A novel method for calculating interpolation points of NURBS curves based on chord length-parameter ratio

Abstract

Abstract Non-uniform rational B-spline (NURBS) curve has been widely used in computer numerical control machining due to its superior properties, such as global smoothness and local support. However, because of the nonanalytic relationship between the spline parameter and the arc length, it can be challenging to obtain accurate parameters when the arc length given. Approximation methods are commonly used because of their simplicity, but they can result in feedrate fluctuations that are difficult to avoid. This paper presents a feedback method to reduce feedrate fluctuations while maintaining simplicity. To obtain the initial parameter value of the next interpolation point, the chord length-parameter ratio (CPR) is defined first, and Newton's divided difference interpolation polynomial is applied to predict the CPR of the current interpolation point based on the CPRs of the historical interpolation points. Then, the initial parameter is corrected using the Second-order Runge-Kutta method with the compensation length, which is derived according to the tangent and chord directions at the initial parameter. The proposed method only requires two calculations of the first derivative of the NURBS curve, which is comparable to the computational complexity of most approximation methods. Numerical simulations are conducted to compare the performance of the proposed method with several approximation methods and a recent feedback method. The results demonstrate that the proposed method performs exceptionally well in terms of both efficiency and accuracy.