A Comparative Study of Dirichlet and Neumann Conditions for Path Planning through Harmonic Functions

by

Madhuri Karnik, Bhaskar Dasgupta and Vinayak Eswaran

Abstract:

Harmonic functions, by virtue of their extrema appearing only on the domain boundary, are known to have an advantage as a global potential function in the potential field based approach for robot path planning. However, a wide range of possibilities exist for the global harmonic function for a particular situation, depending on the boundary conditions. This paper conducts a comparison of two major classes of boundary conditions, namely Dirichlet and Neumann conditions, and attempts to discover their relative merits and demerits. It is found that the Neumann conditions offer a surer and faster approach to the path planning problem, though suffering from the disadvantage of occasional tendency of the planned path to graze along the domain boundary. This minor disadvantage, however, can be remedied by a two-stage strategy, in which the solution with the Neumann condition is used to {\em generate} the Dirichlet boundary conditions for the second stage of solution.

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