Expression of dual Euler parameters using the dual Rodrigues parameters and their application to the screw transformation

Abstract

Dual numbers and dual vectors are widely used in spatial kinematics [3,5- 15,18]. Plücker line coordinates of a straight line can be represented by a dual unit vector located at the dual unit sphere (DUS). By this way, the trajectory of the screw axis of a rigid body in $R^3$ (the real three space) corresponds to a dual curve on the DUS. This correspondence is done through Study Mapping [8,9]. Conversely a dual curve on DUS obtained from the rotations of the DUS represents a rigid body motion in $R^3$ [8]. The dual Euler parameters are used in defining the screw transformation in $R^3$ [8], but originally in this paper these parameters are constructed from the Rodrigues and the dual Rodrigues parameters [15].