辛普森积分法在双NURBS曲线随动插补中的应用

江本赤; 王建彬; 苏学满

doi:10.13433/j.cnki.1003-8728.20180099

辛普森积分法在双NURBS曲线随动插补中的应用

doi: 10.13433/j.cnki.1003-8728.20180099

安徽工程大学机械与汽车工程学院, 安徽芜湖 241000

基金项目:

安徽高校优秀青年人才支持计划重点项目 gxyqZD2016465

安徽工程大学引进人才科研启动基金项目 2016YQQ014

安徽省教育厅自然科学研究重点项目 KJ2015A392

详细信息

作者简介:
江本赤(1979-), 副教授, 博士, 研究方向为智能制造技术、虚拟现实技术等, benchi_ahpu@163.com

中图分类号: TP273
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- 文章访问数: 602
- HTML全文浏览量: 147
- PDF下载量: 185
- 被引次数: 0
出版历程
- 收稿日期: 2017-10-30
- 刊出日期: 2019-01-05

Application of Simpson Integration Method in Dual-NURBS Curve Follow-up Interpolation

School of Mechanical and Automotive Engineering, Anhui Polytechnic University, Anhui Wuhu 241000, China

摘要

摘要: 提出一种精确计算插补步长的双NURBS曲线随动插补算法。首先由曲面数控加工的离散刀位数据分别拟合出刀尖点和刀轴点NURBS曲线，并建立两条曲线插补参数间的随动关系模型; 然后采用辛普森积分法计算出曲线的总弧长，进行插补运动的加减速规划; 再以刀尖点NURBS曲线为基准确定插补参数，采用辛普森法确定各插补周期的进给步长及插补点坐标; 最后依据随动关系模型获得刀轴点NURBS曲线对应的插补参数，完成曲面加工刀路规划的刀具位姿插补。仿真实验表明，与同一参数插补法相比，参数随动法可以获得更加稳定的等距效果，便于实时控制插补过程中的刀轴位置和姿态。
- 辛普森积分 /
- 双非均匀有理B样条 /
- 参数随动 /
- 插补算法
Abstract: A following-up interpolation algorithm for dual-NURBS (Non-uniform rational B-spline) curve was proposed based on precise calculation of interpolation step-length. First, tool-tip and tool-axis NURBS curves are fitted out from the discrete cutter location data for curved surface NC machining, respectively, and after it the follow-up relation model between the two curve interpolation parameters was established. Then, Simpson integral method is used to calculate the total arc length of the curve, and the acceleration/deceleration planning is conducted for interpolation motion. And next, taking tool-tip NURBS curve interpolation parameters as benchmark, and using Simpson method to determine feeding step-length in each interpolation cycle, and the interpolation point coordinates are subsequently gained. The corresponding interpolation parameters of the tool-axis NURBS curve is acquired according to follow-up relationship model, and the tool pose interpolation of the tool-path planning of surface machining is achieved. Simulation experiment shows that compared with the same parameter interpolation method, the follow-up parameters method can obtain more stable isometric effect, which is conducive to real-time control the tool position and posture in the process of interpolation.
- Simpson integration /
- dual-NURBS /
- parameter follow-up /
- interpolation algorithm

