Multi-Camera Robot-World Hand-Eye Calibration by Solving Multi-Unit Dual Quaternion Equations

Abstract

In this paper, we study the solution to the multi-camera robot-world hand-eye calibration problem by employing dual quaternions to represent transformation matrices. This approach yields a system of multi-unit dual quaternion equations of the form $a_d \widetilde{z}_d = (-1)^{\sigma_d} \odot \widetilde{x} b$, $d = 1,\dots,p$. We propose a novel formulation for the subspace constrained least squares solution to $a_d \widetilde{z}_d = \widetilde{x} b$ to avoid discussing the unknown signs $(-1)^{\sigma_d}$ and derive the closed-form expression for the solution. We prove that when the transformation matrix equation associated with the multi-camera robot-world hand-eye calibration admits a solution, the corresponding unit dual quaternion obtained from this matrix equation constitutes a subspace constrained least squares solution for the system of multi-unit dual quaternion equations. We present an algorithm for multi-camera robot-world hand-eye calibration, using the derived closed-form subspace constrained least squares solution to the multi-unit dual quaternion equations. We introduce a correction strategy to handle real-world data scenarios where the basic assumption may not hold. Experimental results demonstrate that the proposed subspace constrained least squares solutions exhibit competitive performance compared to state-of-the-art methods in multi-camera robot-world hand-eye calibration.