• Chinese Journal of Lasers
  • Vol. 50, Issue 14, 1404007 (2023)
Hongfang Chen*, Xingchen Yang, Ao Zhang, and Zhaoyao Shi
Author Affiliations
  • Beijing Engineering Research Center of Precision Measurement Technology and Instruments, Faculty of Materials and Manufacturing, Beijing University of Technology, Beijing 100124, China
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    DOI: 10.3788/CJL230440 Cite this Article Set citation alerts
    Hongfang Chen, Xingchen Yang, Ao Zhang, Zhaoyao Shi. Application of L‑M Algorithm with Trust Region Radius Strategy in Laser Tracer Multi‑Station Measurement[J]. Chinese Journal of Lasers, 2023, 50(14): 1404007 Copy Citation Text show less

    Abstract

    Objective

    High-precision CMM (coordinate measuring machine) is a key equipment for product quality control in high-end precision manufacturing fields such as aerospace, shipbuilding, and engineering machinery, and so on. At present, the primary CMM geometric error detection means are some physical reference methods based on laser interferometers, ballbars, and step gauges. However, the above measurement methods are complicated, and the operation is cumbersome and time-consuming. The model of synthetic error has a certain approximation, and the test conditions are not entirely consistent with the working conditions of CMM. The measuring range of laser tracers is less than 20 m, the measuring accuracy can reach 0.2 μm+0.3 μm/m, and the measuring efficiency is close to that of laser tracker. Therefore, the laser tracer is more suitable for the measurement and compensation of CMM and high-grade CNC (computerized numerical control) equipment volume error. In order to solve the problem of slow convergence of Levenberg-Markuardt (L-M) algorithm in the application of laser tracer multi-station measurement, the L-M algorithm based on the strategy of adding trust region radius was proposed to effectively achieve calibration of laser tracing multi-station measurement technology for CMM.

    Methods

    The solution of the laser tracer multi-station measurement system model was divided into two parts: one is the self-calibration of the laser tracer station, the other is to solve the actual coordinates of the planned measurement points within the CMM space. The L-M algorithm which adds trust region radius strategy was used to solve the laser tracer multi-station measurement system model. Based on L-M algorithm, the determination of the trust region was conducted by adding the positive parameters of the search direction, and the optimal iteration direction and iteration step size were determined, making the algorithm can quickly converge to the global domain optimal solution. The process of trust region determination was to first define the quadratic function, and then consider the ratio of the increment of the quadratic function value to the target function value based on the current positive parameters to determine whether the selection of positive parameters is reasonable. When the absolute value of the ratio is large, the positive parameters should be obtained the smaller to increase the modulus length of the search direction. When the absolute value of the ratio is small, the value of the positive parameter should be increased to limit the modulus length of the search direction. The iterative result can be less than the tolerance error of the laser tracer multi-station measurement system by continuously adjusting the positive parameters through the selection rules of trust region radius, realizing global domain convergence to self-calibrate the laser tracer station and obtain the actual coordinates of the planned measurement points within the CMM space.

    Results and Discussions

    The built laser tracer multi-station system was shown in Fig. 1. The measurement paths were planned based on laser tracer multi-station measurement methods. The L-M algorithm and the L-M algorithm with trust region radius strategy were used to solve this laser tracer multi-station measurement system model. The experimental results show that the average number of iterations in the self-calibration process of laser tracer station based on L-M algorithm is 1553, and the average number of iterations of the L-M algorithm with trust region radius strategy is 9. The latter algorithm greatly improves the convergence speed. The Val parameter is used to represent the accuracy of the algorithm when solving the actual coordinates of the planned measurement points within the CMM space. The average accuracy of the L-M algorithm is 2.30×10-7 mm with a standard deviation of 2.78×10-7 mm, and the average accuracy of the L-M algorithm with trust region radius strategy is 8.75×10-10 mm with a standard deviation of 7.47×10-10 mm. It could be seen that the latter algorithm improves the measurement accuracy.

    Conclusions

    The measurement data were obtained by establishing a laser tracer multi-station measurement system, the nonlinear equations were established based on the spatial distance formula, and the objective function was established through the least square idea. The traditional L-M algorithm is optimized by introducing the trust region radius strategy. By adjusting the positive parameters of the search direction, the trust region determination is determined, so the optimal iteration direction and iteration step size for each iteration are determined. Thus, the algorithm can rapidly converge to the optimal solution in the global domain. And the high-efficiency and high-precision of the volume error measurement of CMM planned measurement points based on the laser tracer multi-station measurement technology can be realized.

    Hongfang Chen, Xingchen Yang, Ao Zhang, Zhaoyao Shi. Application of L‑M Algorithm with Trust Region Radius Strategy in Laser Tracer Multi‑Station Measurement[J]. Chinese Journal of Lasers, 2023, 50(14): 1404007
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