Wireless network positioning as a convex feasibility problem

Abstract

In this semi-tutorial paper, the positioning problem is formulated as a convex feasibility problem (CFP). To solve the CFP for non-cooperative networks, we consider the well-known projection onto convex sets (POCS) technique and study its properties for positioning. We also study outer-approximation (OA) methods to solve CFP problems. We then show how the POCS estimate can be upper bounded by solving a non-convex optimization problem. Moreover, we introduce two techniques based on OA and POCS to solve the CFP for cooperative networks and obtain two new distributed algorithms. Simulation results show that the proposed algorithms are robust against non-line-of-sight conditions.

Upper bounds on position error of a single location estimate in wireless sensor networks

“…In some scenarios, however, other distributions, e.g., an exponential, uniform, or Laplacian distribution, seem to be more reasonable to describe the model in (5) [22][23][24]. In the previous work [9,13,17], it is assumed that all measurement errors are positive and then it is concluded that a target node can definitely be confined to a bounded convex set, derived from the measurements. In fact, the intersection of a number of balls (with centers a i and radiî d i ), which is nonempty for range measurements with positive errors, definitely contains the target node position.…”

“…Based on the geometric interpretation in (7), we can derive estimators to get one point inside the intersection as an estimate. For example, the POCS or outerapproximation approach, picks one feasible point as an estimate [13,15,17]. Regardless of the type of the estimator, if an estimate of the target node position is available, we can define an upper bound on the position error e with respect to the feasible set B [9].…”

“…The geometric approach developed in [9,13] can be used to confine the target node position to a bounded set if certain conditions are satisfied. In fact, regardless of the accuracy of an estimate, one can obtain a bounded set from the measurements in which the target node position resides.…”

“…Based on this interpretation, one can also design a geometric algorithm that gives an estimate inside the feasible set. For example, the wellknown projection onto convex set (POCS) technique has been successfully applied to the positioning problem on the basis of this geometric interpretation (for details of the POCS approach for positioning see, e.g., [13][14][15][16][17]). In the previous study in [9], it has also been argued that the maximum position error (when distance estimation errors are positive) is difficult to compute because the problem is formulated as a nonconvex problem; however, by means of a relaxation technique, an upper bound on the position error can be obtained efficiently [9].…”

Guaranteed confidence region characterization for source localization using RSS measurements

“…Then, using (9), one may compute several estimates of the higher-order statistic E s i=0 w ei t+i (p) of the prediction residuals, for j = 1, . .…”