HTML全文

图 1 双NURBS刀具路径

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图 2 刀尖点NURBS曲线

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图 3 刀轴点NURBS曲线

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图 4 自由曲面刀具路径举例

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图 5 双NURBS曲线刀路举例

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图 6 自由曲面及双NURBS曲线刀路

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图 7 两种积分法的截断误差

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图 8 曲线插补的轮廓误差

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图 9 某双NURBS曲线的曲率变化

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图 10 L_tool取不同数值时的双NURBS曲线

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图 11 双NURBS插补参数间的关系

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图 12 两种方法所得的刀具位姿插补效果

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图 13 两条曲线上对应点的间距变化

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图 14 两种插补加工算法所得的切削纹理

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表 1 刀尖点及法向矢量数据

序号	刀尖点{O_i}			法向矢量{N_i}
序号	x	y	z	i	j	k
1	0	4.65	4.97	0.14	0.33	0.93
2	1.97	4.65	4.79	0.02	-0.04	0.99
3	3.96	4.65	4.91	-0.13	-0.29	0.94
4	5.91	4.65	5.38	-0.29	-0.40	0.86
5	7.78	4.65	6.05	-0.01	-0.43	0.90
6	9.50	4.65	5.65	0.34	-0.29	0.89
7	11.39	4.65	4.80	0.44	-0.10	0.88
8	12.96	4.65	3.84	0.57	0.01	0.81
9	14.77	4.65	3.08	0.16	0.08	0.98
10	16.76	4.65	3.25	-0.30	0.07	0.94
11	18.51	4.65	4.14	-0.56	0.01	0.82
12	20.00	4.65	5.36	-0.68	-0.07	0.72

下载: 导出CSV

表 2 刀轴点及双NURBS曲线的控制顶点

序号	控制顶点{P_i}			刀轴点{Q_i}			控制顶点{R_i}
序号	x	y	z	x	y	z	x	y	z
1	0	4.65	4.97	0.29	5.31	6.83	0.29	5.31	6.83
2	2.28	4.65	4.68	2.01	4.56	6.78	2.20	4.42	6.78
3	3.89	4.65	4.88	3.68	4.05	6.80	3.59	4.03	6.74
4	5.94	4.65	5.25	5.31	3.84	7.11	5.63	3.78	7.09
5	7.68	4.65	6.33	7.74	3.78	7.85	7.69	3.71	8.16
6	9.63	4.65	5.62	10.19	4.06	7.43	10.16	4.02	7.49
7	11.35	4.65	4.89	12.28	4.43	6.57	12.27	4.47	6.67
8	12.98	4.65	3.80	14.11	4.67	5.48	13.79	4.59	5.66
9	14.72	4.65	2.86	15.09	4.81	5.05	15.05	4.85	4.92
10	16.86	4.65	3.19	16.15	4.80	5.15	16.32	4.80	5.14
11	18.32	4.65	3.87	17.38	4.68	5.79	17.31	4.70	5.65
12	20.00	4.65	5.36	18.63	4.50	6.81	18.63	4.50	6.81

下载: 导出CSV

参考文献(15)

