On the Relation Between Kinematic Mappings of Planar and Spherical Displacements

Abstract

Kinematic mappings, which transform the parameters specifying a rigid displacement into a point in a higher dimensional space, have been the focus of recent study for the purpose of classifying rigid motion, aiding mechanism design, and automating robot task planning. Presented here is the procedure which demonstrates that the particular mapping introduced by Blaschke and Mu¨iller (1956) to study planar kinematics, and used subsequently by other researchers, is a limiting case of a mapping based on the Euler parameters of an orthogonal matrix used by Ravani (1982) for spherical kinematics.