Zhuang et al. (1994) 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 for, AX=ZB. They use quaternions to derive explicit linear solution for X and Z. In this paper, we present two new solutions that attempt to solve the homogeneous matrix equation mentioned above: 1) a closed-form method which uses quaternion algebra and a positive quadratic error function associated with this representation; 2) a method based on nonlinear 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 nonlinear optimization method, which solves for rotations and translations simultaneously, seems to be the most stable one with respect to noise and to measurement errors.
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