A Semi-Lagrangian Approach for Time and Energy Path Planning Optimization in Static Flow Fields

In this context, new challenges arise when robotic systems address not just a singular objective but multiple and often conflicting goals. These objectives can range from minimizing travel time and energy consumption simultaneously to optimizing factors like safety and resource allocation [2]. In single-objective approaches, the most commonly prioritized factors are typically the path's length [3, 4] and travel time [5, 6]. However, by incorporating other additional attributes, such as path safety/vulnerability and smoothness [7, 8], we can significantly improve both the quality and the applicability of results. Regarding the more general class of routing problems, where a sequence of visits is demanded, a multi-objective variant of the Orienteering Problem (OP) was proposed in [9], where the goal was to maximize the cumulative reward obtained while concurrently minimizing the exposure to sensors deployed in the environment. Furthermore, it is also imperative to acknowledge that, in numerous domains, environmental dynamics substantially influence the trajectories and behaviors of the vehicles. This is particularly evident in fields such as aerospace, where factors like air density, wind patterns, and gravitational forces intricately shape the aircraft flight paths [10].