复杂曲面非球头刀宽行铣削加工的几何学原理与方法
|
摘要
五轴数控加工是实现复杂曲面零件高效切削的有效手段。球头铣刀因具有适应性强、刀位规划简单等优点而被广泛采用,但是其加工性能较差,加工精度和效率也比较低。与此相比,非球头铣刀不仅能够改善切削性能,而且可以通过调整刀具姿态,使刀触点附近的刀具包络曲面充分逼近理论设计曲面,进而实现复杂曲面零件的宽行加工。但是设计曲面的复杂性和刀具曲面的多样性对处理好干涉避免、误差控制和路径光顺等问题,生成高质量的刀具路径带来了很多困难,因而限制了五轴机床潜力的发挥。
本学位论文以复杂曲面五轴铣削加工为应用背景,研究直纹面侧铣与自由曲面端铣加工刀具路径规划的几何学原理与方法。重点从回转刀具空间运动扫掠体包络面的建模、任意回转刀具五轴侧铣加工刀具路径整体优化、整体叶轮刀具尺寸与路径光顺性优化和平底刀高阶切触加工自由曲面刀具路径规划这四个方面展开研究,主要研究工作及创新性成果如下:
一.提出了刀具空间运动下扫掠体包络面的两种解析计算方法:(1)应用包络条件和刚体运动的物体速度推导出包络面特征线的解析表达式,该方法适用于任意回转刀具,并且具有无须引入附加瞬时标架、公式简洁明了的优点。(2)将回转刀具的切削刃回转面表示为单参数球族的包络面,单参数球族经过空间运动生成一双参数球族,因此刀具空间运动扫掠体的包络面就是这一双参数球族的包络,而刀轴面就是球心面。该方法建立起了包络面与刀轴面之间的关联,为点-刀具包络面法向误差的定义提供了思路。
二.基于双参数球族法扫掠体包络面建模,提出了一种无需构造包络曲面即可计算空间一点到包络面有向距离的新方法,将对包络面的操作转换为对刀轴面的操作,同时推导出点-包络面有向距离关于刀轴面形状控制参数的一阶梯度表达式,刻画了刀轴轨迹微小调整对包络曲面形状和加工误差的影响。通过对设计曲面的离散化表示,将任意回转刀具侧铣加工刀具路径整体优化问题归结为刀具扫掠包络面向设计曲面离散点云的最佳一致逼近问题,应用距离函数的导数信息构造出了高效的数值优化算法,应用于非可展直纹面侧铣加工。该方法不仅可以在遵循轮廓度误差评定准则的前提下极大地减小加工误差,而且可以有效地处理无过切约束。
三.提出了侧铣加工刀具尺寸优化和刀具路径光顺策略,解决了叶轮类零件侧铣刀具路径规划时容易产生干涉和不光顺轨迹的问题。应用点-包络面距离函数,解决了刀具包络曲面与设计曲面的边界对齐问题和刀具与相邻叶片的碰撞检测问题,并建立起以加工误差和干涉避免同时作为约束的优化模型,进而获取侧铣加工最优刀具尺寸。同时,以曲面应变能量表征刀轴轨迹面的光顺性,将其引入到侧铣加工刀具路径整体优化模型中,从而在满足加工精度的前提下得到较平滑的刀具路径。该方法适用于任意回转刀具的侧铣半精加工和精加工,生成的刀具路径已成功应用于整体叶轮叶片的侧铣加工。
四.提出了平底刀高阶切触加工自由曲面的刀具路径规划方法。基于刀具包络曲面的局部重建原理,解析求解了刀触点处的所有二阶切触刀位,在此约束下建立刀具可行空间,不仅极大地降低了干涉检查的计算量,而且将刀具包络曲面与设计曲面的二阶切触特征融入到优化模型之中。在此基础上,既可以通过计算包络曲面与设计曲面沿进给垂直方向的诱导法曲率导数,得到最优切触刀具路径;也可以通过求解以相邻刀具方向夹角的和为目标的最短路径算法,得到最优光顺刀具路径。该方法可以在避免刀具干涉的前提下,有效地光顺刀具路径,并显著提高切削带宽。
本学位论文在对刀具回转面族包络取得规律性认识的基础上,将干涉避免、切削误差和路径光顺等纳入到刀具路径优化的统一模型中,实现了复杂曲面非球头刀的宽行铣削加工,研究结果发展和丰富了五轴加工刀具路径规划的原理与方法。
Five-axis NC machining provides an efficient way to manufacture complex surfaces. The ball-end cutter is widely applied in engineering applications because it is easy to position the cutter. But the lower machining performance and efficiency limit the full use of the five-axis machining. Therefore, the non-ball-end cutters are preferred for complex surface machining. The tool envelope surface is applied to approximate the design surface by adjusting the cutter orientations, so that the cutting performance can be improved and the machining strip width can be increased. However, the complexity of the design surface and the diversity of the cutter surface bring lots of challenges in avoiding the interferences, controlling the machining errors and smoothing the tool orientations during the tool path generation.
