  In this paper, we introduce and explore augmented quaternions and augmented
unit quaternions, and present an augmented unit quaternion optimization model.
An augmented quaternion consist of a quaternion and a translation vector. The
multiplication rule of augmented quaternion is defined. An augmented unit
quaternion consists of a unit quaternion and a translation vector. The
augmented unit quaternions form a Lie group. By means of augmented unit
quaternions, we study the error model and kinematics. Then we formulate two
classical problems in robot research, i.e., the hand-eye calibration problem
and the simultaneous localization and mapping (SLAM) problem as augmented unit
quaternion optimization problems, which are actually real smooth spherical
equality constrained optimization problems. Comparing with the corresponding
unit dual quaternion optimization model, the augmented unit quaternion
optimization model has less variables and removes the orthogonality
constraints.
