基于PH样条的五轴数控插补技术研究
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摘要
目前,具有复杂曲面外形的产品越来越多而且其加工精度要求越来越高,采用三轴联动数控机床加工这类产品时一般需经过多次装夹才能完成,加工速度和加工精度很难保证。因此在具有复杂轮廓外形零部件的高速、高精加工中,五轴联动数控机床的应用最为广泛。在加工自由曲面时,一般采用微段直线逼近曲线的方式,这种加工方法存在NC程序文件庞大导致数据传输负荷大,相邻运动段连接处速度不连续,加工质量差等。样条曲线的插补可使整条曲线的加工速度较均匀,因而可提高表面加工质量。
PH曲线是一类具有优良特性的多项式曲线,可以用于解决曲线弧长的计算问题,且保持与Bézier曲线表示的完全一致性,并且PH曲线往往产生比经典多项式曲线更光顺的轨迹等优点。因此本文主要研究五轴联动数控系统的PH曲线插补技术。
由于在五轴联动加工中,刀轴矢量出现跨象限时,需抬刀处理,加工速度和精度很难保证。为解决此问题,提出了跨象限的处理方法。在此基础上对双转台五轴联动机床的空间运动和基于RPCP的后置处理作了简单的分析和研究;提出了在插补中用预测—校正方法代替通常使用的Taylor展开方法,提高了插补的精度。针对PH曲线的性质和特点,设计、开发了四种空间PH曲线的实时插补算法,并对曲率插补方法的缺点进行了改进;
为解决PH曲线插值中的光滑过渡,提出了5次空间PH曲线的C~1和C~2连续插值方法。针对CAD/CAM系统生成的曲线曲面加工代码文件庞大、速度不连续的缺点,使用了基于数据点长度和角度的方法对数控代码进行了处理,并利用结点消去法对刀位点用PH曲线逼近,实现了微段直线插补指令向PH曲线插补指令的转换;给出了五坐标PH曲线插补的数控代码格式,通过仿真对五坐标PH样条曲线插补算法及曲面加工作了验证。另外,针对跨象限处理方法及RPCP算法,通过仿真和实验验证了其可行性。
More and more manufactured products with sculptured surfaces are expected higher accuracy at the present time. However, those products cannot be machined on the 3-axis machine in only one fixing. The processing speed and processing accuracy cannot be well ensured. So, manufacturing these complicated products in the manner of high speed and high accuracy, machines with five simultaneous motion axes are applied most widely. When machining the model created by free-formed surfaces, many tiny line segments are used to approach these surfaces. This manufacturing method has a large NC file, so it results in great amount data transmission, disconnected speed in joint of motion segments and poor manufacturing quality. Because of the processing speed is uniform in the spline interpolation, it can improve the manufacturing quality.
PH curve is a kind of polynomial curves which has a good characteristic and posses many positive properties. It can solve the problem of computing the arc length, retains full compatibility with standard Bézier represtation and tends to produce fairer loci than ordinary polynomial curves and so on. So the interpolation technology of PH curve in 5-axis CNC system is studied in this paper.
When the tool-axis vector crossing quadrant in five-axis machining the tool needs to be raised that the processing speed and accuracy of products are hard to be sure. In order to solve this problem, cross-quadrant method has been proposed in this paper. Then the structure form of 5-axis machine tool, space-movement and post-processing based on RPCP of 5-axis machine tool is analyzed.
Prediction-correction method is proposed instead of Taylor expression to improve the accuracy of interpolation. For the nature and characteristics of PH curve, four kinds of real-time interpolation arithmetic for spatial PH curve are designed and developed, and also the curvature interpolation is improved.