The tool path optimizations for five-axis flank milling of ruled surfaces and point milling of free-form surfaces are studied in this dissertation. Four aspects are considered including the analytical expression of the swept envelope of a rotary cutter undergoing spatial motion, the global tool path optimization for flank milling of ruled surfaces, cutter size optimization and smooth tool path generation for flank milling of centrigural impellers, and interference-free tool positioning for high-order contact milling of free-form surfaces with a flat-end cutter. The main research work and the novel contributions are listed as follows:
Firstly, two methods for analytically computing the swept envelope undergoing general spatial motions are proposed. (1) Based on the tangency condition in the envelope theory and the representation of the body velocity in spatial kinematics, a closed-form solution of the swept envelope of a general rotary cutter moving along a multi-axis tool path is derived. No additional moving frames or local frames are required. (2) Based on the observation that the cutter surfaces can be treated as the envelope surface of a one-parameter family of spheres, the analytical expression of the envelope of the swept volumes are derived using the envelope theory of sphere congruence. The method associates the tool envelope surface with the tool axis trajectory surface, based on which the measure of the geometric errors is proposed.
Secondly, the signed distance from a point on the design surface to the tool envelope surface is defined without constructing the swept envelope. This transforms the manipulation of the envelope surface to that of the tool axis trajectory surface. Also the first order differential increment of the signed distance with respect to the differential deformation of the tool axis trajectory surface is derived. It quantitatively characterizes the change in the geometric deviation under the deformation of tool trajectory. On this basis, the tool path optimization for flank milling of ruled surfaces is developed in a unified framework, and a sequential approximation algorithm is developed for the optimization. The numerical examples indicate that the proposed method can improve the machining accuracy greatly, and can handle the optimization with non-overcut constraint effectively.
Thirdly, strategies for cutter size optimization and smooth tool path generation are proposed for five-axis flank milling of centrifugal impellers. Aside from the local interference of the cutter with the design surface, the global interferences with the hub surface and the adjacent blade surface are also considered in the optimization models. The distance function is redefined with parameters of both tool trajectory and cutter radius, and used as contraints to avoid interferences. Meanwhile, by characterizing the geometric smoothness by the strain energy of the cutter axis trajectory surface, the tool trajectory is globally smoothed while preserving the geometric accuracy of the machined part. The method applies to both finish and semi-finish flank millings, and the generated tool paths are applied to the practical machining of the centrifugal impellers.
Fourthly, based on the local approximation of the tool envelope surface to the design surface, the strategies for interference-free and smooth tool path generation are proposed for five-axis machining of free-form surfaces with a flat-end cutter. The tool feasibility space is established on the basis of analytically computing the second-order contact cutter locations, so that the geometric contact information between the tool envelope surface and the design surface is included into the optimization models while reducing the computational load. On this basis, the optimal contact tool path can be derived by calculating the relative curvature derivatives between the design surface and the tool envelope surface in the normal section orthogonal to the feed direction. Also, the optimal smooth tool path can be generated by globally minimizing the variation of the tool orientations along the whole tool path. The numerical examples indicate that the proposed method can avoid interferences effectively and improve the machining strip width greatly.
