基于协调映射的网格曲面轨迹规划策略研究
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摘要
近年来,关于自由曲面的刀具轨迹规划问题的研究已经很深入,但是针对离散三角网格曲面的规划方法并不多,通常只有等截面法,往往不能满足实际加工的需要。针对于此,本论文提出了一种基于映射建模的刀具轨迹规划方法,其基本思路是先选择恰当的映射方法将离散三角网格映射到参数曲面,在参数域进行轨迹规划,然后将生成的轨迹映射回原三角网格曲面,即得到所需的刀具轨迹。
首先,探讨了映射方法的选择。此步骤是整个规划方法的基础。协调映射因为具有变形小、网格不会重叠等优点,被作为最佳选择。根据协调映射的理论,建立了平面上圆形域、矩形域、多边形域等参数域的映射模型,可根据轨迹规划的需要来选择映射的边界。由于协调映射时所需处理的数据量很大,本文给出了编程时具体的数据结构,其中大型稀疏矩阵采用三元组压缩存储。
其次是在参数域上规划刀具轨迹,并提出了一种基于平面螺旋线制导的螺旋型刀具轨迹的生成新方法。针对网格曲面,建立了曲面上任一点处的法矢和曲率的计算模型,进而确定网格曲面路径间距及步长,建立相应的计算模型。根据协调映射及相关理论,建立了原网格曲面和参数域曲面的路径间距及步长的对应关系。采用B样条技术,实现了离散型刀触点轨迹的解析表达。
最后,为生成原网格域上的刀具轨迹。根据协调映射求解得到其逆映射,将刀触点轨迹由参数域逆映射回网格曲面。还给出了三角形面积坐标法简化此步骤。根据所选择的刀具——平底刀,推导得到刀触点转化为刀位点的公式,建立计算模型。并对干涉问题进行研究,探讨调整刀轴矢量来解决干涉的问题。最后根据模型将刀触点轨迹转化为刀位点轨迹,即刀具轨迹。
本方法还可以生成等参数线轨迹、Piano轨迹等轨迹,对其具体应用作了进一步的探讨。本文还就所列出的轨迹线的类型给出了相应的例子。随后根据一个实例——自行车座,给出了螺旋型轨迹、等参数线轨迹以及截面线轨迹。通过编程进行了轨迹线的长度和加工时间的仿真,并且就所得试验结果进行分析,给出结论。
Significant research has been carried out to generate tool path for free-form surface recently. However few of them are for triangular meshes except iso-planar tool path. It can't satisfy all demands for machining. So this paper presents a method of generating tool path for triangular meshes based on maps. The method's main process is mapping triangular meshes to parameter region by conformal maps firstly, generating tool path on parameter region secondly and finally mapping tool path back from parameter region to triangular meshes.Firstly, the choice of maps is discussed. The map is the foundation of the whole method. Conformal map is the best choice because it has merits such as little deformation and no grid overlapping, etc. Based on the theory of conformal map, map model is built for planar parametric circle, rectangle and polygon regions. The boundary of parametric region rests with the need of tool path planning. The data structure is presented for the vast data while constructing conformal map. The large-scale sparse matrix is stored as three-number sets.Secondly, tool path is generated in parametric region and a new method for generating spiral tool path guided by spiral in planar is presented. A mathematic model is established to calculate normal vector and curvature of a point on the triangular surface. The calculating model for path interval and step length is developed, too. According to conformal map and relative theory, the relation of path interval between triangular meshes and parametric region is built. Adopting B-spline technology, it is implemented to analytic expression of discrete cutter contact tool path.Finally, tool path is generated on triangular meshes. Cutter contact tool path is mapped from parametric region to triangular meshes by means of conformal map and its reverse map. It can be easily solved by barycenter coordinate, too. The formula is given for calculating cutter location point from cutter contact point. The problem about interference is discussed and solved by adjusting the vector of tool axe. Lastly, cutter location points are calculated according to the calculating model.By using this method, it can generate iso-parametric tool path, Piano tool path, etc. This method's application is deeply discussed and corresponding examples are given. Taking the bicycle seat as an actual example, the spiral tool path, iso-parametric tool path and iso-planar tool path are generated. In addition, every kind of tool path's length and machining time are simulated and the comparison result is analyzed to draw the conclusion.