To solve the smooth transition in interpolation of PH curve, the method of realizing C1 and C~2 continuous interpolation of quintic spatial PH curves is proposed. To deal with the large NC codes that CAD/CAM system generated and velocity discontinuity, NC codes are handled according to the lengths and the angels of lines. Knot removal is used to approximate the discrete points with PH curve. Converting these codes to PH interpolation codes is realized. The NC code format of five-coordinate PH curve is designed. The arithmetic of five-coordinate PH curve interpolation and surface machining are verified by simulation. Simulation and experiment prove that the cross-quadrant method and RPCP algorithm are feasible.
PH曲线是一类具有优良特性的多项式曲线,可以用于解决曲线弧长的计算问题,且保持与Bézier曲线表示的完全一致性,并且PH曲线往往产生比经典多项式曲线更光顺的轨迹等优点。因此本文主要研究五轴联动数控系统的PH曲线插补技术。
由于在五轴联动加工中,刀轴矢量出现跨象限时,需抬刀处理,加工速度和精度很难保证。为解决此问题,提出了跨象限的处理方法。在此基础上对双转台五轴联动机床的空间运动和基于RPCP的后置处理作了简单的分析和研究;提出了在插补中用预测—校正方法代替通常使用的Taylor展开方法,提高了插补的精度。针对PH曲线的性质和特点,设计、开发了四种空间PH曲线的实时插补算法,并对曲率插补方法的缺点进行了改进;
为解决PH曲线插值中的光滑过渡,提出了5次空间PH曲线的C~1和C~2连续插值方法。针对CAD/CAM系统生成的曲线曲面加工代码文件庞大、速度不连续的缺点,使用了基于数据点长度和角度的方法对数控代码进行了处理,并利用结点消去法对刀位点用PH曲线逼近,实现了微段直线插补指令向PH曲线插补指令的转换;给出了五坐标PH曲线插补的数控代码格式,通过仿真对五坐标PH样条曲线插补算法及曲面加工作了验证。另外,针对跨象限处理方法及RPCP算法,通过仿真和实验验证了其可行性。
More and more manufactured products with sculptured surfaces are expected higher accuracy at the present time. However, those products cannot be machined on the 3-axis machine in only one fixing. The processing speed and processing accuracy cannot be well ensured. So, manufacturing these complicated products in the manner of high speed and high accuracy, machines with five simultaneous motion axes are applied most widely. When machining the model created by free-formed surfaces, many tiny line segments are used to approach these surfaces. This manufacturing method has a large NC file, so it results in great amount data transmission, disconnected speed in joint of motion segments and poor manufacturing quality. Because of the processing speed is uniform in the spline interpolation, it can improve the manufacturing quality.
PH curve is a kind of polynomial curves which has a good characteristic and posses many positive properties. It can solve the problem of computing the arc length, retains full compatibility with standard Bézier represtation and tends to produce fairer loci than ordinary polynomial curves and so on. So the interpolation technology of PH curve in 5-axis CNC system is studied in this paper.
When the tool-axis vector crossing quadrant in five-axis machining the tool needs to be raised that the processing speed and accuracy of products are hard to be sure. In order to solve this problem, cross-quadrant method has been proposed in this paper. Then the structure form of 5-axis machine tool, space-movement and post-processing based on RPCP of 5-axis machine tool is analyzed.
Prediction-correction method is proposed instead of Taylor expression to improve the accuracy of interpolation. For the nature and characteristics of PH curve, four kinds of real-time interpolation arithmetic for spatial PH curve are designed and developed, and also the curvature interpolation is improved.
To solve the smooth transition in interpolation of PH curve, the method of realizing C1 and C~2 continuous interpolation of quintic spatial PH curves is proposed. To deal with the large NC codes that CAD/CAM system generated and velocity discontinuity, NC codes are handled according to the lengths and the angels of lines. Knot removal is used to approximate the discrete points with PH curve. Converting these codes to PH interpolation codes is realized. The NC code format of five-coordinate PH curve is designed. The arithmetic of five-coordinate PH curve interpolation and surface machining are verified by simulation. Simulation and experiment prove that the cross-quadrant method and RPCP algorithm are feasible.
