bound Attainment in Range-Only Positioning Using Geometry: The G-WLS

The positioning problem addressed in this article amounts to finding the planar coordinates of a device from a collection of ranging measurements taken from other devices located at known positions. The solution based on weighted least square (WLS) is popular, but its accuracy depends from a number of factors only partially known. In this article, we explore the dependency of the uncertainty from the geometric configuration of the anchors. We show a refinement technique for the estimate produced by the WLS that compensates for the effects of geometry on the WLS and reduces the target uncertainty to a value very close to the Cramer-Rao Lower Bound. The resulting algorithm is called geometric WLS (G-WLS) and its application is particularly important in the most critical conditions for WLS (i.e., when the target is far apart from the anchors). The effectiveness of the G-WLS is proven theoretically and is demonstrated on a large number of experiments and simulations.