Assessing range accuracy for bearings-only geolocation using optimal logarithmic spiral sensor path trajectories

In this paper, we develop a procedure for evaluating the performance of a single moving sensor system to estimate the range to a stationary emitter based on the discrete-time collection of bearing measurements along the trajectory travelled. We describe a numerical procedure to calculate the range p.d.f. from a chosen trajectory assuming a constant quality for the bearing measurements and use this procedure to evaluate the range root mean-square error as a function of the distance travelled. The logarithmic spiral family of trajectories is of particular interest, both from the standpoint of optimal control where such paths are derived, and from experimental biology, where large birds of prey are observed to travel in such paths in their search for food. Our performance analysis of these scenarios indicates why pitch angles less than 45° are to be preferred when a balanced range estimation performance throughout the trajectory is desired.