Hand-Eye Calibration: 4-D Procrustes Analysis Approach

We give a universal analytical solution to the hand-eye calibration problem <inline-formula> <tex-math notation="LaTeX">${AX} = {XB}$ </tex-math></inline-formula> with known matrices <inline-formula> <tex-math notation="LaTeX">${A}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">${B}$ </tex-math></inline-formula> and unknown variable <inline-formula> <tex-math notation="LaTeX">${X}$ </tex-math></inline-formula>, all in the set of special Euclidean group SE(3). The developed method relies on the 4-D Procrustes analysis. A unit-octonion representation is proposed for the first time to solve such a Procrustes problem through which an optimal closed-form eigendecomposition solution is derived. By virtue of such a solution, the uncertainty description of <inline-formula> <tex-math notation="LaTeX">${X}$ </tex-math></inline-formula>, being a sophisticated problem previously, can be solved in a simpler manner. The proposed approach is then verified using simulations and real-world experimentations on an industrial robotic arm. The results indicate that it owns better accuracy and better description of uncertainty and consumes much less computation time.