引文
1宋寒,彭芳瑜,罗忠诚等.五轴联动的数控加工技术的研究及应用.机械工程师. 2007, (5): 120~122
2张翊伟.五轴联动机床.现代制造. 2005, (9): 31~33
3金志平.从CIMT2009看立式加工中心的发展.金属加工. 2009,(13):9~11
4刘士玉.五轴联动车铣中心现状与发展策略.世界制造技术与装备市场. 2008, (3): 93~96
5王永章,杜君文,程国全.数控技术.高等教育出版社, 2001:1~3
6康书杰. NURBS曲线实时插补技术研究.南京航空航天大学硕士论文. 2007: 1~6
7 Y. Altintas, W. K. Munasinghe. Modular CNC Design for Intelligent Machining, Part 2: Modular Integration of Sensor Based Milling Process Monitoring and Control Tasks. Journal of Manufacturing Science and Engineering. 1996, 118(11): 514~521
8 S. Ding, et al. Oriented Bounding Box and Octree Based Global Interference Detection in 5-axis Machining of Free-form Surfaces. Computer-Aided Design. 2004, 36: 1281~1294
9 B. Bahr, et al. A Real-time Scheme of Cubic Parametric Curve Interpolations for CNC Systems. Computers in Industry. 2001, 45: 309~317
10 S. S. Yeh, P. L. Hsu. The Speed-controlled Interpolator for Machining Parametric Curves. Computer-Aided Design. 1999, (31): 349~357
11刘胜涛.五坐标样条曲线插补技术的研究.哈尔滨工业大学硕士论文, 2007: 1~12
12 J. M. Langeron, E. Duc, C. Lartigue. A New Format for 5-axis Tool Path Computa- tion Using Bspline Curves. Computer-Aided Design, 2004, 36(12): 1219~1229
13 R. V. Fleisig, A. D. Spence. A Constant Feed and Reduced Angular Acceleration Interpolation Algorithm for Multi-axis Machining. Computer-Aided Design, 2001, 33(1): 1~15
14 C. C. Lo. Real-time Generation and Control of Cutter Path for 5-axis CNC Machining. International Journal of Machine Tools and Manufacture. 1999, (39): 471~488
15张春良.数字增量式直接函数法的圆弧插补算法的改进.组合机床与自动化加工技术. 2000, (6): 21~23
16帅梅,卜国磊,苗晓燕.五轴五联动数控系统快递推插补算法的研究.西安交通大学学报. 1999, 33(1): 93~96
17黄翔,曾荣,岳伏军. NURBS插补技术在高速加工中的应用研究.南京航空航天大学学报. 2002, 34(1): 82~85
18张伟. CNC系统中任意空间曲线的插补方法.机械. 2002, 29(2): 36~37
19陈良骥,王永章,富宏亚.五轴联动双NURBS曲线的生成与插补方法研究.机械制造. 2006, 44(1): 67~70
20陈良骥.复杂曲面加工中刀具路径生成方法及实时插补技术研究.哈尔滨工业大学博士学位论文. 2006: 83~86
21马雄波.基于PC机的开放式多轴软数控系统关键技术研究与实现.哈尔滨工业大学博士学位论文. 2007: 78~83
22施法中.计算机辅助几何设计与非均匀有理B样条.高等教育出版社, 2001: 306~309
23 Rida T. Farouki, Jairam Manjunathaiah, Guo-Feng Yuan. G codes for the specification of Pythagorean-hodograph tool paths and associated feedrate functions on open-architecture CNC machines.International Journal of Machine Tools & Manufacture 1999, (39):123~142
24 Farouki R T, Sakkalis T. Pythagorean hodographs.Journal of Research and Development, 1990, 34(5): 736~752
25王国瑾,汪国昭,郑建民.计算机辅助几何设计.高等教育出版社, 2001: 287~318
26 R.T. Farouki, T. Sakkalis. Pythagorean-Hodograph Space Curves. Advances in Computational Mathematics, 1994, (2): 41~66
