Application of Hopf mapping in quaternion orientation planning of manipulator

PU Ya-song1,2, ZHANG Wen-bin1, LIN Xiao-jun2, GUO De-wei1, MIN Jie1
(1.School of Engineering, Honghe University, Mengzi 661199, China; 2.Laboratory of Contemporary
Design and Integrated Technology, Northwestern Polytechnical University, Xi'an 710072, China)

Abstract: Aiming at the problems that some quaternion interpolation algorithms were comparatively complex, difficult to find the derivatives, and hard to judge whether the quaternion interpolation was smooth, a study on judging the smoothness of interpolation using quaternion interpolation curve was conducted. It was proposed that Hopf mapping was applied to quaternion interpolation curves, and the fourdimensional interpolated points of quaternion orientation planning of robot manipulator were converted to three-dimensional vectors relying on the dimensionality reduction feature of Hopf mapping. Consequently, quaternion interpolation curves were obtained on the unit sphere, and the judgment was made intuitively whether the four-dimensional interpolation were smooth depending on the interpolation curves in three-dimensional space. The feasibility of the proposed methodology was verified by relevant study case. The results indicate that the suggested methodology avoids the derivation of complex quaternion interpolation function, and can be applied well to judge whether quaternion interpolation is smooth, and the usage is simple. The Spherical and Quadrangle (SQUAD) is more suitable than Spherical Linear Interpolation (SLERP) for quaternion interpolation with more than two orientations.

Key words: robot manipulator orientation planning; unit quaternion; smooth interpolation; Hopf mapping