On geometric upper bounds for positioning algorithms in wireless sensor networks

•This paper studies a technique to upper bound a single position estimate in wireless positioning. In this paper, it is assumed that the measurement noise tends to be positive; hence, it is concluded that the target node can be confined to a closed bounded set. Consequently, geometric upper bound can be defined with respect to the set.•The positive noise assumption is fullfilled in some scenarios, as some practical measurements confirm, although in general it may not be a valid assumption. If the measurement error is negative, one can manually manipulate the measurement to obtain a modified measurement with positive error.•The bound is formulated as nonconvex optimization problems and some relaxation techniques are used to approximately solve the problem.

This paper studies the possibility of upper bounding the position error for range-based positioning algorithms in wireless sensor networks. In this study, we argue that in certain situations when the measured distances between sensor nodes have positive errors, e.g., in non-line-of-sight (NLOS) conditions, the target node is confined to a closed bounded convex set (a feasible set) which can be derived from the measurements. Then, we formulate two classes of geometric upper bounds with respect to the feasible set. If an estimate is available, either feasible or infeasible, the position error can be upper bounded as the maximum distance between the estimate and any point in the feasible set (the first bound). Alternatively, if an estimate given by a positioning algorithm is always feasible, the maximum length of the feasible set is an upper bound on position error (the second bound). These bounds are formulated as nonconvex optimization problems. To progress, we relax the nonconvex problems and obtain convex problems, which can be efficiently solved. Simulation results show that the proposed bounds are reasonably tight in many situations, especially for NLOS conditions.