Type synthesis of primitive Schoenflies-motion generators

A noteworthy type of motion called Schoenflies motion and often termed X-motion for brevity is presented. A specified set of X-motions is endowed with the algebraic structure of a four-dimensional (4D) Lie group. This 4D displacement Lie subgroup includes any translation and any rotation provided that the axis of rotation is parallel to a given direction. In the paper, some preliminary fundamentals about the Lie group of displacements are recalled; the 4D Lie subgroup of X-motion is emphasized. Then serial concatenations of one-dof Reuleaux pairs and hinged parallelograms lead to the enumeration of all possible general architectures of mechanical generators for a given X subgroup. Meanwhile, their corresponding embodiments are graphically displayed for a future use in the structural synthesis of parallel manipulators. These generators are sorted into four classes based on the number of prismatic pairs. In total, forty-three distinct mechanical generators of X-motion are revealed and eighty-two ones having at least one parallelogram are also derived from them. Some chains that are defective generators of X-motion are also identified through an approach based on the group dependency.