The kinematically indeterminate cable suspension manipulator Cablev moves a payload platform in space by three spatially arranged cables with independently controllable winches. As the position of the platform is not fully determined by the lengths of the cables, undesired sway motions of the payload platform may occur. To make the payload platform track prescribed translational and rotational reference trajectories in space, a two-stage control concept is presented. A nonlinear feedforward control that exploits the flatness property of the system generates control inputs for the undisturbed motion along reference trajectories. Sway motions caused by disturbances are actively damped by a linear feedback of measured state variables enabling an asymptotically stable tracking behaviour. Experimental results from the prototype system Cablev are shown.