Nonlinearity and dimensionality have always been computationally challenging problems when it comes to the online implementation of optimization-based control approaches, especially in the presence of disturbances. In this paper, we show how to alleviate the problem via a variable reformulation derived from differential flatness for a quadcopter vehicle. More specifically, we present a robust model predictive control design, to track a predefined trajectory of a quadcopter in the presence of disturbances. The synthesis procedure starts with a coordinate change mediated by the model's flatness property. In this new representation, the dynamics become linear in closed-loop at the price of more convoluted constraint expressions, which are usually disregarded in the literature or simple approximations are proposed. Subsequently, with a proper parameterization to portray the feasible domain, the trajectory tracking problem is transformed into the stabilization of a constrained linear time-invariant system under disturbances, which is then handled by a robust model predictive controller. Simulations and experimental results are presented to analyze and validate the proposed scheme.