Recently, Zhuang, Roth, \& Sudhakar [1] proposed a method that allows simultaneous computation of the rigid transformations from world frame to robot base frame and from hand frame to camera frame. Their method attempts to solve a homogeneous matrix equation of the form AX=ZB. They use quaternions to derive explicit linear solutions for X and Z. In this short paper, we present two new solutions that attempt to solve the homogeneous matrix equation mentioned above: (i) a closed-form method which uses quaternion algebra and a positive quadratic error function associated with this representation and (ii) a method based on non-linear constrained minimization and which simultaneously solves for rotations and translations. These results may be useful to other problems that can be formulated in the same mathematical form. We perform a sensitivity analysis for both our two methods and the linear method developed by Zhuang et al. This analysis allows the comparison of the three methods. In the light of this comparison the non-linear optimization method, which solves for rotations and translations simultaneously, seems to be the most stable one with respect to noise and to measurement errors.

翻译：近期，Zhuang、Roth和Sudhakar [1] 提出了一种方法，可同时计算从世界坐标系到机器人基座坐标系以及从手部坐标系到相机坐标系的刚性变换。该方法尝试求解形式为AX=ZB的齐次矩阵方程，并利用四元数推导出X和Z的显式线性解。本文提出两种针对上述齐次矩阵方程的新解法：（i）一种闭合形式方法，采用四元数代数及与该表示相关的正二次误差函数；（ii）一种基于非线性约束优化的方法，可同时求解旋转与平移。这些结果可能对可表述为相同数学形式的其他问题具有参考价值。我们对本文两种方法及Zhuang等人提出的线性方法进行了灵敏度分析，该分析支持三者之间的比较。基于比较结果，同时求解旋转与平移的非线性优化方法在噪声与测量误差方面表现最为稳定。