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The concept of variable stiffness actuation (VSA) for robotic joints promises advantages regarding robustness, energy efficiency, and task adaptability. The VS joints developed at DLR show very low intrinsic damping for efficient energy storage and retrieval whereas the desired damping behavior for task execution needs to be implemented in control. Robotic arms with multiple VS joints, as for example the DLR Hand Arm System, ask for advanced control algorithms which can cope with the elastic joints and the multi-input multi-output (MIMO) system properties of the mechanical setup. We propose a MIMO controller for flexible joint robots based upon an eigenmode decoupling approach. For robustness reasons, the controller is designed to modify the intrinsic plant properties as little as possible while attaining the desired damping. A gain design algorithm is proposed. The controller is validated in simulations and experiments.