Optimal sensor placement for agent localization

Abstract

In this article we consider deploying a sensor network to help an agent navigate in an area. In particular the agent uses range measurements to the sensors to localize itself. We wish to place the sensors in order to provide optimal localization accuracy to the agent. We begin by considering the problem of placing sensors in order to optimally localize the agent at a single location. The Position Error Bound (PEB), a lower bound on the localization accuracy, is used to measure the quality of sensor configurations. We then present RELOCATE, an iterative algorithm that places the sensors so as to minimize the PEB at that point. When the range measurements are unbiased and have constant variances, we introduce a coordinate transform that allows us to obtain a closed-form solution to minimizing the PEB along one coordinate. We also prove that RELOCATE converges to the global minimum, and we compute the approximate expected rate of convergence of the algorithm. We then apply RELOCATE to the more complex case where the variance of the range measurements depends on the sensors location and where those measurements can be biased. We finally apply RELOCATE to the case where the PEB must be minimized not at a single point, but at multiple locations, so that good localization accuracy is ensured as the agent moves through the area. We show that, compared to Simulated Annealing, the algorithm yields better results faster on these more realistic scenarios. We also show that by optimally placing the sensors, significant savings in terms of number of sensors used can be achieved. Finally we illustrate that the PEB is not only a convenient theoretical lower bound, but that it can actually be closely approximated by a maximum likelihood estimator.