[1]	Laseme A, Xue D Y, Gu P H. Recent development in CNC machining of freeform surfaces:a state-of-the-art review[J]. Computer-Aided Design, 2010, 42(7):641-654 doi: 10.1016/j.cad.2010.04.002
[2]	Li H Y, Chen L J. Study on five-axis NURBS path generation method and direct-interpolation for CNC system[J]. Applied Mechanics and Materials, 2014, 621:162-166 doi: 10.4028/www.scientific.net/AMM.621
[3]	Mohan S, Kweon S H, Lee D M, et al. Parametric NURBS curve interpolators:a review[J]. International Journal of Precision Engineering and Manufacturing, 2008, 9(2):84-92 doi: 10.1007%2Fs00170-012-4179-z
[4]	Wang Y Q, Liu H B, Yu S N. Curvature-based real-time NURBS surface interpolator with look-ahead ACC/DEC control[J]. Mathematics in Computer Science, 2012, 6(3):315-326 doi: 10.1007/s11786-012-0126-5
[5]	Yang J X, Altintas Y. Generalized kinematics of five-axis serial machines with non-singular tool path generation[J]. International Journal of Machine Tools and Manufacture, 2013, 75:119-132. doi: 10.1016/j.ijmachtools.2013.09.002
[6]	Sevilla R, Barbieri E. NURBS distance fields for extremely curved cracks[J]. Computational Mechanics, 2014, 54(6):1431-1446 doi: 10.1007/s00466-014-1067-4
[7]	吴闯, 吴继春, 杨世平, 等.基于NURBS曲线的S型级数式速度规划算法[J].计算机集成制造系统, 2015, 21(12):3249-3255 http://d.old.wanfangdata.com.cn/Periodical/jsjjczzxt201512017 Wu C, Wu J C, Yang S P, et al. S-type series velocity planning algorithm based on NURBS curves[J]. Computer Integrated Manufacturing Systems, 2015, 21(12):3249-3255(in Chinese) http://d.old.wanfangdata.com.cn/Periodical/jsjjczzxt201512017
[8]	Chen L J, Li H Y. 5-axis interpolation and synchronous servo process with NURBS method[J]. International Journal of Advancements in Computing Technology, 2011, 3(2):137-145 doi: 10.4156/ijact
[9]	黄翔, 曾荣, 岳伏军, 等.NURBS插补技术在高速加工中的应用研究[J].南京航空航天大学学报, 2002, 34(1):82-85 doi: 10.3969/j.issn.1005-2615.2002.01.018 Huang X, Zeng R, Yue F J, et al. Application research on NURBS interpolation technique in high speed machining[J]. Journal of Nanjing University of Aeronautics & Astronautics, 2002, 34(1):82-85(in Chinese) doi: 10.3969/j.issn.1005-2615.2002.01.018
[10]	Langeron J M, Duc E, Lartigue C, et al. A new format for 5-axis tool path computation, using Bspline curves[J]. Computer-Aided Design, 2004, 36(12):1219-1229 doi: 10.1016/j.cad.2003.12.002
[11]	Bi Q Z, Wang Y H, Zhu L M, et al. An algorithm to generate compact dual NURBS tool path with equal distance for 5-Axis NC machining[M]//Liu H H, Ding H, Xiong Z H, et al. Intelligent Robotics and Applications. Berlin, Heidelberg: Springer, 2010: 553-564
[12]	Yuen A, Zhang K, Altintas Y. Smooth trajectory generation for five-axis machine tools[J]. International Journal of Machine Tools and Manufacture, 2013, 71:11-19 doi: 10.1016/j.ijmachtools.2013.04.002
[13]	张君, 张立强, 张守军, 等.面向五轴加工的等距双NURBS刀具路径同步插补方法[J].计算机集成制造系统, 2015, 21(6):1523-1528 http://d.old.wanfangdata.com.cn/Periodical/jsjjczzxt201506014 Zhang J, Zhang L Q, Zhang S J, et al. Isometric double-NURBS synchronous interpolation algorithm for five-axis machining[J]. Computer Integrated Manufacturing System, 2015, 21(6):1523-1528(in Chinese) http://d.old.wanfangdata.com.cn/Periodical/jsjjczzxt201506014
[14]	韩江, 江本赤, 夏链, 等.基于轮廓关键点的B样条曲线拟合算法[J].应用数学和力学, 2015, 36(4):423-431 http://d.old.wanfangdata.com.cn/Periodical/yysxhlx201504010 Han J, Jiang B C, Xia L, et al. A B-spline curve fitting algorithm based on contour key points[J]. Applied Mathematics and Mechanics, 2015, 36(4):423-431(in Chinese) http://d.old.wanfangdata.com.cn/Periodical/yysxhlx201504010
[15]	蔡耀雄, 任全伟, 庄清渠.一类四阶微积分方程的四阶差分格式[J].数值计算与计算机应用, 2014, 35(1):59-68 http://d.old.wanfangdata.com.cn/Periodical/szjsyjsjyy201401008 Cai Y X, Ren Q W, Zhuang Q Q. Fourth-order difference method for a class of fourth-order integro-differential equations[J]. Journal on Numerical Methods and Computer Applications, 2014, 35(1):59-68(in Chinese) http://d.old.wanfangdata.com.cn/Periodical/szjsyjsjyy201401008