In this article, a predefined time convergence adaptive tracking control scheme is designed for a class of uncertain robotic manipulators with input saturation. First, a novel auxiliary dynamic system is proposed to handle the influence of input saturation. Radial basis function neural networks are used to approximate the uncertainty of the closed-loop system and the neural adaptive law is designed by using the given time constant so that the neural networks have a fast convergence rate. The adaptive tracking controller is constructed by utilizing a nonsingular terminal sliding mode surface. Different from the finite-time and the fixed-time sliding mode control methods where the upper bound of the convergence time is related to system parameters, the convergence time upper bound of the proposed sliding mode controller is a given constant. Finally, numerical simulations are performed to illustrate that the proposed control scheme possesses the advantages of fast convergence rate and input saturation elimination.