Nowadays, quadcopter unmanned aerial vehicles play important roles in several real-world applications and the improvement of their control performance has become an increasingly attractive topic of a great number of studies. In this paper, we present a new approach for the design and stability analysis of a quadcopter adaptive trajectory tracking control. Based on the quadcopter nonlinear dynamics model which is obtained by using the Euler–Lagrange approach, the tracking controller is devised via the backstepping control technique. Besides, an adaptive law is proposed to deal with the system parameterized uncertainties and to guarantee that the control input is finite. In addition, the vehicle’s vertical descending acceleration is ensured to not exceed the gravitational acceleration by making use of a barrier Lyapunov function. It is shown that the suitable parameter estimator is stable and the tracking errors are guaranteed to be asymptotically stable simultaneously. By prescribing certain flight conditions, we use numerical simulations to compare the control performance of our method to that of existing approaches. The simulation results demonstrate the effectiveness of the proposed algorithm.