Globally Time-Optimal Path Planning for Unmanned Underwater Vehicles in Three-Dimensional Current Fields Using Hamilton-Jacobi Partial Differential Equations
Jeremy Brandman, Colin C. Olson · 2023
A new approach to underwater vehicle path planning, based on solving the Hamilton-Jacobi partial differential equation characterizing a vehicle's minimum travel time, is presented. The travel-time equation is derived as a natural consequence of the dynamic programming principle. In addition, the underlying Hamilton-Jacobi equation is shown to govern the geometric motion by which the reachability front evolves, which results in a straightforward derivation of the level set approach [14]. The viscosity solution of the travel-time equation is computed using the fast sweeping method with the Lax-Friedrichs flux; the resulting algorithm is computationally efficient and straightforward to implement. The method is validated through several examples for which optimal trajectories are derived using the calculus of variations. Additional examples in two and three spatial dimensions are also included.