This paper investigates the finite time prescribed performance control strategy for the trajectory tracking problem of quadrotor UAVs with time-vary disturbance, model uncertainty and constraint of the output error. Firstly, the UAVs dynamic model is decoupled into an sub atitude system and. Then, the finite time convergence of the conversion error is realized, and the output error of the original system is limited by introducing the error conversion function, performance constraint function and designing the fast terminal sliding surface reasonably. The designed controller can guarantee the finite time stability of the system, and make the system have good transient stability performance. Finally, a simulation example shows the effectiveness of the proposed methods.