## 1 Introduction
[15] used a vision-based fast base frame calibration method to obtain the relative pose of the coordinating robot by marking images to meet real-time calibration requirements. Experimental results showed that the accuracy of this calibration method reached 2 mm and 0.1\({}^{\circ}\). To sum up, \(\mathbf{X}\), \(\mathbf{Y}\), and \(\mathbf{Z}\) in these methods are obtained step by step, and there will be cumulative errors in the calibration process, thus affecting the reliability of the calibration results. In order to reduce the errors of the above calibration methods, [PERSON] et al. [21] proposed a linear approximation iterative algorithm to solve the matrix equation \(\mathbf{AXB}\) = \(\mathbf{YCZ}\), achieving simultaneous calibration of hand-eye, tool-flange, and robot-robot for a dual-robot system. [PERSON] et al. [22] and [PERSON] et al. [23] used iterative and probabilistic methods to simultaneously solve \(\mathbf{X}\), \(\mathbf{Y}\), and \(\mathbf{Z}\) in the multi-robot system, respectively. However, it takes a long time for the iterative method to calculate, and the probabilistic method is seriously affected by noise interference. [PERSON] et al. [12] proposed a new dual-robot calibration method by combining the closed-form method based on the Kronecker product and the iterative method of the convex function optimization problem. This method improved the efficiency and accuracy of iteration. [PERSON] et al. [24] proposed a calibration problem based on dual quaternion and singular value decomposition algorithm, and simulation and experiment verified higher calibration accuracy of this method. [PERSON] et al. [25] proposed a combined solution to solve the unknown parameters in the \(\mathbf{AXB}\) = \(\mathbf{YCZ}\) equation based on the dual quaternion closed form solution and Levenberg Marquardt (LM) iterative solution, realizing the calibration of the dual-robot system in orthopedic surgery. Although these non-contact calibration methods can be automated, the dual-robot system using this method is complex with high cost, and sensor measurement errors will cause calibration errors. These problems limit its batch application in industrial production. Contact calibration uses special tools for auxiliary positioning and calculates the relative positional relationship of the base coordinate system. [PERSON] et al. [26] proposed a dual-robot base coordinate calibration method only using a series of \"handclasp\" operations and corresponding joint information. Quaternion and Lagrange multiplier methods are used to carry out the iteration of the orthogonal normalization rotation matrix. However, when these methods are used, the calculation process is complicated, and manual operation will enlarge the calibration error. [PERSON] et al. [27] proposed a method using the unit quaternion to calibrate the robot base coordinate system, which improved the orthogonality of the rotation matrix, and this method measured five different tool center points (TCPs), but the TCP operation was time-consuming and inaccurate. [PERSON] et al. [8] used the three-point calibration method to complete the calibration of the dual-robot base coordinate system for the welding process. By adding a calibration bar at the end of the dual robots, the dual robots were driven to position at three points in space, and the pose transformation matrix of the dual robots was solved. [PERSON] et al. [28] realized the conversion between the base coordinates of the dual robots by measuring the coordinates of three non-collinear calibration points. They proposed an optimization method based on Lie algebra exponential mapping to ensure the orthogonality of the rotation matrix. [PERSON] and [PERSON] [29] proposed a calibration method based on three non-collinear points to establish a new coordinate system, which completes the calibration through transformations between coordinate systems. [PERSON] et al. [30] proposed a three-point calibration method using laser sensor and buzzer. [PERSON] et al.