中文摘要：

机器人手眼标定通常要解一个旋转方程RaRx=RxRb，求解该方程有很多方法，其中利用四元数的方法最为简洁实用。但一般的四元数求解侧重应用，缺乏几何意义的对照，也没有全面分析方程各种解的情况。为了提高机器人手眼标定的效率和精度，在深入研究求解该方程的四元数几何方法的基础上，详细而严格地论证了各种情况下方程的解，不仅给出了四元数矩阵分析与几何解释的有趣对照，而且仿真验证了该算法的正确性。仿真实验表明，了解方程各种解的情形以及几何意义将有助于降低求解的条件数和提高标定的效率，此外，该研究对于发展四元数几何分析也有很大的意义。

英文摘要：

Calibration of robot hand-eye generally needs to solve a rotation equation RaRx=RxRb. Many methods have been proposed, within which quaternion is the most concise one. But common methods using quaternion emphasize particularly on the application, and are short of relevant geometrical insight, and lack of comprehensive analysis of various solutions. In this paper we use quaternion geometry to solve the rotation equation, give proofs of solutions in various conditions in detail, and illuminate interesting insights between the analysis with quaternion matrix and the expression by geometry. Simulations have been tested. Analyzing solutions in various conditions and understanding the relevant geometrical meaning will help to ease the solving conditions and improve the performance of hand-eye calibration. Moreover, the study is important for the development of the quaternion geometrical analysis.