  This paper studies the possibility of upper bounding the position error of an
estimate 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 are positively biased, e.g., in non-line-of-sight
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 worst-case position
error can be defined 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, we propose to get the maximum length
of the feasible set as the worst-case 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 indicate that the proposed bounds are reasonably
tight in many situations.