27 R.T. Farouki, Sagar Shah. Real-time CNC Interpolators for Pythagorean Hodo- graph Curves .Computer Aided Geometric Design, 1996, 13:583~600
28 Rida T. Farouki. Pythagorean-hodograph Curves and Related Topics. Computer Aided Geometric Design 2008, 25: 203~204
29 Farouki et al. Hermite Interpolation by Pythagorean Hodograph Quintics. Mathem- atics of Computation, 1995, 64(212): 299~309
30 Meek et al. Geometric Hermite Interpolation with Tschirnhausen Cubics. Journal of Computational and Applied Mathematics, 1997, 81(2): 299~309
31 Francesca Pelosi et al. A Control Polygon Scheme for Design of Planar C2 PH Quintic Splines Curves. Computer Aided Geometric Design, 2007, (4): 28~52
32 Rida T. Farouki, Yi-Feng Tsai, Guo-Feng Yuan. Contour Machining of Free-formSurfaces with Real-time PH Curve CNC Interpolators. Computer Aided Geometric Design, 1999, (16): 61~76
33韩西安,黄希利,叶正麟. C1五次Pythagorean Bézier样条曲线的构造.计算机学报. 1999, 22(5): 535~539
34韩西安,叶正麟,黄希利. Pythagorean Bézier速端曲线及其等距线.计算数学. 2003, 21(1): 27~36
35韩西安,叶正麟,黄希利. Pythagorean Bézier速端曲线及其性质.计算机辅助设计与图形学学报. 1999, 11(4): 363~366
36马元魁. Pythagorean-Hodograph曲线的理论及应用研究.西北工业大学硕士论文, 2005: 1~15
37吕盼稂,张珊珊,竺长安.基于Pythagorean速端螺线的平稳加工路径.中国机械工程. 2006, 17(23): 2451~2453
38马元魁,张天平,康宝生.球面插值PH曲线.工程数学学报. 2008, 25(3): 442~448
39陈良骥,王永章. 5轴CNC机床空间运动学研究.设计与研究. 2005, (7): 23~25.
40梁全,王永章.五轴数控系统RTCP和RPCP技术应用.组合机床与自动化加工技术. 2008, (2): 62~65
41 Matthias Muller, Gabor Erdos et al. High Accuracy Spline Interpolation for 5-axis Maching. Computer-Aided Design, 2004, (36): 1379~1393
42雍俊海,郑文.一类五次PH曲线Hermite插值的几何方法.计算机辅助设计与图形学学报. 2005, 17(5): 990~995
43吴勃英.数值分析原理.科学出版社. 2003: 303~308
44 Yi-Feng Tsai, Rida T. Farouki et al. Performance analysis of CNC Interpolators for Time-dependent Feedrates Along PH Curves. Computer Aided Geometric Design, 2001, (18): 245~265
45 R. T. Farouki et al. Variable-feedrate CNC Interpolators for Constant Material Removal Rates along Pythagorean-hodograph Curves. Computer Aided Design, 1998, (30): 631~640
46 C. B. U. Perwass, R. T. Farouki, L.Noakes. A Geometric Product Formulation for Spatial Pythagorean Hodograph Curves with Applications to Hermite Interpolation. Computer Aided Geometric Design, 2007, (24): 220~237
47 R.T. Farouki et al. Hermite Interpolation by Rotation-invariant Spatial Pythagorean-hodograph Curves. Advances in Computational Mathematics. 2002, 17:369~383
48 Fletcher Dunn, Ian Parberry. 3D数学基础:图形与游戏开发.史银雪,陈洪,王荣静.清华大学出版社, 2005: 130~173
