CNC工具機之NURBS插值器速度控制規劃

Abstract

隨著加工工件的複雜化，以及自由曲線曲面的廣泛使用，傳統CNC工具機以直線與圓弧來近似產生刀具路徑會產生大量短線短、傳輸量過大、速度不連續及機台震動等缺點，這些並不符合高精高速加工的需求；此外NURBS ( Non-Uniform Rational B-Spline)在CAD/CAM/CAE上已是標準的設計與資料交換格式，而且使用NURBS可以消除上述的缺點，故能接收NURBS-code的CNC工具機已成為未來發展的趨勢。 由於工具機進行加工時除了考量加工工件的品質外，也必須考量機台的動態性能，因此良好的速度規劃可以使工具機滿足機台之加速度、急衝度限制，避免機台產生震動的情形，而且可以達到精準的定位，甚至改善加工工件的表面精度，故本論文的目的為針對CNC工具機之NURBS加工發展出一套速度規劃理論，使其可以滿足上述之條件。 首先，本論文在off-line時針對NURBS曲線作一次掃描，找出曲線上曲率較大之點以及C0連續之點(稱為分段點)，依據弦誤差限制、向心加速度限制、向心急衝度限制與分軸速度差限制求出該分段點的適當進給速率，並且依據數值積分之方法─Simpson’s rule─估算兩鄰近分段點間(稱為NURBS block)的曲線長度；接著由掃描之結果，採用Sine–curve來做速度規劃，並使其滿足切線加速度與切線急衝之度限制；由於在off-line時已建立滿足上述限制條件的速度曲線，故在on-line時僅需依據速度命令，便可由插值演算法─PCI法─得到空間位置命令以輸出給伺服迴路，其計算上相當簡單且迅速，可以滿足加工的即時性。 最後，本論文以三條自由曲線進行電腦模擬並驗證其可行性，由模擬的結果可知，弦誤差、分軸速度、分軸加速度和分軸急衝度等加工性能指標皆能滿足加工之限制條件。

The feedrate is one of the most important factors to machining efficiency and quality. The feedrate scheduling of CNC machining must consider not only chord error but also acceleration and jerk limitation. In this regard, this paper propose a method to construct a velocity profile constrained by desired chord error, max acceleration and jerk limitation, so that one can easily get the needed feedrate at any moment. First, scanning the NURBS curve in the off-line and finding out the points of large curvature and C0 continuity (named as breakpoint) is performed. Then, the suitable feedrate at breakpoints in accordance with chord error limitation, normal acceleration and normal jerk limitation is evaluated. And, the length between two adjacent breakpoints (named as NURBS block) is estimated by using Simpson’s rule of numerical integration. Second, the velocity profile is constructed by using the result of NURBS curve scanning and satisfied the tangent acceleration and jerk limitation. Because of the erection of the velocity profile in the off-line, the position command of servo loop is just generated by interpolator according to the desired feedrate command in the on-line. It is simple and efficient that the calculation for interpolation and curve point generation is extremely, as well as the immediateness for CNC machining can be achieved. Finally, in this paper three free-form NURBS curve is applied to simulation and verification the feasibility of theory. From the result of simulation, the chord error and the velocity component, acceleration component and jerk component on the x- and y-axis of the interpolated data are all satisfied the limitations of CNC machining.