On the Covariance of $\boldsymbol X$ in $\boldsymbol A\boldsymbol X = \boldsymbol X\boldsymbol B$

Hand-eye calibration, which consists in identifying the rigid-body transformation between a camera mounted on the robot end-effector and the end-effector itself, is a fundamental problem in robot vision. Mathematically, this problem can be formulated as: solve for X in AX = XB. In this paper, we provide a rigorous derivation of the covariance of the solution X, when A and Bare randomly perturbed matrices. This line-grained information is critical for applications that require a high degree of perception precision. Our approach consists in applying covariance propagation methods in SE(3). Experiments involving synthetic and real calibration data confirm that our approach can predict the covariance of the hand-eye transformation with excellent precision.