49 Rida T. Farouki, Carlotta Giannelli, Carla Manni, Alessandra Sestini. Identification of Spatial PH Quintic Hermite Interpolants with Near-optimal Shape Measures. Computer Aided Geometric Design 2008, (25): 274~297
50杜建丽,茹少峰,樊少荣等.基于B样条曲线表示的物体轮廓曲线匹配.西北大学学报. 2005, 35(5): 527~530
51张得礼,周来水.数控加工运动的平滑处理.航空学报, 2006, 27(1): 125~130
52郑小军,吴军.数控仿真软件在数控实验中的应用.实验科学与技术, 2009, 7(2): 11~13
53杨胜群. VERICUT7.0中文版数控加工仿真技术.清华大学出版社, 2010: 1~8
54 Rida T. Farouki, Yi-Feng Tsai, Guo-Feng Yuan. Contour Machining of Free-form Surfaces withReal-time PH Curve CNC Interpolators. Computer Aided Geometric Design, 1999, (16): 61~76
2张翊伟.五轴联动机床.现代制造. 2005, (9): 31~33
3金志平.从CIMT2009看立式加工中心的发展.金属加工. 2009,(13):9~11
4刘士玉.五轴联动车铣中心现状与发展策略.世界制造技术与装备市场. 2008, (3): 93~96
5王永章,杜君文,程国全.数控技术.高等教育出版社, 2001:1~3
6康书杰. NURBS曲线实时插补技术研究.南京航空航天大学硕士论文. 2007: 1~6
7 Y. Altintas, W. K. Munasinghe. Modular CNC Design for Intelligent Machining, Part 2: Modular Integration of Sensor Based Milling Process Monitoring and Control Tasks. Journal of Manufacturing Science and Engineering. 1996, 118(11): 514~521
8 S. Ding, et al. Oriented Bounding Box and Octree Based Global Interference Detection in 5-axis Machining of Free-form Surfaces. Computer-Aided Design. 2004, 36: 1281~1294
9 B. Bahr, et al. A Real-time Scheme of Cubic Parametric Curve Interpolations for CNC Systems. Computers in Industry. 2001, 45: 309~317
10 S. S. Yeh, P. L. Hsu. The Speed-controlled Interpolator for Machining Parametric Curves. Computer-Aided Design. 1999, (31): 349~357
11刘胜涛.五坐标样条曲线插补技术的研究.哈尔滨工业大学硕士论文, 2007: 1~12
12 J. M. Langeron, E. Duc, C. Lartigue. A New Format for 5-axis Tool Path Computa- tion Using Bspline Curves. Computer-Aided Design, 2004, 36(12): 1219~1229
13 R. V. Fleisig, A. D. Spence. A Constant Feed and Reduced Angular Acceleration Interpolation Algorithm for Multi-axis Machining. Computer-Aided Design, 2001, 33(1): 1~15
14 C. C. Lo. Real-time Generation and Control of Cutter Path for 5-axis CNC Machining. International Journal of Machine Tools and Manufacture. 1999, (39): 471~488
15张春良.数字增量式直接函数法的圆弧插补算法的改进.组合机床与自动化加工技术. 2000, (6): 21~23
16帅梅,卜国磊,苗晓燕.五轴五联动数控系统快递推插补算法的研究.西安交通大学学报. 1999, 33(1): 93~96
17黄翔,曾荣,岳伏军. NURBS插补技术在高速加工中的应用研究.南京航空航天大学学报. 2002, 34(1): 82~85
18张伟. CNC系统中任意空间曲线的插补方法.机械. 2002, 29(2): 36~37
19陈良骥,王永章,富宏亚.五轴联动双NURBS曲线的生成与插补方法研究.机械制造. 2006, 44(1): 67~70
20陈良骥.复杂曲面加工中刀具路径生成方法及实时插补技术研究.哈尔滨工业大学博士学位论文. 2006: 83~86
21马雄波.基于PC机的开放式多轴软数控系统关键技术研究与实现.哈尔滨工业大学博士学位论文. 2007: 78~83
22施法中.计算机辅助几何设计与非均匀有理B样条.高等教育出版社, 2001: 306~309
