Minimum time heading control of underpowered vehicles in time-varying ocean currents

We present a numerical method for minimum time heading control of fixed speed AUVs (autonomous underwater vehicles) such as gliders in known, spatially complex, 2D, time-varying flow fields. This problem is difficult because locally optimal trajectories abound and, worse, currents stronger than the vehicle can push it far off course. Nevertheless, globally optimal trajectories may be obtained (where they exist) by solving a dynamic HJB (Hamilton Jacobi Bellman) partial differential equation for the time-varying optimal time-to-go function and the associated optimal feedback control law; the local optima and strong currents simply make the control law and the time-to-go function, respectively, discontinuous. In prior work, we found solutions via a variant of the “extremal field” method-essentially the method of characteristics, and equivalent to tracking a 2D “controllability front” backward-in-time from the target set, a line in space-time. In the present work, we exploit a special property of minimum time control to obtain the same globally optimal trajectories, albeit in open loop form, by tracking a 1D “reachability front” forward-in-time from the initial position. The method is further improved by a trimming procedure for locally optimal trajectories. It is tested on a numerically defined flow field from a model of the Adriatic Sea.