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Energy-Time Optimal Path Planning in Dynamic Flows: Theory and Schemes

Doshi, M.M., M.S. Bhabra, and P.F.J. Lermusiaux, 2023. Energy-Time Optimal Path Planning in Dynamic Flows: Theory and Schemes. Computer Methods in Applied Mechanics and Engineering 405: 115865. doi:10.1016/j.cma.2022.115865

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.