Joint map estimation and localization using distance measurements to landmarks with unknown location

In this paper the problem of joint localization and map estimation is studied. The objective is to determine the location of a set of fixed nodes, e.g. access-points or base-stations in a wireless network using only distance measurements acquired by a node moving in the vicinity of the fixed nodes. It is assumed, that no prior information about locations of any of the nodes (fixed or mobile) is available, i.e. no anchor nodes exist. It is shown that, under some mild conditions on the distribution and the number of nodes, the map of the fixed nodes as well as the locations of the mobile node(s) in this map can be identified unambiguously from ranging measurements. A closed form algorithm is derived to estimate the map. Furthermore, the performance of the derived estimator is compared in Monte-Carlo simulations with (i) the Cramér-Rao lower bound (CRLB) and (ii) with the performance of an iterative maximum-likelihood estimator. The derived closed form estimator for the node locations is consistent, but doesn´t achieve the CRLB. The ML-estimator achieves the bound if the number of fixed nodes is at least four, and the distance measurements have been obtained at a minimum of six locations.