## 1 Introduction  
Robotic machining work cells offer numerous advantages, including spacious working spaces, enhanced flexibility, and cost-effectiveness [1, 2]. The large working spaces and flexibility of robotic ability make them suitable as a new intelligent and automated machining platform in machining large-scale products, such as turbine blades and aerospace products [3, 4]. The robotic machining system necessitates a thorough calibration procedure to enhance the precision of coordinate dimensions for improved positioning calibration and alignment of robot measurement and machining systems, which have gained increasing significance as a research area and have been extensively explored by numerous scholars in recent years [5, 6]. Calibrating the coordinate system of a robotic machining system is typically achieved through 3D vision, a laser tracker [7], and contact-based measurement methods. [PERSON] [8] introduced a two-stage method for hand-eye calibration utilizing a laser tracker. They employed the Gauss-Newton (G-N) method for iterative solutions to enhance calibration accuracy, demonstrating its effectiveness and superiority over methods that rely on full pose data. [PERSON] [9] addressed the challenge of dual robot hand-eye calibration by presenting an innovative approach to solving the equation AXB = YCZ with unknown variables. This method combines two simultaneous calibration techniques for pose parameters: one using a closed-form approach based on the Kronecker product and the other employing a numerical iterative approach with the stochastic variance reduced gradient algorithm (SVRG). [PERSON] [10] also introduced a two-step geometric calibration method suitable for freeform surfaces as calibration objects. A coarse-to-fine calibration approach is proposed to simultaneously calibrate the hand-eye pose and kinematic parameters. However, if a calibration method cannot establish an initial solution through a closed-form approach and instead directly resorts to an iterative method, with an arbitrary initial solution chosen for computation, it may lead to non-convergence of the algorithm or convergence to only a local optimum. Both [PERSON] [11] and [PERSON] [12] proposed a method for calibrating the tool frame and the workpiece frame, primarily employing a ruby probe to satisfy the requirements for grinding complex workpieces with a robot. Solving the parameters of the nonlinear system of equations resulting from the sphere-to-sphere calibration method can be a challenging task. Transforming the equation set into a nonlinear least-squares problem, the function may not always be convex. The calibration of the above method ultimately transforms into solving a least squares problem. In solving nonlinear least squares problems, Newton method and quasi-Newton methods [13; 14; 15] are the most classical approaches for solving systems of nonlinear equations Nevertheless, such methods still fall under the category of local optimization techniques. When the function is convex, these methods deliver excellent solving performance, but in the case of non-convex functions, the solution process may converge to a local convergence and fail to achieve the global convergence. The global convergence algorithm for non-convex functions addresses global optimization problems. Non-convex global optimization algorithms can be categorized into two main groups. The first category consists of deterministic global optimization algorithms, such as the parameter-filled method [16], branch-and-bound algorithms [17], and the DIRECT algorithm [18; 19; 20], which do not rely on randomness. The second category comprises stochastic global optimization algorithms, which introduce randomness to facilitate global exploration and escape from local optima. These stochastic optimization algorithms are typically based on heuristic techniques, including the multistart algorithm [21], simulated annealing algorithm [22], firefly algorithm [23], genetic algorithm [24], grey wolf algorithm [25], whale algorithm [26], among others.