Real-Time Path Planning Based on Harmonic Functions under a Proper Generalized Decomposition-Based Framework

Abstract

This paper presents a real-time global path planning method for mobile robots usingharmonic functions, such as the Poisson equation, based on the Proper Generalized Decomposition(PGD) of these functions. The main property of the proposed technique is that the computationalcost is negligible in real-time, even if the robot is disturbed or the goal is changed. The main idea ofthe method is the off-line generation, for a given environment, of the whole set of paths from anystart and goal configurations of a mobile robot, namely the computational vademecum, derived froma harmonic potential field in order to use it on-line for decision-making purposes. Up until now,the resolution of the Laplace or Poisson equations has been based on traditional numerical techniquesunfeasible for real-time calculation. This drawback has prevented the extensive use of harmonicfunctions in autonomous navigation, despite their powerful properties. The numerical techniquethat reverses this situation is the Proper Generalized Decomposition. To demonstrate and validatethe properties of the PGD-vademecum in a potential-guided path planning framework, both realand simulated implementations have been developed. Simulated scenarios, such as an L-Shapedcorridor and a benchmark bug trap, are used, and a real navigation of a LEGO®MINDSTORMS robotrunning in static environments with variable start and goal configurations is shown. This device hasbeen selected due to its computational and memory-restricted capabilities, and it is a good exampleof how its properties could help the development of social robots.