This paper studies the kinematic geometry of a special 6-RUS parallel manipulator for which the axes of all base actuated R-joints coincide, the centers of all U-joints move along the same circular track, and the platform S-joints coincide in pairs. Particularly, the paper presents a geometric algorithm for the computation of the constant-orientation workspace. An already known methodology has been enhanced to include the physical constraint, modelled as three Bohemian dome surfaces, on the U-joint interference. In addition, the singularity loci for a constant orientation are shown to form a quartic surface. The workspace boundaries and the singularity loci are analytically computed and represented as horizontal cross-sections. Important observations are made on the singularities of general parallel mechanisms with pair-wise coincident S-joints. The paper also introduces the phenomenon that the workspace of some parallel mechanisms is divided into regions corresponding to different branch sets.

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