Abstract
This paper studies upper bounds on the position error for a single estimate of an unknown target node position based on distance estimates in wireless sensor networks. In this study, we investigate a number of approaches to confine the target node position to bounded sets for different scenarios. Firstly, if at least one distance estimate error is positive, we derive a simple, but potentially loose upper bound, which is always valid. In addition assuming that the probability density of measurement noise is nonzero for positive values and a sufficiently large number of distance estimates are available, we propose an upper bound, which is valid with high probability. Secondly, if a reasonable lower bound on negative measurement errors is known a priori, we manipulate the distance estimates to obtain a new set with positive measurement errors. In general, we formulate bounds as nonconvex optimization problems. To solve the problems, we employ a relaxation technique and obtain semidefinite programs. We also propose a simple approach to find the bounds in closed forms. Simulation results show reasonable tightness for different bounds in various situations.
Highlights
Position information is often one of the vital requirements for wireless sensor networks (WSNs), especially for location-aware services [1]
The worst-case position error or a reasonable upper bound on the position error can be used in trafficsafety applications to decrease the number of collisions between vehicles [9]
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]
Summary
Position information is often one of the vital requirements for wireless sensor networks (WSNs), especially for location-aware services [1]. By defining a confidence region for an estimate [10], a target node position can be confined to a set, e.g., an ellipsoid, with a certain probability, say, 95% of the cases This approach has been employed in characterizing GPS position errors [11] or in studying a position algorithm [12]. For range-based positioning, it can be concluded that the target node position can be found in a bounded convex set (a feasible set) if the distance measurement error is positive This set, is obtained from the intersection of a number of balls derived from the measurements. Formulating an upper bound that holds with high probability if there is a sufficiently large number of distance estimate between each reference node and the target node and if the distance estimation errors are not always negative.
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