Analysis of Two Spherical Parallel Manipulators With Hidden Revolute Joints

In this paper, we examine two spherical parallel manipulators (SPMs) constructed with legs that include planar and spherical subchains that combine to impose constraints equivalent to hidden revolute joints. The first has supporting serial chain legs constructed from three revolute joints with parallel axes, denoted R$∥$R$∥$R, followed by two revolute joints that have intersecting axes, denoted $RR̂$⁠. The leg has five degrees-of-freedom and is denoted R$∥$R$∥$R-$RR̂$⁠. Three of these legs can be assembled so the spherical chains all share the same point of intersection to obtain a spherical parallel manipulator denoted as 3-R$∥$R$∥$R-$RR̂$⁠. The second spherical parallel manipulator has legs constructed from three revolute joints that share one point of intersection, denoted $RRR̂$⁠, and a second pair of revolute joints with axes that intersect in a different point. This five-degree-of-freedom leg is denoted $RRR̂$-$RR̂$⁠. The spherical parallel manipulator constructed from these legs is 3-$RRR̂$-$RR̂$⁠. We show that the internal constraints of these two types of legs combine to create hidden revolute joints that can be used to analyze the kinematics and singularities of these spherical parallel manipulators. A quaternion formulation provides equations for the quartic singularity varieties some of which decompose into pairs of quadric surfaces which we use to classify these spherical parallel manipulators.