## 5 Calibration Experiment and Algorithm Comparison
### Data Solving and Analysis  
We applied the LM-D algorithm to address the data and conducted a comparative analysis with the L-M algorithm, DIRECT algorithm, and SA algorithm. Each of these methods holds distinct typicality. The L-M algorithm embodies a conventional approach for local optima, the DIRECT algorithm serves as a direct global optimization algorithm, and the SA algorithm stands as an intelligent optimization algorithm. The computer employed for the experiment is an ASUS ROG Zephyrus M16 equipped with an Intel  
Figure 8: Robot TCP calibration experiment using the sphere-to-sphere method. Figure 9: The relation between X, Y parameters and function values of TCP. Core i7-12700H CPU and 16 GB of memory. To ensure solution efficiency, a time constraint is imposed. We set the maximum iteration time to T\({}_{\text{max}}\) = 3600 s. If an algorithm surpasses the 3600 s threshold, it is deemed a timeout, prompting the termination of the iteration, and the final result is generated. Following the establishment of initial parameters, we initiate the iterative calculation process. To further validate the effectiveness of the LM-D algorithm, we conducted experiments using data 1, 2, and 3. These data sets are representative of typical cases. Data 1 is used to verify the convergence and efficiency of various algorithms when there is only a single local optimum. Data 2 and 3 represent cases with multiple local optima. In data 2, the multiple local minima are closely spaced, and the LM-D algorithm demonstrates global convergence while showing faster efficiency compared to the SA algorithm. Data 3, on the other hand, involves multiple local minima that are more widely spaced. Due to the significant differences in the solving principles of the above algorithms, we have plotted separate images for the convergence of each algorithm. The convergence of four different algorithms for processing data 1, 2, and 3 is shown in Figure 10. In this case, only the LM-D algorithm succeeds in achieving global convergence, which highlights its stronger global convergence capability compared to other algorithms. Figure 10: _Cont._  
The optimization results and comparisons for data 1, 2, 3 are presented in Table 1. The L-M algorithm is characterized by its fast convergence speed; however, the final result may either converge globally or become stuck in a local optimum. On the other hand, the LM-D algorithm has a slightly slower average iteration speed compared to the L-M algorithm but is faster than the DIRECT algorithm and the SA algorithm. Additionally, the LM-D algorithm also achieves direct convergence to the global optimum. Nevertheless, employing the DIRECT algorithm and the SA algorithm resulted in excessively prolonged iteration times, and the iteration outcomes failed to converge to the optimal solution. In the case of the DIRECT algorithm, the absence of a more precise initial solution necessitates the definition of a larger constraint range to encompass the global optimum. Boundaries are set as bounds = [0 1000; \(-\)500 500; \(-\)500 500; 0 2000; 0 500; 0 500]. Solving within an expanded constraint space inevitably demands more computational resources, leading to extended computation times. Furthermore, the asymptotically ineffective characteristic of DIRECT restricts convergence. Consequently, within the T\({}_{\text{max}}\) time, the final results fall short of attaining the optimal F(X\({}_{i}\)) compared to the LM-D algorithm, indicating the failure of the algorithm to converge to the optimum solution within the preset time. The SA algorithm, while possessing global optimization capabilities, relies on a Monte Carlo based random approach, introducing inherent uncertainty. Balancing computational efficiency and global convergence during the solving process is challenging. If parameters are adjusted to  
Figure 10: Convergence illustration for data 1, 2, 3 sets using different algorithms: (**a**) Convergence of the three data sets calculated using the L-M algorithm.