• Aleksandr L. Kazakov, Irkutsk National Research Technical University (Irkutsk, Russia)
• Anna A. Lempert, Irkutsk National Research Technical University (Irkutsk, Russia)
• Viet T. Tran, Irkutsk National Research Technical University (Irkutsk, Russia)

This paper addresses the problem of constructing time-optimal paths for a moving vehicle, specifically an autonomous underwater vehicle operating in an aquatic environment. The main difficulties stem from spatially non-uniform currents and the presence of both static and moving obstacles. We propose an approach based on the optical-geometric analogy and the Fermat–Huygens principle. The problem is formulated as a generalized eikonal equation that describes the propagation of a wavefront in a medium with a prescribed velocity vector field. This formulation reduces the original variational problem to a partial differential equation, eliminating the need for an exhaustive search over candidate trajectories. The arrival time field is obtained as the solution to a boundary value problem, and the optimal path is recovered by moving against the gradient of this field. We develop two algorithms for numerical implementation. The first, based on the fast marching method, calculates the minimal time field from the start point to all nodes of a computational grid. The second algorithm reconstructs the optimal trajectory from this field and extracts the section corresponding to a specified time horizon. We perform a series of numerical experiments on four test scenarios of increasing complexity: a uniform flow, two opposing flows, a flow with a vortex and static obstacles, and a scenario that combines moving obstacles with a vortex. In all cases, the algorithm constructs time-optimal paths successfully while ensuring safe obstacle avoidance. The scenario with dynamic obstacles demonstrated the ability to adjust the route dynamically. We compared the proposed approach with the Salp Swarm Algorithm (SSA). The results show that our algorithm finds paths with shorter travel times, even when their geometrical length exceeds that of the routes produced by SSA. Moreover, the computation time is sufficiently low to allow real‐time path planning.

routing problem, dynamic environment, optical-geometric approach, eikonal equation, fast marching method

2026-06-05