This article presents a novel robust centralized controller for impedance control and reference tracking of redundant manipulators. The proposed approach takes advantage of the robustness properties of sliding mode control (SMC) and the prediction capabilities of model predictive control (MPC). SMC theory is employed to compensate unmodeled system dynamics and disturbances, ensuring accurate tracking and enforcement of a desired end-point impedance during interaction with the environment. Differently from other schemes, the sliding manifold is expressed directly in the task space and the approach is generalized to redundant manipulators by projection of the manifolds into joint space. Chattering attenuation is provided by a second-order integral sliding mode control law. These features are exploited by the MPC to guarantee motion and actuation constraint fulfillment based on the nominal feedback linearized robot model. A formal analysis of the control system is given along with the relevant proofs. The resulting model predictive sliding mode controller is able to cope with delays acting on the control input torque. The effectiveness of the approach is validated in simulation on a 4-DOF planar robot, and its viability on real platforms through experiments on a 7-DOF prototype ABB YuMi robot arm.