数控系统中C2连续五次PH曲线插补算法研究

摘要

The traditional interpolation of polynomial curves in CNC system has the disadvantage that arclength polynomial and offset curves do not necessarily have rational form.At present,PH (Pythagorean-Hodograph) curve interpolation can be used to solve this problem.However,in the study of interpolation of PH curves, the C2 continuity of spatial PH curves can not be guaranteed,which will lead to sudden change of motion acceleration and affect the quality of processing.To solve these problems,a C2 continuous quintic PH curve interpolation algorithm was proposed.According to the tangent vector of cubic B-spline curve under the same interpolation points condition,the initial value of quaternion parameter in PH curve equation was obtained.Then the equation was established by integral of velocity curve and interpolation points.The quaternion parameter was solved iteratively after linearization,and therefore the quintic PH curve equation was determined.An interpolation simulation of space spiral curve was performed by MATLAB.The Hausdoff distance error was used to estimate the error between the actual curve and the quintic PH curve,which verifies the fitting approximation effect of the interpolation algorithm in this paper.%数控系统中传统多项式曲线插补存在弧长多项式和偏置曲线不一定具有有理形式的缺点,目前可以采用PH(Pythagorean-Hodograph)曲线插补解决,但在研究PH曲线插补时,无法保证空间PH曲线C2连续,这将导致运动加速度的突变,进而影响加工质量.针对上述问题,提出了C2连续的五次PH曲线插补算法.根据相同插值点下三次B样条曲线的切矢建立方程,得到PH曲线方程中四元数参数的迭代初值,再由速端曲线的积分和插值点间关系建立方程,线性化后迭代求解出四元数参数,从而确定五次PH曲线方程.最后通过MATLAB对空间螺旋线进行插补仿真,采用Hausdoff距离误差来估计实际曲线与五次PH曲线的误差,验证了本文插补算法的拟合逼近效果,具有一定的有效性和实用性.