Speed regulation in steering-based source seeking

The simplest strategy for extremum seeking-based source localization, for sources with unknown spatial distributions and nonholonomic unicycle vehicles without position measurement, employs a constant positive forward speed. Steering of the vehicle in the plane is performed using only the variation of the angular velocity. While keeping the forward speed constant is a reasonable strategy motivated by implementation with aerial vehicles, it leads to complexities in the asymptotic behavior of the vehicle, since the vehicle cannot settle—at best it can converge to a small-size attractor around the source. In this paper we regulate the forward velocity, with the intent of bringing the vehicle to a stop, or as close to a stop as possible. The vehicle speed is controlled using simple derivative-like feedback of the sensor measurement (the derivative is approximated with a washout filter) to which a speed bias parameter V_c is added. The angular velocity is tuned using standard extremum seeking. We prove two results. For V_c in a certain range around zero, we show that the vehicle converges to a ring around the source and on average the limit of the vehicle’s heading is either directly away or towards the source. For other values of V_c>0, the vehicle converges to a ring around the source and it revolves around the source. Interestingly, the average heading of this revolution around the source is more outward than inward—this is possible because the vehicle’s speed is not constant, it is lower during the outward steering intervals and higher during the inward steering intervals. The theoretical results are illustrated with simulations.