This article focuses on the design of a robust model predictive control law for constrained discrete-time time-varying systems with additive uncertainties. The proposed solution to the control problem is a tube-based MPC ensuring robustness and constraints fulfilment. Reachable sets are calculated online taking into account the system dynamics by means of an adaptive local control law and additive uncertainties. The proposed method represents a trade-off between small conservativeness and efficient real-time execution. This approach is applied to solve the trajectory tracking problem of a mobile robot. Simulation results provide a comparison between the tube-based MPC scheme and established motion control algorithms, showing the efficient execution and satisfactory behaviour of the proposed controller.