Block Iterative Method for Robot Path Planning

Harmonic functions are known to have an advantage as a global potential function in the potential field based approach for robot path planning. However, an immense amount of computations are required as the size of the environment get bigger. This paper conducts an experiment to speed up the computation by solving the harmonic functions with faster solver, i.e. two-point Explicit Group (EG) iterative method. It is found that EG offers faster approach to the path planning problem compared to standard point Jacobi and Gauss-Seidel methods. 1.0 INTRODUCTION Harmonic functions are solutions to Laplace’s equation that offer a complete path planning algorithm, and paths derived from them are generally smooth. When applied to path planning of robots, they have the advantage over simple potential field based approach, as they exhibit no spurious local minima. In [1], Connolly and Gruppen reported that harmonic functions have a number of properties useful in robotic applications. The use of potential functions for robot path planning, as introduced by Khatib [5], views every obstacle to be exerting a repelling force on an end effector, while the goal exerts an attractive force. Koditschek [6], using geometrical arguments, showed that, at least in certain types of domains, there exists potential functions which can guide the effector from almost any point to a given point. These potential fields for path planning, however, suffer from the spontaneous creation of local minima. Connolly et al. [7] and Akishita et al. [8] independently developed a global method using solutions to Laplace’s equations for path planning to generate a smooth, collisionfree path. The potential field is computed in a global manner, i.e. over the entire region, and the harmonic solutions to Laplace’s equation are used to find the path lines for a robot to move from the start point to the goal point. Obstacles are considered as current sources and the goal is considered to be the sink, with the lowest assigned potential value. This amounts to using Dirichlet boundary conditions. Then, following the current lines, i.e. performing the steepest descent on the potential field, a succession of points with lower potential values leading to the point with least potential (goal) is found out. It is observed by Connolly et al. [4] that this process guarantees a path to the goal without encountering local minima and successfully avoiding any obstacle. Several other methods are also proposed for solving path planning problem. In [12], an algorithm that employs distance transform method is reported. Jan et al. [13] conducted researches on utilizing geometry maze routing algorithm. The work by Bhattacharya and Gavrilova [14] uses Voronoi Diagram to solve path planning problem.