## 7 Conclusions  
This paper addresses the need for robot machining of large structural components and proposes a rapid workpiece positioning method for robot machining systems. Considering the specific characteristics of the calibration of the measuring head, the LM-D algorithm was designed for solving it, striking a balance between computational efficiency and result accuracy. Ultimately, this method accurately calibrates the parameters of the probe TCP, ensuring precise workpiece coordinate system calibration, and providing a foundation for high-precision robot machining of workpieces. It has significant implications for achieving high-precision robot machining of workpieces. The following conclusions are achieved:  
Firstly, a robotic machining system with milling and polishing capabilities was designed. For structural components with arbitrary positioning, a combined calibration approach was developed, which combines the ease of visual measurement-based positioning with the precision of contact-based measurement positioning. This approach involves the use of low-cost 3D vision products for initial rough positioning of the workpiece coordinate system. After determining the approximate coordinate system of the workpiece, a rapid positioning method using robot-held contact probes is then employed for precise positioning. Secondly, a robot sphere-to-sphere contact calibration model was established, and a system of nonlinear equations for calibrating the robot's contact-based measuring probe was formulated. The research focused on numerical iterative methods for solving this nonlinear equation system. It was concluded that the robot's contact calibration model requires high global convergence and exhibits non-convex and discontinuous characteristics, allowing the establishment of convergence criteria. This provides a theoretical foundation for designing a more suitable method to solve the robot's contact-based calibration model. Thirdly, considering the practical characteristics of measuring head TCP calibration, a combined optimization algorithm, the LM-D algorithm, based on the L-M and DIRECT algorithms, was designed. This algorithm demonstrates excellent global convergence and computational efficiency. In comparison to the other algorithms, the LM-D algorithm combined optimization approach does not rely on initial values, possesses global optimization capabilities, and offers fast iteration speeds, making it effective for solving this type of nonlinear equation system problem. Fourthly, relevant robot calibration experiments were conducted, and experimental data were obtained. After employing various optimization methods to process multiple sets of experimental data and comparing the results, it is evident that the LM-D algorithm, as opposed to other algorithms, exhibits faster computational efficiency, superior global convergence, and robustness in solving TCP problems. The LM-D algorithm proposed in this paper is not limited to measuring head TCP calibration; it can also be attempted for use in similar problems. It has the potential for further expanding its application scope. The accuracy of TCP calibration is greatly influenced by the precision of the robot itself, including factors such as robot deformation under load, joint backlash, and absolute accuracy errors. All of these can reduce the accuracy of contact-based measurements. This is a significant limitation on the precision of contact-based calibration. To further enhance the accuracy of contact-based calibration, it is essential to implement a well-considered combined calibration method and optimize the algorithm for solving the workpiece coordinate system. This will be explored in further research. D.H.: Methodology, validation, resources, writing--original draft. G.Z. : Conceptualization, writing--review and editing, project administration. H.Z. : Conceptualization. H.D. : Supervision, project administration. All authors have read and agreed to the published version of the manuscript.