This paper solves the online obstacle avoidance problem using the Hamilton-Jacobi-Bellman (HJB) theory. Formulating the shortest path problem as a time optimal control problem, the shortest paths are generated by following the negative gradient of the return function, which is the solution of the HJB equation. To account for multiple obstacles, we avoid obstacles optimally one at a time. This is equivalent to following the pseudo-return function, which is an approximation of the true return function for the multi-obstacle problem. Paths generated by this method are near-optimal and guaranteed to reach the goal, at which the pseudo-return function is shown to have a unique minimum. The proposed method is computationally very efficient, and applicable for online applications. Examples for circular obstacles demonstrate the advantages of the proposed approach over traditional path planning methods.

This paper presents a method for generating near-time optimal trajectories in cluttered environments for manipulators with invariant inertia matrices. For one obstacle, the method generates the time-optimal trajectory by minimizing the time-derivative of the return (cost) function for this problem, satisfying the Hamilton-Jacobi-Bellman (HJB) equation. For multiple obstacles, the trajectory is generated using the pseudo return function, which is an approximation of the return function for the multi-obstacle problem. The pseudo return function avoids one obstacle at a time, producing near-optimal trajectories that are guaranteed to avoid the obstacles and satisfy the actuator constraints. An example with circular obstacles demonstrates close correlation between the near-optimal and optimal paths, requiring computational efforts that are suitable for on-line implementations.

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.>

Design methods of robotic manipulators to select the link lengths and actuator sizes for minimum-time motions along specified paths are presented. An exact method is based on a parameter optimization, using the motion time along the path as the cost function. This method serves as a benchmark for a more efficient approximation which selects system parameters so as to maximize acceleration along the path. Examples of the design of a two-link manipulator are presented, demonstrating close correlation between the exact and the approximate methods.>

A potential based method for generating near time optimal paths of manipulators moving in the presence of obstacles is presented. The potential function consists of the navigation function for obstacle avoidance and acceleration potentials that force the path to maximize the acceleration and deceleration near the end points. The parameters of the potential functions are selected to guarantee a unique minimum that coincides with the destination point. The path of minimum potential from the initial point avoids obstacles as well as maximizes the initial acceleration and the final deceleration. The method is demonstrated for a two link planar manipulator, generating paths with optimal motion times substantially lower than those generated by the navigation function alone.>