Citation: Yao, Y.; Zhang, K. An
Improved Self-Born Weighted Least
Square Method for Cylindricity Error
Evaluation. Appl. Sci. 2022, 12, 12319.
https://doi.org/10.3390/
app122312319
Academic Editors: Fang Cheng,
Tegoeh Tjahjowidodo, Qian Wang
and Ziran Chen
Received: 24 October 2022
Accepted: 25 November 2022
Published: 1 December 2022
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Article
An Improved Self-Born Weighted Least Square Method for
Cylindricity Error Evaluation
Yunhan Yao and Ke Zhang *
School of Mechanical Engineering, Shanghai Institute of Technology, Shanghai 201418, China
* Correspondence: zkwy2004@126.com
Abstract:
In order to improve the stability of the evaluation results and the gross error resistance of
the algorithm in view of the widespread gross errors in geometric error evaluation, an improved self-
born weighted least square method (ISWLS) is proposed in this paper. First, the nonlinear cylindrical
axial model is linearized to establish the error equation of the observed values. We use the conditional
equations of the independent observations found as valid information to derive the weights of the
observations. The weights of the observations are subjected to least-square iteration to calculate the
error values and equation parameters. Meanwhile, the ordinal numbers of the independent sets of
equations in the observed equations are updated several times. By updating the ordinal number
information of the conditional equations, the influence of gross error data on the solution of the
equations is minimized. Through a series of experiments, the algorithm is proved to have a strong
resistance to gross differences, and operation time is shorter. According to the evaluation results of
cylindricity error, the uncertainty of cylindricity error was calculated by the Guide to the expression
of uncertainty in measurement method (GUM)and the Monte Carlo method (MCM). Experiments
show that the uncertainty results of the MCM method can verify the results assessed by the GUM
method, which proves that the results of the ISWLS method are effective and robust.
Keywords:
cylindricity error; least square method; gross error; improved self-born weighted least
square method
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), maximum
inscribed 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 same
Appl. Sci. 2022, 12, 12319. https://doi.org/10.3390/app122312319 https://www.mdpi.com/journal/applsci
