A Complete Analytical Solution to Hand-Eye Calibration Using Quaternions and Eigenvector-Eigenvalue Identity

Hand-eye calibration is one of the important problems in the field of robot vision and control, aiming to determine the pose transformation between the robot end-effector and the visual system. The analytical solution of this problem has a more explicit numerical computation process and a more stable solution time compared with iterative or deep learning methods, and theoretically provides more accurate results. When using quaternion parameterization for rigid body motion rotation, the hand-eye calibration problem can be transformed into solving the eigenvector corresponding to the maximum eigenvalue of a symmetric matrix. This paper proposes an analytical solution for hand-eye calibration based on quaternion parameterization. Compared with previous methods, the advantage of this method is that the solution for the required eigenvector is transformed into solving cubic equations through the eigenvector-eigenvalue identity, without SVD or Eigendecomposition, thus making the proposed method a complete analytical solution, and avoids solving irrelevant information. The performance of the proposed method was evaluated through multiple experiments conducted on a synthetic dataset as well as two real-world datasets, and compared to representative analytical methods. The experimental results unequivocally demonstrate that our proposed method achieves comparable accuracy with significantly shorter solution times. Therefore, our complete analytical solution offers an efficient and accurate alternative for addressing the hand-eye calibration problem without SVD or Eigendecomposition.