﻿ 三次三角插值样条曲线的数控插补计算方法 An NC Interpolation Calculation Method of Cubic Trigonometric Cardinal Interpolation Spline

Mechanical Engineering and Technology
Vol.04 No.02(2015), Article ID:15501,7 pages
10.12677/MET.2015.42016

An NC Interpolation Calculation Method of Cubic Trigonometric Cardinal Interpolation Spline

Yihan Zhang, Aiping Song, Zuqi Liu, Lin Qiu

College of Mechanical Engineering, Yangzhou University, Yangzhou Jiangsu

Email: ydzhangyihan@sina.com

Received: Jun. 3rd, 2015; accepted: Jun. 22nd, 2015; published: Jun. 25th, 2015

ABSTRACT

The paper proposes a new spline curve called cubic trigonometric cardinal interpolation spline with adjustable shape. The spline curve realized the unification to accurately represent some common NC tool motion path curves such as straight lines, arcs, ellipses and free curves, etc. An interpolation algorithm of cubic trigonometric cardinal interpolation spline based on data sampling interpolation theory is researched. This curve interpolation method ensures the high-speed of spline curve CNC interpolation while taking into account the smooth movement, using a series of conterminous short straight-line segments to approach the given interpolation spline, and then utilizing the relation between track space and parameter space to control motion acceleration and get the whole discrete interpolation trajectory. This algorithm improves the speed and motion stability of free curve interpolation and guarantees the surface quality of surface numerical control machining.

Keywords:Data Sampling Interpolation Theory, Cubic Trigonometric Interpolation Spline, Tool Motion Path Curves, Track Space

Email: ydzhangyihan@sina.com

1. 引言

2. 三次三角插值样条曲线

(1)

(2)

(，i为曲线段号，)，为由每四个顺序控制点qi-1，qi，qi+1，qi+2所构造的分段三次三角样条曲线，所有曲线段Pi(u)有序组成的曲线P(u)称为三次三角样条曲线。

3. 三次三角插值样条曲线的插补计算方法

3.1. 三次三角插值样条曲线的插补原理

(3)

3.2. 三次三角插值样条曲线插补实现原理和步骤

4. 三次三角插值样条曲线的插补的实现方法

4.1. 插补预处理

(4)

(5)

(6)

(7)

(8)

(9)

(10)

4.2. 三次三角插值样条曲线实时插补计算

(11)

(12)

(13)

(14)

(15)

(16)

(17)

5. 误差控制

(18)

(19)

(20)

(21)

Figure 1. Xiaohegou hydrological station of the water system map

6. 结束语

1) 本文提出了一种自由曲线的简洁表达式，实现了对常见数控加工刀具运动轨迹曲线的统一精确数学表示和自由曲线的快速插补；

2) 基于数据采样插补原理，实现了三次三角插值样条曲线的实时插补计算；

3) 该算法插补弦长的波动较小，具有良好的速度稳定性从而提高加工精度。

An NC Interpolation Calculation Method of Cubic Trigonometric Cardinal Interpolation Spline. 机械工程与技术,02,151-158. doi: 10.12677/MET.2015.42016

1. 1. 宋爱平, 陶建明, 易旦萍, 等 (2015) 可调形三次三角Cardinal插值样条曲线. 计算数学, 1, 34-41.

2. 2. 黄建, 宋爱平, 陶建明, 等 (2012) 一种三次插值样条曲线的插补方法研究. 现代制造工程, 04, 46-49.

3. 3. 谭彦闻, 王小椿 (2008) 数控系统的刀具半径补偿和B样条曲线实时插补算法研究. 北京交通大学, 北京, 39- 42.

4. 4. 荆伟, 傅建中 (2007) 基于DSP的三次B样条曲线插补的实现. 浙江大学, 浙江.

5. 5. Yau, H.-T. and Wang, J.-B. (2007) Fast Bezier interpolator with real-time look ahead function for high-accuracy machining. International Journal of Machine Tools and Manufacture, 47, 66-68.

6. 6. 徐绍华, 田新诚, 方磊 (2007) 三次B样条参数曲线插补算法及其特性分析. 山东大学, 济南.

7. 7. 李芳, 刘凯, 任杰青, 等 (2013) 一种空间B样条曲线变步长离散插补算法. 制造业自动化, 9, 37-40.

8. 8. 叶丽, 谢明红 (2008) 三次Bezier曲线的插补算法及误差分析. 重庆工学院学报, 7, 34-39