Probabilistic Framework for Hand–Eye and Robot–World Calibration $AX = YB$

Hand–eye and robot–world calibration is a problem in which the unknown homogeneous transformations $X$ and $Y$ must be estimated for a loop closure equation $AX = YB$ for a set of transformation measurement pairs $\lbrace (A_{i}, B_{i}) \rbrace$. Previous studies on $AX=YB$ have mainly relied on linear least-squares minimization followed by nonlinear iterative optimization for solution refinement to minimize the distances between $A_{i} X$ and $Y B_{i}$. However, these methods have not been fully clarified, particularly in terms of calibration dependence on the coordination of $A,B,X$, and $Y$ along the system loop, as well as the underlying noise distributions of $A_{i}$ and $B_{i}$. They also lack flexibility in the noise properties of individual measurements; thus, they cannot incorporate the relative reliability between measurements. To address these limitations, we propose a probabilistic framework for hand–eye and robot–world calibration. The proposed framework clarifies the unclear aspects of existing methods by revealing their underlying assumptions regarding system noise. Consequently, it identifies the applicability of distance minimization to a given calibration problem and provides the optimal coordination of transformations for distance minimization. For cases in which distance minimization is inapplicable, an iterative algorithm for the maximum likelihood estimation is proposed, whereby the different noise properties of individual measurements can be accounted for. An estimation uncertainty analysis is presented for the proposed iterative algorithm to quantify the expected estimation accuracy. The presented theories and the proposed algorithm are validated using a set of numerical and hardware experiments. The code for the iterative algorithm and the estimation uncertainty is available at https://github.com/hjhdog1/probabilisticAXYB.