Kinematic Feedback Control Using Dual Quaternions

This paper presents results on kinematic controllers for the stabilization of rigid body displacements using dual quaternions. The paper shows how certain results for quaternion stabilization of rotation can be extended to dual quaternions stabilization of displacements. The paper presents a relevant background material on screw motion and the screw description of lines and twists. Moreover, results are presented on the computation of the exponential functions for dual quaternions for use in numerical integration. The paper presents and analyzes different controllers based on feedback from dual quaternions, where some of the controllers are known from the literature, and some are new. In particular, it is shown which controllers give screw motion, and it is discussed how this will affect the performance of the controlled system compared to other controllers that are not based on screw motion. This analysis is supported by Lyapunov analysis. Also, certain passivity properties for dual quaternions are presented as an extension to previously published results on passivity for quaternions.