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.

翻译：本文介绍并探讨了增广四元数与增广单位四元数，并提出了增广单位四元数优化模型。增广四元数由一个四元数和一个平移向量构成，并定义了其乘法规则。增广单位四元数则由一个单位四元数和一个平移向量组成，且增广单位四元数构成一个李群。借助增广单位四元数，我们研究了误差模型与运动学特性。随后，我们将机器人研究中的两个经典问题——手眼标定问题以及同时定位与地图构建问题——表述为增广单位四元数优化问题，这类问题实际上是实数域上的光滑球面等式约束优化问题。与相应的对偶单位四元数优化模型相比，增广单位四元数优化模型变量更少，且去除了正交性约束。