Integrating Kronecker product and principle of transference to solve the hand–eye calibration problem

Abstract

Hand–eye calibration, a typical problem in robotic kinematics, can be abstracted as solving the kinematic equation $AX=XB$. The current closed-form solutions can be categorized into two aspects: the algebraic representation of the kinematic equations and computational methods. Applying the Lie group’s homomorphism theorem allows one to easily establish a theoretical framework for kinematic equations. While introducing the principle of transference can strengthen the current theoretical framework in terms of both equation representations and computational methods. With the motivation of creating a closer theoretical framework, a new method is proposed in this paper. First, it is applied the principle of transference to verify the Kronecker product of dual matrices. Further applying it to hand–eye calibration to re-prove the sufficient and necessary condition for the existence of a unique solution. The proposed proof process differs from existing work by simultaneously addressing rotation and translation, directly embodying the principle of transference. Subsequently, a simultaneous closed-form solution is proposed, involving geometric interpretation and algebraic analysis. Finally, the feasibility and superiority of the proposed method is verified in experiments. The proposed method achieves a rotation error of $0.07°$ and a translation error of 1.07 mm, outperforming the reference methods.