Based on the modeling of the cutter swept envelope, five-axis non-ball-end milling of complex surfaces with a large machining strip width is focused. A uniform tool path optimization frame is founded with comprehensive consideration of interference avoiding, geometric error controling, and tool trajectory smoothing, etc. The proposed methods enrich the tool path planning strategies for five-axis machining.
本学位论文以复杂曲面五轴铣削加工为应用背景,研究直纹面侧铣与自由曲面端铣加工刀具路径规划的几何学原理与方法。重点从回转刀具空间运动扫掠体包络面的建模、任意回转刀具五轴侧铣加工刀具路径整体优化、整体叶轮刀具尺寸与路径光顺性优化和平底刀高阶切触加工自由曲面刀具路径规划这四个方面展开研究,主要研究工作及创新性成果如下:
一.提出了刀具空间运动下扫掠体包络面的两种解析计算方法:(1)应用包络条件和刚体运动的物体速度推导出包络面特征线的解析表达式,该方法适用于任意回转刀具,并且具有无须引入附加瞬时标架、公式简洁明了的优点。(2)将回转刀具的切削刃回转面表示为单参数球族的包络面,单参数球族经过空间运动生成一双参数球族,因此刀具空间运动扫掠体的包络面就是这一双参数球族的包络,而刀轴面就是球心面。该方法建立起了包络面与刀轴面之间的关联,为点-刀具包络面法向误差的定义提供了思路。
二.基于双参数球族法扫掠体包络面建模,提出了一种无需构造包络曲面即可计算空间一点到包络面有向距离的新方法,将对包络面的操作转换为对刀轴面的操作,同时推导出点-包络面有向距离关于刀轴面形状控制参数的一阶梯度表达式,刻画了刀轴轨迹微小调整对包络曲面形状和加工误差的影响。通过对设计曲面的离散化表示,将任意回转刀具侧铣加工刀具路径整体优化问题归结为刀具扫掠包络面向设计曲面离散点云的最佳一致逼近问题,应用距离函数的导数信息构造出了高效的数值优化算法,应用于非可展直纹面侧铣加工。该方法不仅可以在遵循轮廓度误差评定准则的前提下极大地减小加工误差,而且可以有效地处理无过切约束。
三.提出了侧铣加工刀具尺寸优化和刀具路径光顺策略,解决了叶轮类零件侧铣刀具路径规划时容易产生干涉和不光顺轨迹的问题。应用点-包络面距离函数,解决了刀具包络曲面与设计曲面的边界对齐问题和刀具与相邻叶片的碰撞检测问题,并建立起以加工误差和干涉避免同时作为约束的优化模型,进而获取侧铣加工最优刀具尺寸。同时,以曲面应变能量表征刀轴轨迹面的光顺性,将其引入到侧铣加工刀具路径整体优化模型中,从而在满足加工精度的前提下得到较平滑的刀具路径。该方法适用于任意回转刀具的侧铣半精加工和精加工,生成的刀具路径已成功应用于整体叶轮叶片的侧铣加工。
四.提出了平底刀高阶切触加工自由曲面的刀具路径规划方法。基于刀具包络曲面的局部重建原理,解析求解了刀触点处的所有二阶切触刀位,在此约束下建立刀具可行空间,不仅极大地降低了干涉检查的计算量,而且将刀具包络曲面与设计曲面的二阶切触特征融入到优化模型之中。在此基础上,既可以通过计算包络曲面与设计曲面沿进给垂直方向的诱导法曲率导数,得到最优切触刀具路径;也可以通过求解以相邻刀具方向夹角的和为目标的最短路径算法,得到最优光顺刀具路径。该方法可以在避免刀具干涉的前提下,有效地光顺刀具路径,并显著提高切削带宽。
本学位论文在对刀具回转面族包络取得规律性认识的基础上,将干涉避免、切削误差和路径光顺等纳入到刀具路径优化的统一模型中,实现了复杂曲面非球头刀的宽行铣削加工,研究结果发展和丰富了五轴加工刀具路径规划的原理与方法。
Five-axis NC machining provides an efficient way to manufacture complex surfaces. The ball-end cutter is widely applied in engineering applications because it is easy to position the cutter. But the lower machining performance and efficiency limit the full use of the five-axis machining. Therefore, the non-ball-end cutters are preferred for complex surface machining. The tool envelope surface is applied to approximate the design surface by adjusting the cutter orientations, so that the cutting performance can be improved and the machining strip width can be increased. However, the complexity of the design surface and the diversity of the cutter surface bring lots of challenges in avoiding the interferences, controlling the machining errors and smoothing the tool orientations during the tool path generation.
