This paper addresses the altitude trajectory tracking problem of quadrotor under unknown disturbances taking into account actuators saturation. In this work, a new adaptive sliding mode control (ASMC) is developed to guarantee a high robustness with a less chattering effect against unknown bounds of lumped disturbances. Compared to classical SMC which requires a priori knowledge of disturbance bounds. The proposed approach allows to avoid these drawbacks through a dynamic adaptive law with minimal effort, high tracking accuracy and without gain overestimation that will automatically deal with the input saturation constraints. The stability analysis proved by Lyapunov theory demonstrates that the control strategy guarantees that all the signals of the closed-loop system are bounded and that the tracking errors are bounded in finite time. Experimental results are given to illustrate the effectiveness of the proposed method, using the Parrot Mambo mini-drone.