Quaternions and Euler Parameters — A Brief Exposition

The quaternion concept has found successful applications in many areas of the physical sciences. In the kinematics and dynamics of spatial mechanical systems and synthesis of mechanisms, quaternion theory may be found under the guise of Euler parameters, dual numbers, dual quaternions, rotation tensors, screw axis calculus, etc. Quaternion algebra has been applied to obtain analytical solutions, and to classify single- and multi-degree-of-freedom motions of many closed loop spatial mechanisms. The resulting systems of algebraic equations are generally extremely complex and difficult to interpret or transform to computer programs. The objective of this paper is to look at some of the basic quaternion algebra and identities, and their corresponding matrix representations to aid in the development of mechanism anaysis capabilities and computer algorithms.