A discrete trajectory planner is described which has been shown to be capable of planning optimal trajectories for a six-degree-of-freedom robot with minimum time/energy objective functions and realistic constraints on the joint velocities, joint torques, joint jerks, and hand velocity. It is also flexible enough to handle other objective functions and constraints, provided they can be satisfactorily discretized. An alternative formulation of the discrete trajectory planner is also provided for the efficient generation of optimal trajectories when equal time intervals are required. Once an optimal trajectory is planned offline with the discrete trajectory planner, the open-loop solution can be fed to the online feedback controller for trajectory tracking. Computation time taken by the discrete trajectory planner is acceptable for offline planning purposes. Numerical examples are presented for the Stanford manipulator using all six degrees of freedom.>

A minimum time trajectory planner is proposed for a manipulator arm. A totally discrete approach is adopted, in contrast to other models which use continuous-time but resort to discretization in the computation. The Neuman and Tourassis discrete-dynamic robot model is used to model the robot dynamics. The proposed trajectory planner includes joint-torque constraints to fully utilize the joint actuators. Realistic constraints such as the joint-jerk and joint-velocity constraints are incorporated into the model. The nonlinear optimization problem associated with the planner is partially linearized, which enables the iterative method of approximate programming to be used in solving the problem. Numerical examples for a two-link revolute arm are presented to demonstrate the use of the proposed trajectory planner. It is numerically verified that the convergence of the iterative algorithm is quadratic, and the trajectory planner therefore is computationally efficient. The use of a near-minimum time-cost function is also shown to yield a solution close to that obtained with the true minimum time-cost function.>

A numerical method is introduced for solution of the Dirichlet problem within a circle. The data prescribed on the original boundary is used to compute the solution at mesh points on a concentric circle of slightly smaller radius, thereby defining another boundary value problem over the smaller circle. The solution for the entire region is obtained by successive contractions of the boundary in this way. Appropriate approximating relations are presented and their stability is discussed.