Time-optimal control of two-link manipulators
Time-optimal maneuvers of rigid Two-Link Manipulators (TLM) are analyzed using Pontryagin's Minimum Principle. The problem is formulated using Optimal Control Theory and ideas developed in the Calculus of Variations. The dynamics of rigid TLM are obtained using the Lagrange equations of motion. The optimal solutions with bang-bang controls are found by solving the corresponding nonlinear Two-Point Boundary Value Problems (TPBVP). Since the problem is very sensitive to the unknown initial costates, a strategy which combines the Forward-Backward Method (FBM) and the Shooting Method (SM) is proposed and used successfully. The FBM is a numerical procedure which finds a non-optimal solution that satisfies the state equations and the initial and the final boundary conditions. This solution is used to obtain approximate locations of the switch times and the initial values of the costates. The initial costates are required to initiate the SM which then is used to find the optimal solution that satisfies the state equations, the costate equations, and all the boundary conditions. Usefulness of the FBM for generating the initial costates such that the SM converges is demonstrated by numerical examples. Linear and nonlinear systems with single, or double controls are considered. Several rigid TLM with different mechanical properties undergoing various maneuvers are discussed in detail. It is shown that for the time-optimal maneuvers of the TLM the number of switches is related to the magnitude of the maneuver and is not constant as suggested in earlier works. In general, for the cases investigated, shorter maneuvers required three switches while longer maneuvers required four switch times. The effects on TLM time-optimal maneuvers of physical parameters such as geometry, link masses and masses at both link tips as well as changes in the limits of control forces are also examined. It is shown that the maneuver time can be further shortened by altering the lengths of the links and the ratio of the controls applied at the two joints. For analysis of the effects of flexibility of the manipulator, the forces obtained from optimal solution of the rigid TLM are applied to the flexible manipulator and the response is simulated by the nonlinear finite element method. The effects of flexibility are measured in terms of the amount of vibration generated and the distance between the tip of the manipulator at the end of the maneuver and the target point. The performance of such forces to control the flexible TLM is found to be satisfactory if the slenderness of the links is sufficiently small.