Efficient Guidance in finite time flow fields

We study path planning for small vehicles in strong, spatially complex, time-varying flow fields. Of particular interest is how optimal trajectories relate to flow structures and might be approximated heuristically. Toward this end, we focus on cases where the only concern is the position at some fixed final time, and the control effort. This allows a natural coordinate transformation for the optimal control problem in terms of the so-called flow map. In the transformed coordinates the flow is zero, but the control input (the velocity of the vehicle relative to the flow) is multiplied (and, in more than 1 dimension, rotated) by a time-varying matrix-the Jacobian of the flow map. The definition of what we call the pulled back end cost function provides additional insight and leads to a simple but effective “Lagrangian heuristic control” law, which, in 1d at least, reduces to the optimal control for the case of linear time-invariant flows and quadratic end costs. We demonstrate this control and compare it to the optimal control by solving the associated Hamiltonian Jacobi Bellman (HJB) equation backwards in time with an adaptive 1d grid.