Approaching Dual Quaternions From Matrix Algebra

Dual quaternions give a neat and succinct way to encapsulate both translations and rotations into a unified representation that can easily be concatenated and interpolated. Unfortunately, the combination of quaternions and dual numbers seems quite abstract and somewhat arbitrary when approached for the first time. Actually, the use of quaternions or dual numbers separately is already seen as a break in mainstream robot kinematics, which is based on homogeneous transformations. This paper shows how dual quaternions arise in a natural way when approximating 3-D homogeneous transformations by 4-D rotation matrices. This results in a seamless presentation of rigid-body transformations based on matrices and dual quaternions, which permits building intuition about the use of quaternions and their generalizations.