Optimal obstacle avoidance based on the Hamilton-Jacobi-Bellman equation

This paper presents a method for generating shortest paths in cluttered environments, based on the Hamilton-Jacobi-Bellman (HJB) equation. Formulating the shortest obstacle avoidance problem as a time optimal control problem, the shortest paths are generated by following the negative gradient of the return function, which satisfies the HJB equation. A method to generate near-optimal paths is also presented, based on a psuedo return function. Paths generated by this method are guaranteed to reach the goal, at which the psuedo return function is shown to have a unique minimum. The computation required to generate the near-optimal paths is substantially lower than those of traditional potential field methods, making it applicable to on-line obstacle avoidance. Examples with circular obstacles demonstrate close correlation between the near-optimal and optimal paths, and the advantages of the proposed approach over traditional potential field methods. >

[1]  S. Dreyfus.  Dynamic Programming and the Calculus of Variations , 1960 .

[2]  N. Nahi.  On design of time optimal systems via the second method of Lyapunov , 1964 .

[3]  E B Lee,et al.  Foundations of optimal control theory , 1967 .

[4]  Lamberto Cesari,et al.  Optimization-Theory And Applications , 1983 .

[5]  R. A. Jarvis,et al.  Collision-free trajectory planning using distance transforms , 1985 .

[6]  Oussama Khatib,et al.  Real-Time Obstacle Avoidance for Manipulators and Mobile Robots , 1986 .

[7]  P. Khosla,et al.  Artificial potentials with elliptical isopotential contours for obstacle avoidance , 1987, 26th IEEE Conference on Decision and Control.

[8]  Pradeep K. Khosla,et al.  Superquadric artificial potentials for obstacle avoidance and approach , 1988, Proceedings. 1988 IEEE International Conference on Robotics and Automation.

[9]  J. Brian Burns,et al.  Path planning using Laplace's equation , 1990, Proceedings., IEEE International Conference on Robotics and Automation.

[10]  D. Koditschek,et al.  Robot navigation functions on manifolds with boundary , 1990 .

[11]  Jean-Claude Latombe,et al.  Numerical potential field techniques for robot path planning , 1991, Fifth International Conference on Advanced Robotics 'Robots in Unstructured Environments.

[12]  Daniel E. Koditschek,et al.  Exact robot navigation using artificial potential functions , 1992, IEEE Trans. Robotics Autom..

[13]  Pradeep K. Khosla,et al.  Real-time obstacle avoidance using harmonic potential functions , 1991, IEEE Trans. Robotics Autom..

[14]  Paul M. Griffin,et al.  Path planning for a mobile robot , 1992, IEEE Trans. Syst. Man Cybern..

[15]  S. Arimoto,et al.  Path Planning Using a Tangent Graph for Mobile Robots Among Polygonal and Curved Obstacles , 1992 .

[16]  Z. Shiller,et al.  On time-optimal feedback control , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[17]  Zvi Shiller,et al.  Time-optimal obstacle avoidance , 1995, Proceedings of 1995 IEEE International Conference on Robotics and Automation.

[18]  Z. Shiller,et al.  Time-optimal obstacle avoidance for robotic manipulators , 1995 .