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.