In this paper, three control schemes are proposed and experimentally compared on the R4 redundantly actuated parallel manipulator for applications with very high accelerations. First, a proportional-integral-differential (PID) in operational space is proposed to adequately take into consideration the actuation redundancy. Because of its lack of performance, a dual-space feedforward control scheme based on the dynamic model of R4 is proposed. The improvements obtained with this controller allowed the implementation of an experiment, which consisted in the tracking of a trajectory with a maximum acceleration of more than 100G. However, such a controller may have loss of performance in case of any operational change (such as different payloads). Therefore, a dual-space adaptive control scheme is proposed. The stability analysis of the R4 parallel robot when controlled by the proposed dual-space adaptive controller is provided. The objective of this paper is to show that the proposed dual-space adaptive controller not only maintains its good performance independently of the operational conditions but also has a better performance than both the PID and the dual-space feedforward controllers, even when the latter is best configured for the given case (which confirms its applicability in an industrial environment).