Inversion Symmetry of the Euclidean Group: Theory and Application to Robot Kinematics

Abstract

Just as the 3-D Euclidean space can be inverted through any of its points, the special Euclidean group $\bf SE(3)$ admits an inversion symmetry through any of its elements and is known to be a symmetric space. In this paper, we show that the symmetric submanifolds of $\bf SE(3)$ can be systematically exploited to study the kinematics of a variety of kinesiological and mechanical systems and, therefore, have many potential applications in robot kinematics. Unlike Lie subgroups of $\bf SE(3)$ , symmetric submanifolds inherit distinct geometric properties from inversion symmetry. They can be generated by kinematic chains with symmetric joint twists. The main contribution of this paper is: 1) to give a complete classification of symmetric submanifolds of $\bf SE(3)$ ; 2) to investigate their geometric properties for robotics applications; and 3) to develop a generic method for synthesizing their kinematic chains.