The geometric structure of unit dual quaternion with application in kinematic control

In this paper, the geometric structure, especially the Lie-group related properties, of unit dual quaternion is investigated. The exponential form of unit dual quaternion and its approximate logarithmic mapping are derived. Correspondingly, Lie-group and Lie-algebra on unit dual quaternions and the approximate logarithms are explored, respectively. Afterwards, error and metric based on unit dual quaternion are given, which naturally result in a new kinematic control model with unit dual quaternion descriptors. Finally, as a case study, a generalized proportional control law using unit dual quaternion is developed.