A Novel Fixed-Time Stability Theorem and Its Application to Nonlinear Manipulator Trajectory Tracking Control

An adaptive fixed-time robust trajectory tracking control method is developed for the control problem of nonlinear robotic manipulator systems under friction and external disturbances. First, a novel fast fixed-time stable system is presented by combining the fixed-time stability theory. Second, based on this fixed-time stable system, a new fixed-time sliding mode (FTSM) surface and reaching law are proposed to achieve accurate trajectory tracking accuracy. At the same time, a faster convergence velocity can be achieved no matter whether the system state is near or far from the origin. The theoretical analysis gives the upper bound of the convergence time of the sliding mode (SM) surface. Then, the adaptive law is designed to overcome the problem of the unknown upper bound of the composite perturbation, and a rigorous theory analyzes the stability of the control method to prove that the system state convergence time is independent of the initial conditions of the system and can reach near the zero point within a fixed time upper bound. Finally, the proposed control scheme is simulated and compared with different control schemes to verify its effectiveness.