Time-optimal path planning in dynamic flows using level set equations: theory and schemes

We develop an accurate partial differential equation-based methodology that predicts the time-optimal paths of autonomous vehicles navigating in any continuous, strong, and dynamic ocean currents, obviating the need for heuristics. The goal is to predict a sequence of steering directions so that vehicles can best utilize or avoid currents to minimize their travel time. Inspired by the level set method, we derive and demonstrate that a modified level set equation governs the time-optimal path in any continuous flow. We show that our algorithm is computationally efficient and apply it to a number of experiments. First, we validate our approach through a simple benchmark application in a Rankine vortex flow for which an analytical solution is available. Next, we apply our methodology to more complex, simulated flow fields such as unsteady double-gyre flows driven by wind stress and flows behind a circular island. These examples show that time-optimal paths for multiple vehicles can be planned even in the presence of complex flows in domains with obstacles. Finally, we present and support through illustrations several remarks that describe specific features of our methodology.