This paper deals with the control of a 6-DOF cable-suspended parallel robot (CSPR) able to perform tasks such as pick-and-place trajectories over a large workspace. In order to maximize the ratio between the robot workspace and its overall dimensions, actuation redundancy can be used. The control of such a redundantly actuated CSPR turns to be challenging as a realtime embeddable algorithm for distributing the cable tensions should be used, together with a suitable control scheme. This paper proposes a computationally efficient tension distribution algorithm implemented within a dual-space feedforward scheme in order to properly control the moving platform. Experimentations are performed on a large 6-DOF CSPR prototype equipped with 8 actuators.