首先,探讨了映射方法的选择。此步骤是整个规划方法的基础。协调映射因为具有变形小、网格不会重叠等优点,被作为最佳选择。根据协调映射的理论,建立了平面上圆形域、矩形域、多边形域等参数域的映射模型,可根据轨迹规划的需要来选择映射的边界。由于协调映射时所需处理的数据量很大,本文给出了编程时具体的数据结构,其中大型稀疏矩阵采用三元组压缩存储。
其次是在参数域上规划刀具轨迹,并提出了一种基于平面螺旋线制导的螺旋型刀具轨迹的生成新方法。针对网格曲面,建立了曲面上任一点处的法矢和曲率的计算模型,进而确定网格曲面路径间距及步长,建立相应的计算模型。根据协调映射及相关理论,建立了原网格曲面和参数域曲面的路径间距及步长的对应关系。采用B样条技术,实现了离散型刀触点轨迹的解析表达。
最后,为生成原网格域上的刀具轨迹。根据协调映射求解得到其逆映射,将刀触点轨迹由参数域逆映射回网格曲面。还给出了三角形面积坐标法简化此步骤。根据所选择的刀具——平底刀,推导得到刀触点转化为刀位点的公式,建立计算模型。并对干涉问题进行研究,探讨调整刀轴矢量来解决干涉的问题。最后根据模型将刀触点轨迹转化为刀位点轨迹,即刀具轨迹。
本方法还可以生成等参数线轨迹、Piano轨迹等轨迹,对其具体应用作了进一步的探讨。本文还就所列出的轨迹线的类型给出了相应的例子。随后根据一个实例——自行车座,给出了螺旋型轨迹、等参数线轨迹以及截面线轨迹。通过编程进行了轨迹线的长度和加工时间的仿真,并且就所得试验结果进行分析,给出结论。
Significant research has been carried out to generate tool path for free-form surface recently. However few of them are for triangular meshes except iso-planar tool path. It can't satisfy all demands for machining. So this paper presents a method of generating tool path for triangular meshes based on maps. The method's main process is mapping triangular meshes to parameter region by conformal maps firstly, generating tool path on parameter region secondly and finally mapping tool path back from parameter region to triangular meshes.Firstly, the choice of maps is discussed. The map is the foundation of the whole method. Conformal map is the best choice because it has merits such as little deformation and no grid overlapping, etc. Based on the theory of conformal map, map model is built for planar parametric circle, rectangle and polygon regions. The boundary of parametric region rests with the need of tool path planning. The data structure is presented for the vast data while constructing conformal map. The large-scale sparse matrix is stored as three-number sets.Secondly, tool path is generated in parametric region and a new method for generating spiral tool path guided by spiral in planar is presented. A mathematic model is established to calculate normal vector and curvature of a point on the triangular surface. The calculating model for path interval and step length is developed, too. According to conformal map and relative theory, the relation of path interval between triangular meshes and parametric region is built. Adopting B-spline technology, it is implemented to analytic expression of discrete cutter contact tool path.Finally, tool path is generated on triangular meshes. Cutter contact tool path is mapped from parametric region to triangular meshes by means of conformal map and its reverse map. It can be easily solved by barycenter coordinate, too. The formula is given for calculating cutter location point from cutter contact point. The problem about interference is discussed and solved by adjusting the vector of tool axe. Lastly, cutter location points are calculated according to the calculating model.By using this method, it can generate iso-parametric tool path, Piano tool path, etc. This method's application is deeply discussed and corresponding examples are given. Taking the bicycle seat as an actual example, the spiral tool path, iso-parametric tool path and iso-planar tool path are generated. In addition, every kind of tool path's length and machining time are simulated and the comparison result is analyzed to draw the conclusion.