The tool path optimizations for five-axis flank milling of ruled surfaces and point milling of free-form surfaces are studied in this dissertation. Four aspects are considered including the analytical expression of the swept envelope of a rotary cutter undergoing spatial motion, the global tool path optimization for flank milling of ruled surfaces, cutter size optimization and smooth tool path generation for flank milling of centrigural impellers, and interference-free tool positioning for high-order contact milling of free-form surfaces with a flat-end cutter. The main research work and the novel contributions are listed as follows:
Firstly, two methods for analytically computing the swept envelope undergoing general spatial motions are proposed. (1) Based on the tangency condition in the envelope theory and the representation of the body velocity in spatial kinematics, a closed-form solution of the swept envelope of a general rotary cutter moving along a multi-axis tool path is derived. No additional moving frames or local frames are required. (2) Based on the observation that the cutter surfaces can be treated as the envelope surface of a one-parameter family of spheres, the analytical expression of the envelope of the swept volumes are derived using the envelope theory of sphere congruence. The method associates the tool envelope surface with the tool axis trajectory surface, based on which the measure of the geometric errors is proposed.
Secondly, the signed distance from a point on the design surface to the tool envelope surface is defined without constructing the swept envelope. This transforms the manipulation of the envelope surface to that of the tool axis trajectory surface. Also the first order differential increment of the signed distance with respect to the differential deformation of the tool axis trajectory surface is derived. It quantitatively characterizes the change in the geometric deviation under the deformation of tool trajectory. On this basis, the tool path optimization for flank milling of ruled surfaces is developed in a unified framework, and a sequential approximation algorithm is developed for the optimization. The numerical examples indicate that the proposed method can improve the machining accuracy greatly, and can handle the optimization with non-overcut constraint effectively.
Thirdly, strategies for cutter size optimization and smooth tool path generation are proposed for five-axis flank milling of centrifugal impellers. Aside from the local interference of the cutter with the design surface, the global interferences with the hub surface and the adjacent blade surface are also considered in the optimization models. The distance function is redefined with parameters of both tool trajectory and cutter radius, and used as contraints to avoid interferences. Meanwhile, by characterizing the geometric smoothness by the strain energy of the cutter axis trajectory surface, the tool trajectory is globally smoothed while preserving the geometric accuracy of the machined part. The method applies to both finish and semi-finish flank millings, and the generated tool paths are applied to the practical machining of the centrifugal impellers.
Fourthly, based on the local approximation of the tool envelope surface to the design surface, the strategies for interference-free and smooth tool path generation are proposed for five-axis machining of free-form surfaces with a flat-end cutter. The tool feasibility space is established on the basis of analytically computing the second-order contact cutter locations, so that the geometric contact information between the tool envelope surface and the design surface is included into the optimization models while reducing the computational load. On this basis, the optimal contact tool path can be derived by calculating the relative curvature derivatives between the design surface and the tool envelope surface in the normal section orthogonal to the feed direction. Also, the optimal smooth tool path can be generated by globally minimizing the variation of the tool orientations along the whole tool path. The numerical examples indicate that the proposed method can avoid interferences effectively and improve the machining strip width greatly.
Based on the modeling of the cutter swept envelope, five-axis non-ball-end milling of complex surfaces with a large machining strip width is focused. A uniform tool path optimization frame is founded with comprehensive consideration of interference avoiding, geometric error controling, and tool trajectory smoothing, etc. The proposed methods enrich the tool path planning strategies for five-axis machining.
