A semi-Lagrangian approach for time and energy path planning optimization in static flow fields

Efficient path planning for autonomous mobile robots is a critical problem across numerous domains, where optimizing both time and energy consumption is paramount. This paper introduces a novel methodology that considers the dynamic influence of an environmental flow field and geometric constraints, including obstacles and forbidden zones, enriching the complexity of the planning problem. Here, we formulate it as a multi-objective optimal control problem, and propose a novel transformation called Harmonic Transformation, applying a semi-Lagrangian scheme to solve it. The set of Pareto efficient solutions is obtained considering two distinct approaches: ($i$) a deterministic method referred to as Concurrent Policy Iteration (CPI); and ($ii$) an evolutionary-based one, called Multi-objective Evolutionary Policy Iteration (MEPI). Both methods were designed to make use of the proposed Harmonic Transformation. Through an extensive analysis of these approaches, comparing them with the state-of-the-art literature, we demonstrate their efficacy in finding optimized paths. Generally speaking, the Pareto Set of solutions found in our experiments indicates that the CPI demonstrated better performance in finding solutions close to the time-optimal one, whereas the MEPI was most successful in finding solutions close to the energy-optimal solution.