Geometrical distribution of rotational axes of 3-[P][S] parallel mechanisms

A 3-[P][S] parallel mechanism consists of three limbs and each limb can generate a planar-spherical ([P][S]) kinematic bond. Typically, the 3-[P][S] parallel mechanism family includes four types of architectures, namely, 3-RPS, 3-PRS, 3-RRS and 3-PPS, where R denotes a revolute pair, P a prismatic pair, and S a spherical joint. The 3-[P][S] parallel mechanism has received extensive attention due to its practical potential. But little is known about the geometrical distribution of the axes of the two rotational DOF (degrees of freedom) of the 3-[P][S] parallel mechanism. Consequently, although the kinematic derivations of the 3-[P][S] parallel mechanism are correct, the interpretation of the actual instantaneous rotation is not clear. This fact may hinder its application. This paper concentrates on the identification of the rotational axes of the 3-[P][S] parallel mechanism with different limb arrangements. First, the geometrical condition for the axis of a feasible rotation of a rigid body constrained by a force is discussed using screw theory. Then, the 3-[P][S] PMs are classified into four categories and seven subcategories based on the geometrical condition of their LPs (limb planes) and spherical joint centers, The instantaneous and finite rotational axes of the seven subcategories of 3-[P][S] parallel mechanism are identified using reciprocal screw theory. The results apply to all 3-[P][S] PMs.