C-TOL: Convex triangulation for optimal node localization with weighted uncertainties

Abstract

There is always a need to reduce localization error in any wireless sensor network (WSN), and our aim is to observe the impact of localization uncertainty on network awareness. When nodes are deployed in a 2D plane and their l2-norm ranged triangulations are found, usually the unweighted localization uncertainty values become absurdly large with large triangulation cases. Moreover, there is no regard for the disparity between the lengths of any two links on the localization uncertainty. The upper bound of uncertainty keeps on rising with formation of asymmetric node triangulations with longer internodal distances and sharper vertices. To address this gap, a convex combination weighted approach (C-TOL, standing for Convex-Triangulation for Optimal node Localization) for solving the localization uncertainty problem is described here. The advantage of the proposed method is shown with the help of rigorous mathematical analysis of weighted uncertainty behaviour. The relationship of sensor node symmetry with triangulation uncertainty is formulated algebraically by considering both symmetric as well as asymmetric triangulations. Cramer Rao bound is derived to justify estimation under triangulation uncertainty. This approach paves the way for the WSN to prioritize different kinds of triangulations. Numerical results reveal that the weighted method prefers triangulations with more symmetry; hence it consistently achieves significantly lower values of mean and standard deviations than the existing unweighted localization technique, especially for densely connected sensor networks. Moreover, the proposed method shows robust localization performance for sparsely deployed networks as well, when compared to recent methods in literature.