基于切比雪夫多项式的数控机床几何误差参数化建模  

Chebyshev Polynomials Based Parametric Modeling of Geometric Errors for CNC Machine Tools

作　　者：郭然[1] 付国强[2] 孙磊[2] 傅建中[2] 

机构地区：[1]浙江衢州广播电视大学机械系,衢州324000 [2]浙江大学机械工程学院,杭州310027

出　　处：《农业机械学报》2015年第5期336-343,共8页Transactions of the Chinese Society for Agricultural Machinery

基　　金：国家自然科学基金资助项目(51175461)

摘　　要：为了快速精确地建立机床几何误差项数学模型,提出了一种基于切比雪夫多项式的参数化建模方法。首先针对测量得到的机床基本几何项数据,将机床相应运动轴进给量转化为切比雪夫变量。其次将切比雪夫变量代入不同阶次的切比雪夫多项式得到相应的值。然后根据切比雪夫基函数值和切比雪夫变量用多元线性回归方法获得相应的系数,得到关于切比雪夫基函数的数学模型。最后将运动轴进给量与切比雪夫变量之间的转化关系代入得到基本几何误差项的数学模型。建模过程简单且易程序化,切比雪夫多项式的高逼近精度使得建立的模型精度高。将所有几何误差项参数化模型代入机床几何误差模型综合数学模型,从而可得到机床工作空间几何误差场分布。以MV-5A三轴立式加工中心为例,将各个几何误差项参数化模型代入机床几何误差模型中得到该机床综合几何误差数学模型,进而得到该机床工作空间几何误差场分布,为机床设计和误差补偿提供了理论依据。In order to establish the mathematical models of geometric errors of machine tools rapidly and precisely, one parametric modeling approach based on Chebyshev polynomials was proposed. Firstly, according to the measured data of basic geometric errors of machine tools, the displacement of the axis was converted to Chebyshev variable. Secondly, the values of Chebyshev polynomials with different orders were calculated with Chebyshev variable. Then, coefficients of Chebyshev basic functions were calculated by using multiple linear regression methods according to Chcbyshev variable and Chebyshev values, and the model about Chebyshev basic functions was obtained. At last, the mathematical models of basic geometric errors were established by inputting the conversation of the displacement of the axis and Chebyshev variable. The modeling was simple and programming; and the high approximating accuracy of Chebyshev polynomials made models precise. The comprehensive mathematical models were established by inputting parametric models of geometric errors into geometric models of machine tools. Then, the geometric error fields of working zone for machine tools were obtained. Taking the MV - 5A three-axis machine tool for an example, the parametric models of all geometric errors were inputted to geometric error model of this machine tool to obtain the comprehensive mathematical model of geometric errors. Then, geometric error fields of working zone of this machine tool were computed, which laid the foundation for machine design and the error compensation.

关 键 词：数控机床 几何误差 参数化建模 切比雪夫多项式 

分 类 号：TH161.21[机械工程—机械制造及自动化] TG659[金属学及工艺—金属切削加工及机床]