Least Square Cooperative Localization

Location awareness is becoming essential for emerging wireless applications where most network activities require the location information of network nodes, e.g., routing between nodes in ad-hoc sensor networks, positioning vehicles on the road, or tracking targets in underwater acoustic sensor networks. In particular, cooperation among nodes is highly beneficial for the localization accuracy and coverage in harsh environments. In this paper, we study least square (LS) cooperative localization in the presence of arbitrary non-line-of-sight (NLOS) ranging bias. To develop the network position error bound (PEB), we first derive the Fisher information matrix (FIM) for a general NLOS bias model and show that a Gaussian bias due to NLOS effects is the worst case that produces the extremal FIM, whereas a constant bias or equivalently full line-of-sight is the best situation leading to the largest FIM in a sense of Löwner partial ordering. We then analyze the asymptotic performance, such as uniform convergence, consistency, and efficiency, of LS cooperative localization to quantify the deviations of localization accuracy for LS, squared-range LS, and squared-range weighted LS solutions from the fundamental limit (i.e., Cramér-Rao lower bound) due to their practical tractability. We also propose a simple distributed algorithm for LS cooperative localization by integrating squared-range relaxation into Gaussian variational message passing on the localization network. To account for stochastic natures of node locations and populations, we further characterize the network PEB for Gilbert's disk localization network, where anchors and/or agents are randomly distributed in the network according to point processes.