The task of planning and controlling robot motion in practical applications is often complicated by the effects of model uncertainties and environment disturbances. We present in this paper a systematic approach for generating robust motion control strategies to satisfy high level specifications of safety, target attainability, and invariance, under unknown but bounded, continuous disturbances. The motion planning task is decomposed into the two sub-problems of finite horizon reach with avoid and infinite horizon invariance. The set of states for which each of the sub-problems is robustly feasible is computed via iterative reachability calculations under a differential game framework. We discuss how the results of this computation can be used to inform selections of control inputs based upon state measurements at run-time and provide an algorithm for implementing the corresponding feedback control policies. Finally, we demonstrate an experimental application of this method to the control of an autonomous helicopter in tracking a moving ground vehicle.