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

Abstract

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.

A Novel Robust Approach for Correspondence-Free Extrinsic Calibration

“…• the Batch1 (marked as B1) and Batch2 (B2) correspondence-free methods from [13] are used in the first experiment. • the joint method (KR) [7], the iterative methods (NLQ by [4] and ATA) [8]), the SOCP optimization method (SOCP) from [10], the global optimization method using multivariate polynomial (GPOLY 1 ) [11] and the DUAL method proposed in [2] are used in the second experiment. Regarding the initial value for iterative methods, we use the default initialization routine of NLQ and ATA, see [4] and [8].…”

“…The second category of methods solves the hand-eye calibration jointly by estimating the rotation and translation simultaneously [4], [6]- [8]. The introduction of the screw theory in [9] provides a geometric view to analyze the restrictions of the hand-eye calibration problem, which later motivated the invention of the dual quaternion approach [6].…”

“…Due to noise, neither the constraint of the unit quaternion (||q|| = 1) nor the orthogonality constraint of SO(3) can be maintained, which requires normalization or orthogonalization. To suppress the influence of noise, iterative nonlinear optimization methods [4], [8] have been proposed. However, iterative approaches rely on good initial values for accuracy and fast convergence, while obtaining favorable initial values may be difficult in practice.…”

“…The nonlinear iterative optimization starts with an initial value which is computed with the first-order method [13]. Recalling the definition of mean in (8)…”

Medical Robotics

“…14 Robotic hand-eye calibration is important when moving towards more autonomous actuation, and an improved algorithm to achieve this is proposed and evaluated based on the adjoint transformation of twist motions. 19 In a twist on the theme of this section, in Singh et al, 20 the effect that manual and robotic suturing tasks have on the prefrontal cortex is assessed with near-infrared spectroscopy. The insight provided by the end-users-surgeons in this case-is a key to successful design of robotic systems and workflows, and it has been shown that recovery time, operation time and cost can all be reduced for minimally invasive reduction of intra-articular fractures.…”