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Hand-eye calibration aims at determining the unknown rigid transformation between the coordinate systems of a robot arm and a camera. Existing hand-eye algorithms using closed-form solutions followed by iterative non-linear refinement provide accurate calibration results within a broad range of robotic applications. However, in the context of surgical robotics hand-eye calibration is still a challenging problem due to the required accuracy within the millimetre range, coupled with a large displacement between endoscopic cameras and the robot end-effector. This paper presents a new method for hand-eye calibration based on the adjoint transformation of twist motions that solves the problem iteratively through alternating estimations of rotation and translation. We show that this approach converges to a solution with a higher accuracy than closed form initializations within a broad range of synthetic and real experiments. We also propose a stereo hand-eye formulation that can be used in the context of both our proposed method and previous state-of-the-art closed form solutions. Experiments with real data are conducted with a stereo laparoscope, the KUKA robot arm manipulator, and the da Vinci surgical robot, showing that both our new alternating solution and the explicit representation of stereo camera hand-eye relations contribute to a higher calibration accuracy.

Adjoint Transformation Algorithm for Hand–Eye Calibration with Applications in Robotic Assisted Surgery