Optimal Beacon Placement for Self-Localization Using Three Beacon Bearings

Autonomous vehicles need to localize themselves within the environment in order to effectively perform most tasks. In situations where a Global Navigation Satellite System such as the Global Positioning System cannot be used for localization, other methods are required. One self-localization method is to use signals transmitted by beacons at known locations to determine the relative distance and bearing of the vehicle from the beacons. Estimation performance is influenced by the beacon-vehicle geometry and the investigation into the optimal placement of beacons is of interest to maximize the estimation performance. In this article, a new solution to the optimal beacon placement problem for self-localization of a vehicle on a two-dimensional plane using angle-of-arrival measurements is proposed. The inclusion of heading angle in the estimation problem differentiates this work from angle-of-arrival target localization, making the optimization problem more difficult to solve. First, an expression of the determinant of the Fisher information matrix for an arbitrary number of beacons is provided. Then, a procedure for analytically determining the optimal angular separations for the case of three beacons is presented. The use of three beacons is motivated by practical considerations. Numerical simulations are used to demonstrate the optimality of the proposed method.