The regulation of the link positions of compliant robots, damping out undesired link oscillations while preserving the system's inherent elasticity is still a challenging task in practical applications. This task becomes even harder to be tackled in the case of compliant robots driven by agonistic-antagonistic variable stiffness actuators in which there are two motors associated with each joint of the system. In this work, leveraging on the physical realization of the elastic mechanism of such actuators, we propose a novel control law able to simultaneously achieve a good regulation performance and a desired damped behavior at the link, while preserving the elastic structure of the system as well as the possibility of adjusting the passive stiffness at the joints. Simulations on the agonistic-antagonistic actuators of the forearm and wrist joints of the Hand Arm System from the German Aerospace Center (DLR) and experiments on a planar platform with an analogous actuation unit, namely qbMove Advanced, validate the proposed method.