The paper develops DILOC, a emph{distributive}, emph{iterative} algorithm that locates M sensors in $mathbb{R}^m, mgeq 1$, with respect to a minimal number of m+1 anchors with known locations. The sensors exchange data with their neighbors only; no centralized data processing or communication occurs, nor is there centralized knowledge about the sensors locations. DILOC uses the barycentric coordinates of a sensor with respect to its neighbors that are computed using the Cayley-Menger determinants. These are the determinants of matrices of inter-sensor distances. We show convergence of DILOC by associating with it an absorbing Markov chain whose absorbing states are the anchors. We introduce a stochastic approximation version extending DILOC to random environments when the knowledge about the intercommunications among sensors and the inter-sensor distances are noisy, and the communication links among neighbors fail at random times. We show a.s. convergence of the modified DILOC and characterize the error between the final estimates and the true values of the sensors locations. Numerical studies illustrate DILOC under a variety of deterministic and random operating conditions.
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