Rotation interpolation based on the geometric structure of unit quaternions

In this paper smooth rotation interpolation is revisited before several quaternion based algorithms are developed. The start point is the geometric structure of unit quaternions, and the aim is to simplify the computation of existing methods and to enhance the interpolating accuracy of approximate methods. All unit quaternions are defined as a group, whose relation with SO(3) and S³ are revealed, and whose Lie group structure can be established then. In solving the two-point interpolation problem with minimum acceleration constraint, numerical method and approximate method using cubic polynomial splines are presented. Three suboptimal analytical solutions are developed later on, utilizing quartic splines and projection skills on SO(3) or S³ respectively. Numerical results show that the new methods can have better performance over existing algorithms.