A classic hand-eye system involves hand-eye calibration and robot-world and hand-eye calibration. Insofar as hand-eye calibration can solve only hand-eye transformation, this study aims to determine the robot-world and hand-eye transformations simultaneously based on the robot-world and hand-eye equation. According to whether the rotation part and the translation part of the equation are decoupled, the methods can be divided into separable solutions and simultaneous solutions. The separable solutions solve the rotation part before solving the translation part, so the estimated errors of the rotation will be transferred to the translation. In this study, a method was proposed for calculation with rotation and translation coupling; a closed-form solution based on Kronecker product and an iterative solution based on the Gauss–Newton algorithm were involved. The feasibility was further tested using simulated data and real data, and the superiority was verified by comparison with the results obtained by the available method. Finally, we improved a method that can solve the singularity problem caused by the parameterization of the rotation matrix, which can be widely used in the robot-world and hand-eye calibration. The results show that the prediction errors of rotation and translation based on the proposed method be reduced to $0.26^\circ$ and $1.67$ mm, respectively.

Although the registration of a robot is crucial in order to identify its pose with respect to a tracking system, there is no reported solution to address this issue for a hybrid robot. Different from classical registration, the registration of a hybrid robot requires the need to solve an equation with three unknowns where two of these unknowns are coupled together. This property makes it difficult to obtain a closed-form solution. This paper is a first attempt to solve the registration of a hybrid robot. The Degradation-Kronecker (D-K) method is proposed as an optimal closed-form solution for the registration of a hybrid robot in this paper. Since closed-form methods generally suffer from limited accuracy, a purely nonlinear (PN) method is proposed to complement the D-K method. With simulation and experiment results, it has been found that both methods are robust. The PN method is more accurate but slower as compared to the D-K method. The fast computation property of the D-K method makes it appropriate to be applied in real-time circumstances, while the PN method is suitable to be applied where good accuracy is preferred.

When computer vision technique is used in robotics, robotic hand–eye calibration is a very important research task. Many algorithms have been proposed for hand–eye calibration. Based on these algorithms, we introduce a new hand–eye calibration algorithm in this paper, which employs the screw motion theory to establish a hand–eye matrix equation by using quaternion and gets a simultaneous result for rotation and translation by solving linear equations. The algorithm proposed in this paper has high accuracy and stable computational efficiency and can be understood easily. Both simulations and real experiments show the superiority of our algorithm over the comparative algorithms.