Abstract: Aiming at the overshoot quantitative constraint problem of quadrotor UAV with time-varying disturbance, a neural network adaptive control method with preset performance based on a new Lyapunov function with time-varying obstacles was proposed. Firstly, the overshoot constraint problem of the quadrotor UAV was analyzed, and a new time-varying barrier Lyapunov function was proposed for the overshoot constraint problem. The new time-varying barrier Lyapunov function can impose continuous asymmetric constraints on the system, so as to constrain the behavior of the system more accurately and enrich the setting form of the preset performance boundary. Secondly, a new tubular preset performance boundary function is designed to impose quantitative constraints on the overshoot of the system output and meet the steady-state performance requirements. On this basis, the feedback control law and the neural network adaptation law are designed by the backstepping method to ensure the performance constraints of the system. Finally, based on the Lyapunov function stability theory, the uniform ultimate boundedness of all closed-loop signals is proved, and the effectiveness of the proposed method is verified by numerical simulation. Simulation results show that the proposed control law can achieve quantitative constraints on the overshoot of the quadrotor UAV. Considering the realistic conditions and safety factors, such as when the UAV passes through a narrow passage and when the UAV is hanging to carry goods, it is often necessary to impose certain constraints on the overshoot of the system. In the current control methods, the transient performance of the system is generally adjusted by adjusting the parameters, although there are many studies on adjusting the overshoot of the system. However, none of these studies can achieve a quantitative constraint on the overshoot. Therefore, in this paper, we focus on the quantitative constraint problem of the overshoot of the quadrotor UAV from the basic viewpoint of constraint control. Firstly, a barrier Lyapunov function is designed, which does not depend on the switching function and can impose continuous asymmetric constraints on the system. Then, a new tubular preset performance boundary function is designed based on the asymmetric barrier Lyapunov function, which can predefine a more accurate and flexible tubular working region. Moreover, the system position error is limited to the working region, and then the overshoot of the system is quantitatively constrained. Secondly, the neural network adaptive control technology is introduced, the RBF neural network is selected as the observer of the system, and its adaptive law is designed to estimate the multi-source time-varying disturbance of the UAV. The simulation verification is carried out by setting up the comparison experiment of two scenarios. The results show that the system error can be constrained in the predefined tubular working area, and the proposed control strategy can quantitatively constrain the overshoot of the quadrotor UAV system, and has high steady-state accuracy and robustness. Compared with other traditional control methods, the proposed control strategy can reduce the overshoot of the quadrotor UAV system. The control method proposed in this paper can design the transient and steady-state performance of the system in advance without adjusting the parameters.