引文
[1] 周济 周艳红 编著。数控加工技术,2002。国防工业出版社
[2] 柯映林等著。反求工程CAD建模理论、方法和系统。2006,北京,机械工业出版社。
[3] Hugues Hoppe, Tony Derose, Tom Duchamp. Surface reconstruction from unorganized points. Computer Draphics, 1992,26(2):71-78.
[4] 王平江,黄雪梅,陈吉红,周济。曲面激光密集测量三维数据的三角片逼近算法。工程图学学报,1998,1:17-28
[5] Loney GC, Ozsoy TM. NC machining of free form surfaces. Computer-Aid Design 1987;19(2)85-90
[6] Elber G, Cohen E. Toolpath generating for free form surface. Computer-Aid Design 1994;26(6)490-6
[7] His-Yung Feng, Zhengji Teng. Iso-planar piecewise linear NC tool path generation from discrete measured data points. Computer-Aid Design. 37(2005) 55-64.
[8] Bobrow JE. NC machining tool path generation from CSG part representations. Computer-Aid Design 1985;17, 69-76
[9] Park SC, Choi BK. Tool path planning for direction-parallel area milling. Computer-Aid Design 2000;32(1): 17-25
[10] S. Ding, M.A Mannan, A.N. Poo, D.C.H. Yang, Z. Han. Adaptive iso-planar tool path generation for machining of free-form surfaces. Computer-Aid Design 35(2003) 141-153
[11] Sang C. Park. Tool-path generation for Z-constant contour machining. Computer-Aid Design 2003,35(1): 27-36
[12] K.Suresh, D. C. H. Yang. Constant scallop-height machining of free-form surfaces. Journal of Engineering for Industry, 1994, 116(5): 253-259
[13] 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。11(2001),16-17.
[14] 任秉银,唐余勇。数控加工中的几何建模理论及其应用。2000。哈尔滨工业大学出版社。
[15] Beom-Sahng Ryuh, Sang Min Park, Gordon R.Pennock. An automatic tool changer and integrated software for a robotic die polishing station. Mechanism and machine Theory, 2005. Article in press.
[16] D.C.H. Yang, J.-J. Chuang et. Boundary-conformed toolpath generation for trimmed free-form surfaces. Computer-Aid Design 2003,35: 127-129
[17] 苏云玲。三元整体叶轮的几何造型与五坐标数控加工。大连:大连理工大学硕士学位论文。2003。
[18] 周艳红。自由曲面三坐标加工走刀步长的确定。华中理工大学学报,1998,26(8),7-9
[19] 彭群生,胡国飞。三角网格的参数化。计算机辅助设计与图形学学报。2004,16期6卷。731-738。
[20] Eck M, DeRose T, et al. Multiresolution Analysis of arbitrary meshes[A]. In Proceeding of SIGGRAPH' 95[C], 1995: 173-182.
[21] 周昆,鲍虎军,石教英。统一的数字几何处理框架。计算机学报。2002,25(9),904-909.
[22] 谭家万,金一丞,石教英。任意拓扑三角形网格的全局参数化。中国图像图形学报。2003,8(6),686-691
[23] Lee A W.F, Sweldens W, et al. MAPS: Multiresolution Adaptive Parameterization of surfaces. In: Proceeding of SIGGRAPH' 98[C], 1998:95-104.
[24] Kun Zhou, Hujun Bao, Jiaoying Shi. 3D surface filtering using spherical harmonics. Computer-aided design, 36 (2004):363-375
[25] Alexa M, Merging polyhedron shapes with scattered features [J].The visual Computer. 2000,16(1):26-37
[26] Peter Lindstrom, Greg Turk. Fast and memory efficient polygonal simplification. IEEE Visualization 98 Conference Proceedings: October 1998: 276-86
[27] 王海霞。RE中基于流形网格的四边形区域划分方法研究。大连:大连理工大学硕士学位论文。2004。
[28] Floater M S. Parametdzation and smooth approximation of surface triangulations [J]。Computer-Aid Design 1997;14(3): 231-250
[29] Hormann K, Greiner G. MIPS: An efficient global Parametrization method [M].In: Laurent P J.Sablonniere P, Schumaker L L,eds. Curve and surface Design. Nashville Tennessee: Vanderbilt University Press, 1999.153-162.