23 Rida T. Farouki, Jairam Manjunathaiah, Guo-Feng Yuan. G codes for the specification of Pythagorean-hodograph tool paths and associated feedrate functions on open-architecture CNC machines.International Journal of Machine Tools & Manufacture 1999, (39):123~142
24 Farouki R T, Sakkalis T. Pythagorean hodographs.Journal of Research and Development, 1990, 34(5): 736~752
25王国瑾,汪国昭,郑建民.计算机辅助几何设计.高等教育出版社, 2001: 287~318
26 R.T. Farouki, T. Sakkalis. Pythagorean-Hodograph Space Curves. Advances in Computational Mathematics, 1994, (2): 41~66
27 R.T. Farouki, Sagar Shah. Real-time CNC Interpolators for Pythagorean Hodo- graph Curves .Computer Aided Geometric Design, 1996, 13:583~600
28 Rida T. Farouki. Pythagorean-hodograph Curves and Related Topics. Computer Aided Geometric Design 2008, 25: 203~204
29 Farouki et al. Hermite Interpolation by Pythagorean Hodograph Quintics. Mathem- atics of Computation, 1995, 64(212): 299~309
30 Meek et al. Geometric Hermite Interpolation with Tschirnhausen Cubics. Journal of Computational and Applied Mathematics, 1997, 81(2): 299~309
31 Francesca Pelosi et al. A Control Polygon Scheme for Design of Planar C2 PH Quintic Splines Curves. Computer Aided Geometric Design, 2007, (4): 28~52
32 Rida T. Farouki, Yi-Feng Tsai, Guo-Feng Yuan. Contour Machining of Free-formSurfaces with Real-time PH Curve CNC Interpolators. Computer Aided Geometric Design, 1999, (16): 61~76
33韩西安,黄希利,叶正麟. C1五次Pythagorean Bézier样条曲线的构造.计算机学报. 1999, 22(5): 535~539
34韩西安,叶正麟,黄希利. Pythagorean Bézier速端曲线及其等距线.计算数学. 2003, 21(1): 27~36
35韩西安,叶正麟,黄希利. Pythagorean Bézier速端曲线及其性质.计算机辅助设计与图形学学报. 1999, 11(4): 363~366
36马元魁. Pythagorean-Hodograph曲线的理论及应用研究.西北工业大学硕士论文, 2005: 1~15
37吕盼稂,张珊珊,竺长安.基于Pythagorean速端螺线的平稳加工路径.中国机械工程. 2006, 17(23): 2451~2453
38马元魁,张天平,康宝生.球面插值PH曲线.工程数学学报. 2008, 25(3): 442~448
39陈良骥,王永章. 5轴CNC机床空间运动学研究.设计与研究. 2005, (7): 23~25.
40梁全,王永章.五轴数控系统RTCP和RPCP技术应用.组合机床与自动化加工技术. 2008, (2): 62~65
41 Matthias Muller, Gabor Erdos et al. High Accuracy Spline Interpolation for 5-axis Maching. Computer-Aided Design, 2004, (36): 1379~1393
42雍俊海,郑文.一类五次PH曲线Hermite插值的几何方法.计算机辅助设计与图形学学报. 2005, 17(5): 990~995
43吴勃英.数值分析原理.科学出版社. 2003: 303~308
44 Yi-Feng Tsai, Rida T. Farouki et al. Performance analysis of CNC Interpolators for Time-dependent Feedrates Along PH Curves. Computer Aided Geometric Design, 2001, (18): 245~265
45 R. T. Farouki et al. Variable-feedrate CNC Interpolators for Constant Material Removal Rates along Pythagorean-hodograph Curves. Computer Aided Design, 1998, (30): 631~640
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