Quaternions in Kinematics

This paper examines Hamilton’s quaternions, dual quaternions and Clifford’s biquaternions in order to show how they are related to the kinematics of rotations in space, spatial displacements and rotations in four dimensional space. The three algebras are constructed in the same way as the even Clifford Algebras for these spaces, and their quaternion products are shown to provide a formula for spherical triangulation, spatial triangulation of lines, and double triangulation of two spherical triangles, respectively. In the process, we obtain the spherical triangle of relative rotations axes and the spatial triangle of relative screw axis, which are important generalizations of the planar pole triangle in Kinematics. We conclude with applications of double quaternions as approximations to spatial movement, which simplify calculations that rely on distance metrics for spatial positions, such as spatial motion interpolation. • A comparative study of Hamilton’s quaternions, dual quaternions and double quaternions. • Definition of these sets of quaternions as even Clifford algebras of E 3 , P 3 and E 4 . • The quaternion product is shown to yield triangulation formulas for the three cases. • The quaternion product is also shown to yield Rodrigues’s formulas for the three cases. • There is a benefit to considering spatial displacements as four dimensional rotations.