[30] 周昆,潘志庚,马小虎,石教英。调和映射的构造及其在图形学中的应用。中国图像图形学报。1998,3(7)578-582。
[31] 施法中。计算机辅助几何设计与非均匀有理B样条,北京:北京航天航空大学出版社。1994。
[32] 徐士良主编。C常用算法程序集。北京:清华大学出版社。年代,第二版。
[33] Sun Yuwen, Guo Dongming, Jia zhenyuan, Wang Haixia. Iso-parametric tool path generation from triangular meshes for freeform surface machining.
[34] 数学辞海。第一卷。北京,中国科学技术出版社。2002。
[35] 孙玉文,刘伟军,王越超。基于三角网格曲面模型的刀位轨迹计算方法。机械工程学报,2002,38(10),50-53。
[36] H Biermann, A Levin, D Zorin. Piecewise smooth subdivision surfaces with normal control [C]. In: K Akeley ed. Proceeding of SIGGRAPH 2000, Boston, MA: Addision Wesley Professional, 2000:113-120.
[37] D L Page, A Koschan, Y Sun et al. Robust crease detection and curvature estimation of piecewise smooth surfaces from triangle mesh approximations using normal voting[C]. In: C E Bordley, A P Danyluk eds. Proceedings of the International Conference on Computer Vision and Pattern Recognition 2001, San Francisco, CA: Morgan Kaufmann, 2001:162-167.
[38] G Taubin. Estimating the tensor of curvature of a surface from a polyhedral approximation[C]. In WEL Grimson ed. Proceedings of the Fifth International Conference on Computer Vision, Los Alamitos: IEEE computer Society Press, 1995:902-907.
[39] 神会存,李建华,周来水。三角网格模型顶点法矢于理算曲率计算。计算机工程与应用,2005(26):12-15。
[40] Moreton H P, Sequin C H. Functional optimization for fair surface design[J]. In Computer Graphics Proceedings, Annual Conference Series, ACM SIGGRAPH, Chicago, Illinois, 1992. 167-176.
[41] Mayer U F. Numerical solutions for the surface diffusion flow in three space dimensions[OL], http://www.math.utah.edu/～mayer/math/Mayer07.pdf, 2001
[42] Desbrun M, Meyer M,Schroder P, et al. Implicit fairing of irregular meshes using diffusion and curvature flow[A]. Computer Graphics Proceedings, Annual Conferences Series, ACM SIGGRAPH, Los Angeles, California, 1999.317-324
[43] Cohen-Steiner D, Morvan J M. Restricted Delaunay triangulations and normal cycle[A]. Proceedings of the 19th Conference on Computational Geometry, San Diego, California,2003,53:312-321
[44] Jack Goldefather, Victoria Interrante. A novel cubic-order algorithm for approximating principal direction vectors. ACM Transactions on Graphics 23 (2004), no. 1, 45-63
[45] Holger Theisel, Christian Rossl, Thaleb Zayer, Hans-peter Seidel. Norma based estimateion of the curvature tensor for triangular meshes. 12th Pacific Conference on Computer Graphics and Applications (2004), 288-297
[46] Szymon Rusinkiewicz, Princeton University. Estimating curvatures and their deriveatives on triangle meshes.
[47] Anshuman Razdan, MyungSoo Bea. Curvature estimateion scheme for triangle meshes using biquadratic Bezier patches.
[48] Yuan-Shin Lee. Admissible tool orientation control of gouging avoidance for 5-axis complex surface machining. Computer-Aided Design 29(7), 1997. 507-521
[49] Byoung K Choi, Dac H Kim and Robert B Jerard. C-space approach to tool-path generation for die and mould machining. Computer-Aided design. 1997,29(9). 657-669.
