基于Catmull-Clark模式的细分曲面NC刀具轨迹生成技术
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摘要
细分方法是一种新的离散造型技术,通过定义控制网格和细分规则来表示造型曲面。它不仅具备B样条曲面的局部支承性、仿射不变性等良好性质,而且具有参数曲面所没有的任意拓扑性等特点,正逐渐成为几何造型的有力工具。曲面的细分算法采用逐次细分,从离散到离散,最终得到所需要的曲面,避免了以往样条曲面从离散到连续,再从连续到离散的过程。在诸多细分模式中,Catmull-Clark细分曲面是B样条曲面的推广,这样的细分曲面具有易于与NURBS曲面融合的特点,能更好的与数控加工相结合。细分曲面的数控加工,是细分曲面应用于工程生产的基础。而精确的刀具轨迹规划算法是数控加工的灵魂。本文就是以Catmull-Clark细分曲面为切入点,提出了一种基于Catmull-Clark模式的细分曲面NC刀具轨迹生成算法,旨在将细分曲面应用于生产加工,为细分曲面在工业领域的发展打下基础。主要工作如下:
1.提出了一种基于Catmull-Clark模式的细分曲面NC刀具轨迹生成算法:首先计算Catmull-Clark细分曲面初始控制网格顶点的极限点和法矢量,然后由极限点出发,按给定精度、以球头刀半径长度沿其法矢量方向向外等距,获得等距曲面,最后在等距曲面上生成NC刀具轨迹。该算法可以将细分曲面应用于CAD/CAM系统,适用于任意拓扑的四边形网格模型。2.应用开发工具Visual C++6.0和OpenGL,对Catmull-Clark模式的细分曲面进行了实现;采用四叉树结构和翼边结构相结合的方法,对Catmull-Clark细分曲面等距面进行了实现;采用截平面法,生成了Catmull-Clark细分曲面NC刀具轨迹。
3.等距曲面误差作为细分曲面等距的一个基本问题,本文对其进行了分析,提出了两种控制方法:第一种方法以相邻点法矢夹角为基准,控制等距面的精度。第二种以控制网格点与非控制网格点的等距距离差为基准,获得具有给定精度的等距面。
Subdivision scheme is a new discrete modeling technology which describes the modeling surface through controlling mesh and subdivision rules. It has many advantages such as localness and affine-invariance of traditional B-spline surface and adaptability to arbitrary topology which parameter surfaces do not have, and is becoming one of the most powerful geometric modeling tools. Subdivision scheme for surface is a gradual subdivision which obtains final surface by discrete-to-discrete method, and the disadvantage of conventional method from discrete to continuous and then from continuous to discrete is avoided. The Catmull-Clark subdivision scheme is generalization of B-spline surfaces, which can be blent with NURBS surfaces easily and be unified with the NC processing commendably. Numerical Control of subdivision surface is the base of subdivision surface applied to industry manufacture. And an accurate programming method of Numerical Control tool path is a soul of Numerical Control.The paper focuses on the Catmull-Clark subdivision surfaces. Based on the Catmull-Clark subdivision scheme , a new approach to generate NC tool path for subdivision surfaces is presented. The aim is to apply subdivision surfaces to product manufacture, and to make preparations for development of subdivision surfaces in industry field. The main contents are as follows:
1. Based on the Catmull-Clark subdivision scheme, a new approach to generate NC tool path for subdivision surfaces is presented: First calculate the limit points and the normal vectors at these points from the initial control mesh . Second generate offset surface with controlling accuracy by using the limit points, the normal vectors and the radius of the ball end milling cutter. Next generate NC cutting tool path from the offset subdivision surface. This method applies subdivision surfaces to CAD/ CAM system , and adapts to quadrilateral models of arbitrary topology.
2. Catmull-Clark subdivision scheme is implemented by developing tool Visual C++ 6.0 and OpenGL; Offset surface of Catmull-Clark subdivision surface is implemented by combinative method of Quadtree structure and Winged Edge structure; NC tool path of Catmull-Clark subdivision surface is generated by section plane method.
3. Offset error, which is a basis problem in subdivision suifaces, is also analyzed in the paper. And two methods about analysis and control are presented. The first one is to control precision of offset surfaces by criterion based on angle between normal vectors of two neighbor vertexes. The second one is to obtain offset surfaces whose precision has been given by criterion based on distance between control mesh vertexes and non-control mesh vertexes.
