This paper addresses an improved adaptive fast nonsingular terminal sliding mode control(IAFNTSMC) method for a class of nonlinear systems with complicated disturbances and uncertainties under the conditions of time-varying output constraints and quantized input signal. The robust adaptive method is jointly used to deal with the disturbances and uncertainties, and the controller is designed by combining the bakcstepping control strategy with the terminal sliding mode strategy. A novel time-varying barrier Lyapunov function is constructed to time-varyingly constrain the output error of the system during operation. Also, a novel sliding mode surface construction scheme is proposed to improve the error convergence rate of the closed-loop system. The proposed method guarantees that all signals in the closed-loop systems are bounded, and the output tracking error can converge to the given boundary constraints. Finally, a numerical example is given to illustrate the effectiveness of the proposed scheme.