Maximum likelihood localization: When does it fail?

Maximum likelihood is a criterion often used to derive localization algorithms. In particular, in this paper we focus on a distance-based algorithm for the localization of nodes in static wireless networks. Assuming that Ultra Wide Band (UWB) signals are used for inter-node communications, we investigate the ill-conditioning of the Two-Stage Maximum-Likelihood (TSML) Time of Arrival (ToA) localization algorithm as the Anchor Nodes (ANs) positions change. We analytically derive novel lower and upper bounds for the localization error and we evaluate them in some localization scenarios as functions of the ANs’ positions. We show that particular ANs’ configurations intrinsically lead to ill-conditioning of the localization problem, making the TSML-ToA inapplicable. For comparison purposes, we also show, through some examples, that a Particle Swarm Optimization (PSO)-based algorithm guarantees accurate positioning also when the localization problem embedded in the TSML-ToA algorithm is ill-conditioned.