This paper deals with dual-space control of R4 redundantly actuated parallel manipulator for very high acceleration applications. This controller consists in a PID in the Cartesian space complied with a feedforward of the desired acceleration in both Cartesian and articular spaces for tracking performance improvements: models show that this "dual-space" control strategy is an e cient way to implement computed torque control. For comparison purposes, experiments were made with a Cartesian PID until 20G. Experimental results show that the proposed control scheme is considerably better than the PID in the Cartesian space, and that a good tracking performance could be achieved even for the very high acceleration of 40G (equivalent to more than 425 pick-and-place cycles per minute).