1.提出了一种基于Catmull-Clark模式的细分曲面NC刀具轨迹生成算法:首先计算Catmull-Clark细分曲面初始控制网格顶点的极限点和法矢量,然后由极限点出发,按给定精度、以球头刀半径长度沿其法矢量方向向外等距,获得等距曲面,最后在等距曲面上生成NC刀具轨迹。该算法可以将细分曲面应用于CAD/CAM系统,适用于任意拓扑的四边形网格模型。2.应用开发工具Visual C++6.0和OpenGL,对Catmull-Clark模式的细分曲面进行了实现;采用四叉树结构和翼边结构相结合的方法,对Catmull-Clark细分曲面等距面进行了实现;采用截平面法,生成了Catmull-Clark细分曲面NC刀具轨迹。
3.等距曲面误差作为细分曲面等距的一个基本问题,本文对其进行了分析,提出了两种控制方法:第一种方法以相邻点法矢夹角为基准,控制等距面的精度。第二种以控制网格点与非控制网格点的等距距离差为基准,获得具有给定精度的等距面。
Subdivision scheme is a new discrete modeling technology which describes the modeling surface through controlling mesh and subdivision rules. It has many advantages such as localness and affine-invariance of traditional B-spline surface and adaptability to arbitrary topology which parameter surfaces do not have, and is becoming one of the most powerful geometric modeling tools. Subdivision scheme for surface is a gradual subdivision which obtains final surface by discrete-to-discrete method, and the disadvantage of conventional method from discrete to continuous and then from continuous to discrete is avoided. The Catmull-Clark subdivision scheme is generalization of B-spline surfaces, which can be blent with NURBS surfaces easily and be unified with the NC processing commendably. Numerical Control of subdivision surface is the base of subdivision surface applied to industry manufacture. And an accurate programming method of Numerical Control tool path is a soul of Numerical Control.The paper focuses on the Catmull-Clark subdivision surfaces. Based on the Catmull-Clark subdivision scheme , a new approach to generate NC tool path for subdivision surfaces is presented. The aim is to apply subdivision surfaces to product manufacture, and to make preparations for development of subdivision surfaces in industry field. The main contents are as follows:
1. Based on the Catmull-Clark subdivision scheme, a new approach to generate NC tool path for subdivision surfaces is presented: First calculate the limit points and the normal vectors at these points from the initial control mesh . Second generate offset surface with controlling accuracy by using the limit points, the normal vectors and the radius of the ball end milling cutter. Next generate NC cutting tool path from the offset subdivision surface. This method applies subdivision surfaces to CAD/ CAM system , and adapts to quadrilateral models of arbitrary topology.
2. Catmull-Clark subdivision scheme is implemented by developing tool Visual C++ 6.0 and OpenGL; Offset surface of Catmull-Clark subdivision surface is implemented by combinative method of Quadtree structure and Winged Edge structure; NC tool path of Catmull-Clark subdivision surface is generated by section plane method.
3. Offset error, which is a basis problem in subdivision suifaces, is also analyzed in the paper. And two methods about analysis and control are presented. The first one is to control precision of offset surfaces by criterion based on angle between normal vectors of two neighbor vertexes. The second one is to obtain offset surfaces whose precision has been given by criterion based on distance between control mesh vertexes and non-control mesh vertexes.
引文
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[51]Kuijt F,Damme V R.Stability of subdivision schemes.Memorandum No.1469,Faculty of Mathematical Sciences,University of Twente,1998.
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[2]殷国富,陈永华.计算机辅助设计技术与应用.北京:科学出版社,2000.
[3]王国瑾,汪国昭,郑建民.计算机辅助几何设计.北京:高等教育出版社,2001.
[4]施法中.计算机辅助几何设计与非均匀有理B样条.北京:高等教育出版社,2001.
[5]DeRose T,Kass M,Truong T.Subdivision surfaces in character animation.Proceedings of the ACM SIGGRAPH Conference on Computer Graphics,Orlando,1998:85-96.
[6]周海.细分曲面造型技术研究:(博士学位论文).南京:南京航空航天大学,2004.
[7]Zorin D.Overview of subdivision schemes.Proceedings of the ACM SIGGRAPH Conference on Computer Graphics,New York,2000:65-84.
[8]Peters J.Local smooth surface interpolation:a classification.CAGD,1990,7(1-4):191-195.
[9]Halstead M,Kass M,DeRose T.Efficient,fair interpolation using Catmull-Clark surfaces.Computer Graphics,1993,27(3):35-44.
[10]Nasri A.Polyhedral subdivision methods for free-form surfaces.ACM Transaction on Graphics,1987,6(1):29-73.
[11]Nasri A.Surface interpolation on irregular networks with normal conditions,CAGD,1991,8(1):89-96.
[12]Zorin D.Stationary subdivision and multiresolution surface representations:[dissertation].Pasadena:Caltech of Pasadena,1997.
[13]Hoppe H,DeRose T,Duchamp T et al.Piecewise smooth surface construction.In Computer Graphics Proceedings,ACM SIGGRAPH,1994:295-302.
[14]Nasri A.Interpolation of open B-spline curves by recursive subdivision surfaces.Mathematics of Surfaces Ⅶ,1997:173-188.
[15]Nasri A.Recursive subdivision of polygonal complexes and its applications in computer-aided geometric design.CAGD,2000,17:595-619.
