Circumnavigation Using Distance Measurements Under Slow Drift

Consider an agent A at an unknown location, undergoing sufficiently slow drift, and a mobile agent B that must move to the vicinity of and then circumnavigate A at a prescribed distance from A. In doing so, B can only measure its distance from A, and knows its own position in some reference frame. This paper considers this problem, which has applications to surveillance and orbit maintenance. In many of these applications it is difficult for B to directly sense the location of A, e.g. when all that B can sense is the intensity of a signal emitted by A. This intensity does, however provide a measure of the distance. We propose a nonlinear periodic continuous time control law that achieves the objective using this distance measurement. Fundamentally, a) B must exploit its motion to estimate the location of A, and b) use its best instantaneous estimate of where A resides, to move itself to achieve the circumnavigation objective. For a) we use an open loop algorithm formulated by us in an earlier paper. The key challenge tackled in this paper is to design a control law that closes the loop by marrrying the two goals. As long as the initial estimate of the source location is not coincident with the intial position of B, the algorithm is guaranteed to be exponentially convergent when A is stationary. Under the same condition, we establish that when A drifts with a sufficiently small, unknown velocity, B globally achieves its circumnavigation objective, to within a margin proportional to the drift velocity.