Source Seeking With Nonholonomic Unicycle Without Position Measurement---Part I: Tuning of Forward Velocity

We consider the problem of seeking the source of a scalar signal using an autonomous vehicle modeled as the nonholonomic unicycle and equipped with a sensor of that scalar signal but not possessing the capability to sense either the position of the source nor its own position. We assume that the signal field is the strongest at the source and decays away from it. The functional form of the field is not available to our vehicle. We employ extremum seeking to estimate the gradient of the field in real time and steer the vehicle towards the point where the gradient is zero (the maximum of the field, i.e., the location of the source). We have developed two control strategies - one which keeps the angular velocity at a constant nonzero value and tunes the forward velocity, and the other which keeps the forward velocity at a constant positive value and tunes the angular velocity. In this paper we present the former method, which is implementable with mobile robots and some underwater vehicles. In a future companion paper we will present the latter method, which is more suitable for aerial vehicles. Our approach in this paper employs periodic forward-backward movement of the unicycle, with a tunable bias term, which is appropriately combined with extremum seeking to produce a net effect of "drifting" towards the source. In addition to simulation results we present a local convergence proof via averaging, which exhibits a delicate periodic structure with two sinusoids of different frequencies - one related to the angular velocity of the unicycle and the other related to the probing frequency of extremum seeking