Rotational-translational actuator (RTAC) systems can capture the major working principle of dual-spin spacecrafts and can be used to study the despin maneuver. In this paper, we consider the problem of globally stabilizing RTAC systems, which work in the vertical plane and are affected by nonvanishing matched disturbances by employing continuous control inputs. To this end, a nonlinear continuous control approach is synthesized to effectively damp out the platform oscillations, while ensuring that the eccentric ball's rotation angle returns to zero without residual swing. To our knowledge, this paper yields the first continuous controller, which is designed and analyzed without linearizing/approximating the original nonlinear dynamical equations to globally stabilize both the rotation and translation of RTAC systems influenced by nonvanishing matched disturbances. We implement experiments to show that the proposed control approach can achieve superior performance vis-à-vis existing methods and it admits for good robustness.