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This paper presents a new family of fully-parallel robots producing motions of the Schoenflies displacements subgroup (three translations and one rotation about a given axis in world coordinates) for high-speed handling and machining. First, the structure's ability to provide this four-degrees-of-freedom motion is presented. Then, two of its possible designs are analyzed: the symmetrical and the asymmetrical one. Constructive designs are then presented. Some of the prototype's preliminary control results are given to prove the H4 robot's efficiency.