[50] Mahadevan Balasubramaniam, Sanjay E. Sarma, Krzyztof Marciniak. Collision-free finishing toolpaths from visibility data. Computer-Aided design.35(2003) 359-374.
[51] 刘亚珍。NURBS曲面数控加工刀具可行方向计算研究。大连:大连理工大学硕士学位论文。2005。
[52] V. Pateloup, E. Duc, P. Ray. Corner optimization for pocket machining. International Journal of Machine Tools & Manufacture 44(2004) 1343-1353
[53] Bo H.Kim, Byoung K. Choi. Machining efficiency comparison direction-parallel tool path with contour-parallel tool path. Computer-Aid Design 2002,34,89-95
[2] 柯映林等著。反求工程CAD建模理论、方法和系统。2006,北京,机械工业出版社。
[3] Hugues Hoppe, Tony Derose, Tom Duchamp. Surface reconstruction from unorganized points. Computer Draphics, 1992,26(2):71-78.
[4] 王平江,黄雪梅,陈吉红,周济。曲面激光密集测量三维数据的三角片逼近算法。工程图学学报,1998,1:17-28
[5] Loney GC, Ozsoy TM. NC machining of free form surfaces. Computer-Aid Design 1987;19(2)85-90
[6] Elber G, Cohen E. Toolpath generating for free form surface. Computer-Aid Design 1994;26(6)490-6
[7] His-Yung Feng, Zhengji Teng. Iso-planar piecewise linear NC tool path generation from discrete measured data points. Computer-Aid Design. 37(2005) 55-64.
[8] Bobrow JE. NC machining tool path generation from CSG part representations. Computer-Aid Design 1985;17, 69-76
[9] Park SC, Choi BK. Tool path planning for direction-parallel area milling. Computer-Aid Design 2000;32(1): 17-25
[10] S. Ding, M.A Mannan, A.N. Poo, D.C.H. Yang, Z. Han. Adaptive iso-planar tool path generation for machining of free-form surfaces. Computer-Aid Design 35(2003) 141-153
[11] Sang C. Park. Tool-path generation for Z-constant contour machining. Computer-Aid Design 2003,35(1): 27-36
[12] K.Suresh, D. C. H. Yang. Constant scallop-height machining of free-form surfaces. Journal of Engineering for Industry, 1994, 116(5): 253-259
[13] 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。11(2001),16-17.
[14] 任秉银,唐余勇。数控加工中的几何建模理论及其应用。2000。哈尔滨工业大学出版社。
[15] Beom-Sahng Ryuh, Sang Min Park, Gordon R.Pennock. An automatic tool changer and integrated software for a robotic die polishing station. Mechanism and machine Theory, 2005. Article in press.
[16] D.C.H. Yang, J.-J. Chuang et. Boundary-conformed toolpath generation for trimmed free-form surfaces. Computer-Aid Design 2003,35: 127-129
[17] 苏云玲。三元整体叶轮的几何造型与五坐标数控加工。大连:大连理工大学硕士学位论文。2003。
[18] 周艳红。自由曲面三坐标加工走刀步长的确定。华中理工大学学报,1998,26(8),7-9
[19] 彭群生,胡国飞。三角网格的参数化。计算机辅助设计与图形学学报。2004,16期6卷。731-738。
[20] Eck M, DeRose T, et al. Multiresolution Analysis of arbitrary meshes[A]. In Proceeding of SIGGRAPH' 95[C], 1995: 173-182.
[21] 周昆,鲍虎军,石教英。统一的数字几何处理框架。计算机学报。2002,25(9),904-909.
[22] 谭家万,金一丞,石教英。任意拓扑三角形网格的全局参数化。中国图像图形学报。2003,8(6),686-691
[23] Lee A W.F, Sweldens W, et al. MAPS: Multiresolution Adaptive Parameterization of surfaces. In: Proceeding of SIGGRAPH' 98[C], 1998:95-104.
