Simultaneous Hand–Eye, Tool–Flange, and Robot–Robot Calibration for Comanipulation by Solving the $\mathbf{AXB=YCZ}$ Problem

Multirobot comanipulation shows great potential in surpassing the limitations of single-robot manipulation in complicated tasks such as robotic surgeries. However, a dynamic multirobot setup in unstructured environments poses great uncertainties in robot configurations. Therefore, the coordination relationships between the end-effectors and other devices, such as cameras (hand–eye calibration) and tools (tool–flange calibration), as well as the relationships among the base frames (robot–robot calibration) have to be determined timely to enable accurate robotic cooperation for the constantly changing configuration of the systems. We formulated the problem of hand–eye, tool–flange, and robot–robot calibration to a matrix equation
$\mathbf{AXB=YCZ}$
. A series of generic geometric properties and lemmas were presented, leading to the derivation of the final simultaneous algorithm. In addition to the accurate iterative solution, a closed-form solution was also introduced based on quaternions to give an initial value. To show the feasibility and superiority of the simultaneous method, two nonsimultaneous methods were compared through thorough simulations under various robot movements and noise levels. Comprehensive experiments on real robots were also performed to further validate the proposed methods. The comparison results from both simulations and experiments demonstrated the superior accuracy and efficiency of the proposed simultaneous calibration method.