U. A. Khan, S. Kar, and J. M. F. Moura, "Distributed sensor localization in random environments using minimal number of anchor nodes," IEEE Transactions on Signal Processing, vol. 57, no. 5, pp. 2000-2016, May 2009. Nominated for IEEE Signal Processing Society, Young Author Best Paper Award.

## Abstract

The paper develops DILOC, a \emph{distributed}, \emph{iterative} algorithm to locate~$M$ sensors (with unknown locations) in~$\mathbb{R}^m, m\geq 1$, with respect to a minimal number of~$m+1$ anchors with known locations. The sensors and anchors, nodes in the network, exchange data with their neighbors only; no centralized data processing or communication occurs, nor is there a centralized fusion center to compute the sensors' locations. DILOC uses the barycentric coordinates of a node with respect to its neighbors; these coordinates are computed using the Cayley-Menger determinants, i.e., the determinants of matrices of inter-node distances. We show convergence of DILOC by associating with it an absorbing Markov chain whose absorbing states are the states of the anchors. We introduce a stochastic approximation version extending DILOC to random environments, i.e., when the communications among nodes is noisy, the communication links among neighbors may fail at random times, and the inter-nodes distances are determined with errors. We show a.s.~convergence of the modified DILOC and characterize the error between the true values of the sensors' locations and their final estimates given by DILOC. Numerical studies illustrate DILOC under a variety of deterministic and random operating conditions.