[24] Kun Zhou, Hujun Bao, Jiaoying Shi. 3D surface filtering using spherical harmonics. Computer-aided design, 36 (2004):363-375
[25] Alexa M, Merging polyhedron shapes with scattered features [J].The visual Computer. 2000,16(1):26-37
[26] Peter Lindstrom, Greg Turk. Fast and memory efficient polygonal simplification. IEEE Visualization 98 Conference Proceedings: October 1998: 276-86
[27] 王海霞。RE中基于流形网格的四边形区域划分方法研究。大连:大连理工大学硕士学位论文。2004。
[28] Floater M S. Parametdzation and smooth approximation of surface triangulations [J]。Computer-Aid Design 1997;14(3): 231-250
[29] Hormann K, Greiner G. MIPS: An efficient global Parametrization method [M].In: Laurent P J.Sablonniere P, Schumaker L L,eds. Curve and surface Design. Nashville Tennessee: Vanderbilt University Press, 1999.153-162.
[30] 周昆,潘志庚,马小虎,石教英。调和映射的构造及其在图形学中的应用。中国图像图形学报。1998,3(7)578-582。
[31] 施法中。计算机辅助几何设计与非均匀有理B样条,北京:北京航天航空大学出版社。1994。
[32] 徐士良主编。C常用算法程序集。北京:清华大学出版社。年代,第二版。
[33] Sun Yuwen, Guo Dongming, Jia zhenyuan, Wang Haixia. Iso-parametric tool path generation from triangular meshes for freeform surface machining.
[34] 数学辞海。第一卷。北京,中国科学技术出版社。2002。
[35] 孙玉文,刘伟军,王越超。基于三角网格曲面模型的刀位轨迹计算方法。机械工程学报,2002,38(10),50-53。
[36] H Biermann, A Levin, D Zorin. Piecewise smooth subdivision surfaces with normal control [C]. In: K Akeley ed. Proceeding of SIGGRAPH 2000, Boston, MA: Addision Wesley Professional, 2000:113-120.
[37] D L Page, A Koschan, Y Sun et al. Robust crease detection and curvature estimation of piecewise smooth surfaces from triangle mesh approximations using normal voting[C]. In: C E Bordley, A P Danyluk eds. Proceedings of the International Conference on Computer Vision and Pattern Recognition 2001, San Francisco, CA: Morgan Kaufmann, 2001:162-167.
[38] G Taubin. Estimating the tensor of curvature of a surface from a polyhedral approximation[C]. In WEL Grimson ed. Proceedings of the Fifth International Conference on Computer Vision, Los Alamitos: IEEE computer Society Press, 1995:902-907.
[39] 神会存,李建华,周来水。三角网格模型顶点法矢于理算曲率计算。计算机工程与应用,2005(26):12-15。
[40] Moreton H P, Sequin C H. Functional optimization for fair surface design[J]. In Computer Graphics Proceedings, Annual Conference Series, ACM SIGGRAPH, Chicago, Illinois, 1992. 167-176.
[41] Mayer U F. Numerical solutions for the surface diffusion flow in three space dimensions[OL], http://www.math.utah.edu/～mayer/math/Mayer07.pdf, 2001
[42] Desbrun M, Meyer M,Schroder P, et al. Implicit fairing of irregular meshes using diffusion and curvature flow[A]. Computer Graphics Proceedings, Annual Conferences Series, ACM SIGGRAPH, Los Angeles, California, 1999.317-324
[43] Cohen-Steiner D, Morvan J M. Restricted Delaunay triangulations and normal cycle[A]. Proceedings of the 19th Conference on Computational Geometry, San Diego, California,2003,53:312-321
[44] Jack Goldefather, Victoria Interrante. A novel cubic-order algorithm for approximating principal direction vectors. ACM Transactions on Graphics 23 (2004), no. 1, 45-63
[45] Holger Theisel, Christian Rossl, Thaleb Zayer, Hans-peter Seidel. Norma based estimateion of the curvature tensor for triangular meshes. 12th Pacific Conference on Computer Graphics and Applications (2004), 288-297
[46] Szymon Rusinkiewicz, Princeton University. Estimating curvatures and their deriveatives on triangle meshes.
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