Development of a CNC interpolation scheme for CNC controller based on Runge-Kutta method

Abstract

The parametric interpolators of modern CNC machines use Taylor's series approximation to generate successive parameter values for the calculation of x, y, z coordinates of tool positions. In order to achieve greater accuracy, higher order derivatives are required at every sampling period which complicates the calculation for contours represented by NURBS curve. In addition, this method calculates the chordal error in a given segment through estimation of the curvature neglecting a fraction of the error. In order to avoid calculating higher derivatives and make the calculations simpler, this paper proposes the classical fourth-order Runge-Kutta (RK) method for the determination of successive tool positions requiring the calculation of the first derivatives only. Furthermore, a method of estimating the chordal error on the average value of parameters at the end points of a given curve segment is proposed here that does not require the calculation of curvature at every segment. Finally, a variable feedrate interpolation scheme is designed combining the RK method of parameter calculation and the proposed method of chordal error calculation. Results show that reduced chordal error and feedrate fluctuations are achievable with the proposed interpolator compared to the conventional interpolator based on Taylor's approximation with higher order terms.