<p>A key challenge in robot hand-eye calibration lies in the fact that the accuracy of the estimated transformation is highly sensitive to input data quality. To mitigate the influence of outliers of the pose data, we propose an adaptive method for selecting and fitting pose pairs of unit dual quaternions generated from robotic motions. Addressing the scale disparity between rotational and translational components in rigid motions, we introduce the concept of the weighted norm for dual quaternions. Leveraging this novel norm, a direct numerical method is developed for projecting a dual quaternion onto the set of unit dual quaternions. Then, the hand-eye calibration problem can be modeled by a nonconvex nonsmooth optimization. The alternating direction method with multipliers (ADMM) is employed for solving the optimization problem. Under mild conditions, the global convergence of ADMM is analyzed. Numerical experiments demonstrate that the proposed method is effective and promising. In contrast to conventional least squares based estimations, the proposed enhanced method exhibits superior robustness when handling corrupted pose pairs containing outliers.</p>