[16]Levin A.Interpolation nets of curves by smooth subdivision surfaces.In Computer Graphics Proceedings,Annual Conference Series,ACM SIGGRAPH,1999:57-64.
[17]Biermann H,Levin A,Zorin D.Piecewise smooth subdivision surfaces with normal control.In Computer Graphics Proceedings,Annual Conference Series,ACM SIGGRAPH,2000:113-120.
[18]Litke N,Levin A,Schroder P.Fitting subdivision surfaces.Proceedings of Scientific Visualization,2001.
[19]Sederberg T,Zheng J,Swell D.Non-uniform recursive subdivision surfaces.Proceedings of the ACM SIGGRAPH Conference on Computer Graphics,Orlando,1998:387-394.
[20]Litke N,Levin A,Schroder P.Trimming for subdivision surfaces.CAGD,2001,18:463-481.
[21]张大涌.先进制造技术.北京:机械工业出版社,1995.
[22]申丽国,张昆,黄征.国外数控技术的研究动向和发展趋势.机械工业自动化,1996,2:19-23.
[23]周艳红,周云飞,李作清等.复杂型腔CNC直接加工路线规划原理与算法.华中理工大学学报,1995,23(6):1-9.
[24]陈桦.超高速数控机床控制系统的发展.制造技术与机床,2000,5:12-14.
[25]张俊,魏红根.数控技术发展趋势—智能化数控系统.制造技术与机床,2000,4:7-9.
[26]Ishida J.the General B-spline Interpolation Method and Its Application to the Modification of Curves and Surfaces.Computer-Aided Design,1998,29(11):55-62.
[27]上海交大生产系统与控制十时微 研究所.新一代数控系统的研究开发.机电一体化,2000,(3):7-11.
[28]Hwangt J S,Chang T C,Three-axis Machining of Compound Surfaces Using Flat and Filleted Endmills.Computer-Aided Design,1998,30(8):641-647.
[29]杨文玉,熊有伦.一种提取曲面加工可行刀具方向的方法.电加工与模具,2000,3:53-55.
[30]阎光荣,许鹤峰,白俊涛.高速加工的CAM技术与模具加工.电加工与模具,2000,3:1-5.
[31]Bobrow J E.NC machining tool path generation from CSG part representations.Computer-Aided Design,1985,17:69-76.
[32]Ding S,Manan M,Poo A et al.Adaptive iso-planar tool path generation for machining of free-form surfaces.Computer-Aided Design,2003,35:141-153.
[33]Park S.Tool-path generation for Z-constant contour machining.Computer-Aided Design,2003,35(1):27-36.
[34]Feng His-Yung,Teng Zhengji.Iso-planar piecewise linear NC tool path generation from discrete measured data points.Computer-Aid Design,2005,37:55-64.
[35]Loney G C,Ozsoy T M.NC machining of free from surfaces.Computer-Aided Design,1987,19(2):85-90.
[36]Elber G,Cohen E.Tool path generation for freeform surface models[J].Computer Aided Design,1994,26(6):490-495.
[37]Suresh K,Yang D.Constant scallop-height machining of free-from surfaces.Journal of Engineering for Industury,1994,116(5):253-259.
[38]高军.自由曲面数控加工刀具轨迹优化算法的研究.机械加工与自动化,2001,11:16-17.
[39]任秉银,唐余勇.数控加工中的集合建模理论及其应用.哈尔滨:哈尔滨工业大学出版社,2000.
[40]Ryuh B,Park S,Pennock G R.An automatic tool changer and integrated software for a robotic die polishing station.Mechanism and machine Theory,2005.
[41]Kurgano J.Generation of NC tool path for Subdivision Surfaces.http://www.cim.pe.u tokyo.ac.jp/~joe/MonthlyReport/Generation_of_NC_for Subdivision Surfaces.pdf.
[42]王占东.细分曲面关键技术研究:(博士学位论文).南京:南京航空航天大学,2003.
[43]丁俊勇,胡事民,周登文.Loop细分曲面的等距曲面的逼近.计算机学报,2003,26(7):789-795.
[44]朱翔.Loop细分曲面数控加工刀具轨迹的生成.清华大学学报,2003,43(4).
[45]周海.细分曲面造型技术研究:(博士学位论文).南京:南京航空航天大学,2004.
[46]钟大平.细分曲面的NC刀具轨迹生成算法及实现.东南大学学报,2004,34(1).
[47]梁伟文.Loop细分曲面的数控粗加工刀具路径生成方法.深圳职业技术学院学报,2007,3.
[48]李桂清.细分曲面造型及应用:(博士学位论文).北京:中国科学院计算机技术研究所,2001.
[49]Loop C.Smooth subdivision surfaces based on triangles:[dissertation].Utah:Univ.of Utah,1987.
[50]Zorin D.Smoothness of stationary subdivision on irregular meshes.Constructive Approxination,2000,16(3):359-398.
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