Energy–time optimal path planning in dynamic flows: Theory and schemes

Abstract

We obtain, solve, and verify fundamental differential equations for energy–time path planning in dynamic flows. The equations govern the energy–time reachable sets, optimal paths, and optimal controls for autonomous vehicles navigating to any destination in known dynamic environments, minimizing both energy usage and travel time. Based on Hamilton–Jacobi theory for reachability and the level set method, the resulting methodology computes the Pareto optimal solutions to the multi-objective path planning problem, numerically solving the exact equations governing the evolution of reachability fronts and optimal paths in the augmented energy and physical-space domain. Our approach is applicable to path planning in various dynamic flow environments and energy types. We first validate the methodology through a benchmark case of crossing a steady jet for which we compare our results to semi-analytical optimal energy–time solutions. We then consider unsteady flow environments and solve for energy–time optimal missions in a quasi-geostrophic double-gyre flow field. Results show that our theory and schemes can provide all the energy–time optimal solutions and that these solutions can be strongly influenced by unsteady flow conditions.