Distributed Cooperative Localization of Wireless Sensor Networks with Convex Hull Constraint

Localization of wireless sensor networks is aimed at determining the positions of all sensors in a network, usually given a few connected anchor nodes´ positions and certain relative measurements, where the latter could be pairwise distance measurements among directly connected neighbors as considered in this paper. In this paper we investigate neighborhood collaboration based distributed cooperative localization of all sensors in a particular network with the so-called `convex hull constraint´: all nodes in such a network are either position-known anchors or sensors to be localized, and every sensor is inside the convex hull of its neighbors. For such a practically widely seen thus important class of localizable wireless sensor networks, we propose three iterative self-positioning algorithms, for independent implementation at all individual sensors of the considered network. Analysis and simulation study show that when iteratively running at all sensors of the considered network, i) the first one of our proposed iterative self-positioning algorithms leads to global convergence, where the converged solution is the correct positions of all sensors in the absence of measurement error, but might not be optimum if there exist measurement errors; ii) the second algorithm suffers from local convergence, but once correctly converged the converged solution would be the least squares (LS) solution; iii) the third algorithm, a combined version that switches between the former two algorithms´ iterations efficiently and independently at individual sensors based on locally collected information, globally converges to the LS solution, as long as the measurement errors are sufficiently small such that the converged solution by the first algorithm is well inside the correctly converging area of the second algorithm.