Numerical streamline path planning based on log-space harmonic potential function: A simulation study

This paper presents a simulation validation of streamline-based path planning by log-space harmonic potential function method proposed in [1] for the numerical solution of a boundary value problem for Laplace equation associated with Dirichlet or Neumann boundary conditions in a closed region. Log-space iterative method resolves the precision problem of numerically computing the gradient of harmonic potential, or the streamline path to be followed. We begin with properties of harmonic functions and then summarize three basic numerical methods of Jacobi iteration, Gauss-Seidel iteration and SOR iteration used to solve the linear equations resulting from finite difference approximation to Laplace equation. For real-time mobile robot navigation, the iterative method is an anytime smooth path planning method that allows parallel implementation on GPU and incorporations of acceleration techniques and ordering/reordering of grids. A specific scheme implemented in this paper is a parallelized log-space Jacobi method to provide anytime and numerical-precision valid gradient of harmonic potential field for smooth path generation. Concrete numerical examples of smooth path planning in an experiment environment are presented to compare the paths generated via Laplace equation with Dirichlet or Neumann boundary conditions using log-space Jacobi method and the path generated by analytical potentials to show the advantages of iteration methods.