Abstract: Aiming at the problems of feedrate fluctuation and computational efficiency in Non-Uniform Rational B-Spline curve (NURBS) interpolation,a double secant interpolation algorithm of NURBS and a fast calculation and derivation algorithm of NURBS were proposed.In the double secant interpolation algorithm,the secant lines which were generated by the required length for zero-feedrate fluctuation were used to approximate the desired NURBS curve based on the first interpolation by the second order Taylor method.In the fast calculation and derivation algorithm,all non-zero derivatives of the numerators and denominators at knots in NURBS expression were calculated and stored in advance,and any order derivative of NURBS could be fast calculated in real-time interpolation by Taylor formula,thus the computations of B-spline basis functions could be avoided,and the computational efficiency could be improved.A NURBS interpolator based on the proposed algorithms was implemented on a CNC platform developed by the authors,and the feasibility and applicability were verified by simulation analysis and experimental results.

Key words: non-uniform rational B-spline, interpolation, computer numerical control, feedrate fluctuation, feedrate

摘要： 针对NURBS插补中的速度波动与计算效率两大问题,提出无速度波动的NURBS割线二次插补算法与NURBS快速求值求导算法。在割线二次插补法中,采用二阶Taylor法对NURBS曲线进行一次插补,在此基础上使用根据无速度波动要求给定插补步长的割线逼近原曲线,从而计算插补点,以消除因截断误差和弦线逼近偏差引起的速度波动。在NURBS快速求值求导算法中,预先计算并存储NURBS表达式中分子式与分母式在节点值处的各阶非零导数,实时插补中使用Taylor公式快速计算NURBS各阶导数,从而避免计算B样条基函数,达到提高计算效率的目的。在自主研发的数控平台上实现了基于所提算法的NURBS插补器,并通过仿真分析与加工实验验证了该插补器是有效且可行的。

关键词: 非均匀有理B样条, 插补, 数控系统, 速度波动, 进给速度