Exponential Displacement Coordinates by Means of the Adjoint Representation

Abstract

This paper introduces methods to determine the exponential coordinates of first kind and of second kind for an arbitrary spatial displacement given in terms of the left adjoint representation. Due to the algebraic properties of the $(6 \times 6)$-matrix group, the obtained formulae are structure-preserving generalizations of their purely-rotative counterparts in coherence with the principle of transference. While the exponential coordinates of first kind represent the line-geometric parameters of a displacement (rotation and translation along a unit spear), the exponential coordinates of second kind coincide with dual Euler angles and with the parameters according to the kinematic convention by Sheth and Uicker. In either case, a spatial displacement is specified via six independent scalars, in form of a dual angle and a dual unit vector ($2+4=6$) for the first kind and in form of three dual angles ($3\cdot 2 =6$) along three sequentially orthogonal axes for the second kind. From a practical viewpoint, the parametrization method enables an automated parametrization of the kinematics of an arbitrary mechanism. From a theoretical viewpoint, the reported methods are relevant due to their structural simplicity and coherence.