Absolute laser ranging is widely used in scientific research and industry. Several types of absolute distance measurement methods can be used, such as the time-of-flight method, phase ranging method, multiwavelength interferometry, and dual-comb ranging. The polarization laser ranging method is a phase-ranging method that establishes the functional relationship among the laser polarization, modulation frequency, and measurement distance. As laser polarization is less susceptible to interference from environmental disturbances, the polarization laser ranging method can achieve higher precision than the phase-ranging method. Laser polarization rangefinders have an irreplaceable role in fields such as assembly and inspection of large structural parts and field baseline construction. However, owing to technological monopoly and blockades, no independently developed laser polarization rangefinders are available in China. Research in universities and research institutes has mainly focused on system implementation and error analysis. The swing method is the only polarization laser ranging algorithm used in these studies. Waveform distortion would reduce the swing method accuracy; therefore, research on polarization laser ranging algorithms is necessary to improve ranging accuracy.

Based on the principle of polarization modulation ranging, the reason for waveform distortion and the influence of the local minimum point frequency accuracy on the ranging accuracy are studied. To obtain the minimum points of the demodulated signal, the swing and least square methods for polarization laser ranging are compared. The swing method is affected by waveform distortion. The least square method is widely used in curve fitting, which can effectively reduce the influence of waveform distortion and increase the accuracy by using higher-order fitting polynomials. The least square method uses all data to fit the curve; however, the polarization laser ranging algorithm focuses solely on the local minimum ranges. In terms of fitting efficiency and accuracy, it is not completely applicable to resolve the local minimum points of the demodulated signal. The moving least square (MLS) method is an optimization algorithm of the least square method that can simultaneously realize interval fitting and high-precision fitting near special points. This is suitable for high-precision curve fitting using the local minimum range data. In this study, a polarization-ranging algorithm based on the improved moving least square (IMLS) algorithm is proposed. The algorithm flow is as follows. First, the algorithm achieves global fit and determines the intervals containing the minimum points. Second, it performs curve fitting using the MLS algorithm along the curve in one of the intervals until it finds the minimum point in this interval. Then, the influence radius is selected automatically according to the mean square deviation, and the frequency corresponding to the local minimum point is calculated. Finally, the distance is calculated using two adjacent frequency values of the local minimum points. A flowchart of the improved MLS algorithm for polarization laser ranging is shown in Fig. 4.

Distance-measurement experiments are conducted to verify the accuracy of the IMLS algorithm. First, the influence of the key IMLS algorithm parameters, such as the weight function and influence radius, is tested. Thus, the influence radius is extremely important. If the value is too small, it can cause matrix singularity and fitting anomaly, as shown in Fig.3. The experimental results listed in Table 3 indicate that the higher the influence radius, the smaller the IMSL residual error with the same weight function. However, to ensure the movement characteristics of the MLS algorithm and decrease the number of calculations, the influence radius should be less than 10% of the sweep interval. The comparison results under comprehensive parameters are listed in Table 4, and indicate that the measurement mean square deviation is subject to the common influence of both the weight functions and the influence radius. The measurement mean square deviations of the normal weighted function, cubic spline function, and Gaussian function are 0.111, 0.237, and 0.177, respectively. The comparative results of the IMLS algorithm, swing method, and least square method are presented in Table 5. When the measured distance is 11.94 m, the minimum errors of the swing method, least square method, and IMLS algorithm are 0.219, 0.317, and 0.111 mm, respectively. The accuracy of the IMLS algorithm is higher than those of the swing and least square methods.

In this study, the IMLS method is applied to implement a polarization laser ranging algorithm. The experimental results show that the IMLS algorithm can reduce the impact of waveform distortion and improve the precision of the polarization ranging system. The IMLS algorithm has a higher precision than the swing and least square methods. The proposed IMLS algorithm is suitable for determining the polarization modulation range and provides a reference for research on polarization laser ranging algorithms.