On the Comparisons of Unit Dual Quaternion and Homogeneous Transformation Matrix

This paper reveals the differences and similarities between two popular unified representations, i.e. the UDQ (unit dual quaternion) and the HTM (homogeneous transformation matrix), for transformation in the solution to the kinematic problem, in order to provide a clear, concise and self-contained introduction into dual quaternions and to further present a cohesive view for the UDQ and HTM representations as used in robotics. Specifically, after investigating some fundamental algebraic properties of the UDQ, it is revealed that the kinematical equations represented by the UDQ and the HTM are accordant, and afterwards the direct relationship of UDQ-based error kinematical models in spatialframe and in body-frame are further discussed, with conclusion that either error kinematic model can be chosen for designing kinematical control laws. Finally, the comparative study on the proportional control algorithms based on the logarithmical mapping of the HTM and the UDQ shows that the UDQ-based control law is indeed higher in computational efficiency.