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The harmonic functions have proved to be a powerful technique for motion planning in a known environment. They have two important properties: given an initial point and an objective in a connected domain, a unique path exists between those points. This path is the maximum gradient path of the harmonic function that begins in the initial point and ends in the goal point. The second property is that the harmonic function cannot have local minima in the interior of the domain (the objective point is considered as a border). This paper proposes a new method to solve Laplace's equation. The harmonic function solution with mixed boundary conditions provides paths that verify the smoothness and safety considerations required for mobile robot path planning. The proposed approach uses the Finite Elements Method to solve Laplace's equation, and this allows us to deal with complicated shapes of obstacles and walls. Mixed boundary conditions are applied to the harmonic function to improve the quality of the trajectories. In this way, the trajectories are smooth, avoiding the corners of walls and obstacles, and the potential slope is not too small, avoiding the difficulty of the numerical calculus of the trajectory. Results show that this method is able to deal with moving obstacles, and even for non-holonomic vehicles. The proposed method can be generalized to 3D or more dimensions and it can be used to move robot manipulators.
motion planning; harmonic functions; finite elements