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.