慢刀伺服加工中的Zernike多项式局部拟合算法

Zernike Polynomial Local Fitting Algorithm for Slow Tool Servo Machining

针对慢刀伺服加工前后的曲面拟合问题,将Zernike多项式与移动最小二乘法结合,提出Zernike多项式局部拟合算法。使用Gram⁃Schmidt正交化构造正交基底函数,解决了拟合计算中出现的病态矩阵及矩阵求逆运算量大等问题。局部拟合中支持域半径对拟合精度影响显著,基于此提出了支持域半径优化算法。以渐进多焦点曲面、环曲面、正弦阵列面为例,采用与慢刀伺服加工关系密切的刀触点精度及拟合优度R⁃square作为评价标准,在MATLAB软件中进行了数值仿真。结果表明,Zernike多项式局部拟合算法各项标准均优于移动最小二乘法,并且算法在经过半径优化后,不仅进一步提高了精度、改善了拟合优度,还改善了刀触点误差的离散程度。

In order to solve the surface fitting problem before and after slow tool servo machining,this paper combines Zernike polynomial with moving least squares and proposes Zernike polynomial local fitting algorithm.In this paper,the Gram⁃Schmidt orthogonalization algorithm is used to construct the orthogonal basis function.This method solves the problems of the ill⁃conditioned matrix problem and the large amount of matrix inversion in the calculation process of the fitting algorithm.The support domain radius in the local fitting has a significant influence on the fitting accuracy.Based on this,the support domain radius optimization algorithm is proposed.Taking progressive⁃addition lenses,toric surface and sinusoidal array surface as examples,the accuracy of the cutting location points and the goodness of fit R⁃square,which are closely related to the slow tool servo machining,are used as the evaluation criteria.Numerical simulation is carried out in MATLAB software,the results show that the Zernike polynomial local fitting algorithm is better than the moving least squares method,and after the support domain radius optimization,the algorithm not only further improves the accuracy and the goodness of fit,but also improves the discrete degree on the cutting location points error.