以弧長為參數的 NURBS 曲線插值器設計

Abstract

現今的 CNC 工具機已經可以直接接受參數式曲線進行插補，已解決 CAD/CAM 與 CNC 之間大量資料傳輸的問題，並可進一步改善其加工品質。然而，由於 NURBS 曲線本身複雜的數學表示式，使得 NURBS 插值器需要以數值方法求得每一個插值點的參數值，這對即時環境下而言是相當大的負擔。本論文提出以預處理器在非即時環境下對 NURBS 曲線資訊進行預處理，參數區間依所設定的誤差容許值 被分割成許多小區間，以數值方法計算各段弧長值與總長度，建立原參數 與弧長 的函數 ，再利用重參數化的流程步驟產生以弧長為參數的 NURBS 曲線表示式，一方面充分利用了總長計算過程的中介資訊，一方面在即時插補時，可以快速地求得每一個位置命令點，避免了複雜的 NURBS 曲線微分項與疊代計算。 本論文所提出之插值方法與其他既有之插值方法經過模擬測試比較，顯示本方法在進給率均方根誤差的表現上最小；而平均即時插值計算的時間，本論文之方法也比其他插值方法更有效率，最多有近五倍之快速，且本論文之方法在數值計算上相當地穩定，亦不會有累積誤差的問題。

Modern CNC systems are designed with the function of arbitrary parametric curves to save massive data communication between CAD/CAM and CNC systems and improve their machining quality. However, the computation loads of NURBS interpolator by numerical methods are relatively large. In this paper, a real-time NURBS motion command generator is presented; this method efficiently integrates the data processing of a NURBS path in a CNC controller, from pre-processing to real-time interpolation. The key point is to generate the arc-length parameterized NURBS curves, in the real-time NURBS path interpolation, the new setting path points can be calculated fast and directly using the arc-length parameterized NURBS curves without the need for any time-consuming computation of NURBS derivatives and iteration. The comparative results of different interpolation algorithms show that the mean-square feedrate deviation of the proposed method is the smallest among all algorithms. Furthermore, comparison of the average computing times of different algorithms show the excellent performance of the arc-length parameterized NURBS path interpolation method – using the same CPU clock, the proposed method is approximately five times faster. The proposed method is numerically stable and does not have the problems of error accumulation.