In this paper, we first study the projections onto the set of unit dual quaternions, and the set of dual quaternion vectors with unit norms. Then we propose a power method for computing the dominant eigenvalue of a dual quaternion Hermitian matrix, and show its convergence and convergence rate under mild conditions. Based upon these, we reformulate the simultaneous localization and mapping (SLAM) problem as a rank-one dual quaternion completion problem. A two-block coordinate descent method is proposed to solve this problem. One block subproblem can be reduced to compute the best rank-one approximation of a dual quaternion Hermitian matrix, which can be computed by the power method. The other block has a closed-form solution. Numerical experiments are presented to show the efficiency of our proposed power method.

翻译：本文首先研究了对偶单位四元数集合及单位范数对偶四元数向量集合上的投影问题。随后提出了一种用于计算对偶四元数埃尔米特矩阵主特征值的幂法，并在温和条件下证明了其收敛性及收敛速率。基于此，我们将同时定位与地图构建（SLAM）问题重构为秩一对偶四元数补全问题。为解决该问题，提出了一种双块坐标下降法，其中一块子问题可归结为计算对偶四元数埃尔米特矩阵的最佳秩一逼近，该逼近可通过幂法实现；另一块则具有闭式解。数值实验验证了所提幂法的有效性。