Parallel mechanisms generating 3-DoF finite translation and (2 or 1)-DoF infinitesimal rotation

The product of two Schoenflies motion subgroups of the group of general displacements characterizes a noteworthy type of 5-dimensional (5D) displacement set called double Schoenflies or X–X motion. It includes any spatial translation and any two sequential rotations whose axes are parallel to two given independent vectors. The canonical irreducible representation of the 5D set of X–X motion is utilized to study various intersection sets of three distinct X–X motions. Three remarkable possibilities: 3-DoF translation motion, 3-DoF translation and 1-DoF infinitesimal rotation, and 3-DoF translation and 2-DoF infinitesimal rotation are discussed. Then we disclose the corresponding mechanisms with a mixture of finite and infinitesimal DoFs. Novel parallel mechanisms with 3-DoF finite translation and 2- or 1-DoF infinitesimal rotation are proposed. Parallel mechanisms whose infinitesimal mobility is specific of a singularity locus are derived, too. Meanwhile, it is established that the well-known SNU 3-UPU parallel manipulator with 3-DoF finite translation also has a mobility of 2-DoF infinitesimal rotation for a set of singular postures including its home configuration.