# An Improved Self-Born Weighted Least Square Method for Cylindicity Error Evaluation
## 1 Introduction  
Cylindricity error is an important basis for the acceptance of shaft parts. Accurate cylindricity error evaluation not only provides a reliable guarantee for improving the machining accuracy and assembly accuracy of parts but is also a prerequisite for stably improving production efficiency [1]. The spatial coordinate information of the measured point is measured, and the data are analyzed by an error evaluation algorithm to calculate the cylindricity error of the part [2]. The cylindricity error evaluation methods include the minimum zone cylinder method (MZC), least square cylinder method (LS), maximumscribed cylinder method (MIC), and minimum circumscribed cylinder method (MCC). The minimum zone cylindrical method satisfies the minimum condition defined by the cylindricity error and is recognized as an arbitration method in case of inconsistent errors. Since it is an unconstrained nonlinear optimization problem, it cannot be solved directly by a computer. The least square method is widely used in the field of error evaluation by instruments such as the coordinate measuring machine (CMM), which has the advantages of mature theory, simple calculation, and stable evaluation results [3]. Since least square does not meet the minimum conditions defined by international standards, more in-depth research is required. In recent years, many scholars have successfully applied the genetic algorithm [4], ant colony algorithm [5], artificial immune algorithm [6], artificial fish swarm algorithm [7], particle swarm algorithm [8], and other intelligent algorithms in geometric error assessment. In the field of geometric error evaluation, good results have been achieved. At the sametime, more and more algorithms are proposed according to the characteristics of cylindricity error evaluation. [PERSON] et al. [9] proposed a cylindricity error evaluation method based on the sequential quadratic programming algorithm, which uses coordinate simplification and the sequential quadratic programming algorithm to improve the accuracy of cylindricity error evaluation. [PERSON] et al. [10] established an improved integrated learning particle swarm optimization algorithm applied to the cylindricity error evaluation problem to solve the local and globally optimal solutions of the cylindricity error evaluation results. [PERSON] et al. [11] proposed a cylindricity error evaluation method based on incremental optimization. [PERSON] et al. [12] used the Hybrid Greedy Differential Evolution-Sine-Cosine Algorithm (HGSCADE) to solve optimization problems and evaluate cylindricity errors. [PERSON] et al. [13] proposed a step-acceleration-based optimization algorithm to solve the efficiency problem of the crankshaft cylindricity error evaluation algorithm. The above methods for geometric error assessment have achieved good assessment accuracy, but the assessment methods are complicated, and further research and improvement are needed to improve the stability and robustness of the methods. Based on the analysis of the observed data in production and experiments, statisticians observe that the probability of gross deviations is 1% to 10% of the total [14]. These gross errors seriously affect the error evaluation results. In the field of geodesy, to eliminate or attenuate the effect of gross errors on parameter estimation, [PERSON] et al. [15] proposed robust estimation to detect and eliminate gross errors at long distances. [PERSON] et al. [16] discussed the method of gross difference localization in detail: the \"good\" and \"bad\" points of gross difference are eliminated, but the \"good\" points are often eliminated as well. [PERSON] et al. [17] studied the effectiveness of robust estimation in weakening and eliminating gross variances. The self-born weighted least square (SWLS) method is a robust estimation method with excellent robustness [18], which can effectively reduce the effect of gross differences, and the method is mainly applied in the field of mapping.