This article provides a new method for trajectory tracking for quadcopters following a cascaded control approach with formal closed-loop tracking guarantees. An outer-loop model predictive controller generates twice differentiable acceleration references, which provide attitude, angular velocity, and acceleration references for a nonlinear inner-loop controller. The model predictive controller allows for tracking of references while explicitly considering that the thrust of the quadcopter is upper and lower limited. It is proven that the overall strategy renders the trajectory tracking errors uniformly almost globally asymptotically stable. Via a numerical case study, the advantages of the novel method are highlighted.