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 considers geometric constraints, including obstacles and forbidden zones, enriching the complexity of the planning problem. We formulate it as a multi-objective optimal control problem, propose a novel transformation called Harmonic Transformation, and apply a semi-Lagrangian scheme to solve it. The set of Pareto efficient solutions is obtained considering two distinct approaches: a deterministic method and an evolutionary-based one, both of which are designed to make use of the proposed Harmonic Transformation. Through an extensive analysis of these approaches, we demonstrate their efficacy in finding optimized paths.