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Generators of the product of two Schoenflies motion groups
Authors:Chung-Ching Lee  Jacques M Herv
Institution:aDept. Tool and Die-Making, National Kaohsiung University of Applied Sciences, 415 Chien Kung Road, Kaohsiung, 80782 Taiwan, ROC;bEcole Centrale Paris, Grande Voie des Vignes, F-92295, Chatenay-Malabry, France
Abstract:Schoenflies motion is often termed X-motion for conciseness. A set of X-motions with a given direction of its axes of rotations has the algebraic properties of a Lie group for the composition product of rigid-body motions or displacements. The product of two X-subgroups, which is the mathematical model of a serial concatenation of two kinematic chains generating two distinct X-motions, characterizes a noteworthy type of 5-dimensional (5D) displacement set called double Schoenflies motion or X–X motion for brevity. This X–X motion set is a 5D submanifold of the displacement 6D Lie group. Such a motion type includes any spatial translation (3T) and any two sequential rotations (2R) provided that the axes of rotation are parallel to two fixed independent vectors. This motion set also contains the rotations that are products of the foregoing two rotations. In the paper, some preliminary fundamentals on the 4D X-motion are recalled; the 5D set of X–X motions is emphasized. Then implementing serial arrays of one-dof Reuleaux pairs and hinged parallelograms, we enumerate all serial mechanical generators of X–X motion, which have no redundant internal mobility. Based on the group-theoretic concepts, one can differentiate two families of irreducible representations of an X–X motion. One family is realized by twenty-one open chains including the doubly planar motion generators as special cases. The other is generally classified into eight major categories in which one hundred and six distinct open chains generating XX motion are revealed and nineteen more ones having at least one parallelogram are derived from them. Meanwhile, these kinematic chains are graphically displayed for a possible use in the structural synthesis of parallel manipulators.
Keywords:Double Schoenflies motion  Schö  nflies  X  X motion  Lie group  Mechanical generator  Kinematic bond  Reuleaux pair  Parallelogram
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