引文
[1]周济,周艳红.数控加工技术.北京:国防工业出版社. 2005.
[2]丁汉,朱利民.复杂曲面数字化制造的几何学理论和方法.北京:科学出版社. 2011.
[3] Abdel-Malek K,Yeh HJ. Geometric representation of the swept volume using Jacobian rank-deficiency conditions. Computer Aided Design,1997,29:457–468.
[4] Abdel-Malek K, Seaman W, Yeh HJ. NC-verification of up to 5 axis machining processes using manifold stratification. ASME Journal of Manufacturing Science and Engineering, 2000, 122:1–11.
[5] Yang JZ, Abdel-Malek K. Verification of NC machining processes using swept volumes. International Journal of Advanced Manufacturing Technology, 2006, 28(1-2): 82-91.
[6] Blackmore D, Leu MC, Wang KK. Applications of Flows and Envelopes to NC Machining. 1992, Annals of CIRP, 41(1):493-496.
[7] Blackmore D, Leu MC, Wang LP. The sweep-envelope differential equation algorithm and its application to NC-machining verification. Computer Aided Design, 1997, 29(9): 629–637.
[8] Wang L, Leu MC, Blackmore D. Generating sweep solids for NC verification using the SEDE method. Proceedings of the Fourth Symposium on Solid Modeling and Applications. Atlanta, Georgia, 1997, 364-375.
[9] Blackmore D, Samulyak R, Leu MC. Trimming swept volumes. Computer Aided Design, 1999, 31(3): 215–224.
[10] Roth D, Bedi S, Ismail F, et al. Surface swept by a toroidal cutter during 5-axis machining. Computer Aided Design 2001,33:57–63.
[11] Wang WP, Wang KK. Geometric modeling for swept volume of moving solids. IEEE Computer Graphics and Applications, 1986, 6(12):8-17.
[12] Chung YC, Park JW, Shin H, et al. Modeling the surface swept by a generalizedcutter for NC-verification. Computer Aided Design 1998, 30: 587–593.
[13] Sheltami K, Bedi S, Ismail F. Swept volumes of toroidal cutters using generating curves. International Journal of Machine Tools and Manufacture, 1998, 38:855-70.
[14] Pottmann H, Peternell M. Envelopes-computational theory and applications. Proceedings of Spring Conference on Computer Graphics and its Applications. Budmerice, Slovakia, 2000, 3–23.
[15]方向,彭群生.多轴数控加工中刀具包络面的计算.计算机辅助设计与图形学学报,2000,12(8): 609-613.
[16]严思杰,周云飞,陈学东.五轴NC加工中刀具运动包络面的计算.中国机械工程,2005,16(23):2120-2124.
[17]严思杰,周云飞,陈学东.五轴NC加工干涉检查与避免算法研究.中国机械工程,2006,17(17):1822-1825.
[18] Chiou CJ,Lee YS. Swept surface determination for five-axis numerical control machining. International Journal of Machine Tools & Manufacture,2002,42: 1497-1507.
[19] Chiou CJ. Accurate tool position for five-axis ruled surface machining by swept envelope approach. 2004,Computer-Aided Design,36(10):967–974.
[20] Chiou CJ,Lee YS. Optimal tool orientation for five-axis tool-end machining by swept envelope approach. Trans. Of ASME,Journal of Manufacturing Science & Engineering,2005,127:810-818.
[21] Weinert K,Du Shangjian,Damm P,et al. Swept volume generation for the simulation of machining processes. International Journal of Machine Tools & Manufacture,2004,44:617-628.
[22] Du Shangjian,Surmann T,Webber O,et al. Formulating swept profiles for five-axis tool motions. International Journal of Machine Tools & Manufacture,2005,45: 849-861.
[23] Aras E. Generating Cutter Swept Envelopes in Five-Axis Milling by Two-Parameter Families of Spheres, Computer-Aided Design, 2009, 41(2):95–105.
[24] Aras E. Cutter-workpiece engagement identification in multi-aixs milling [Dissertation]. Vancouver: The University of British Columbia, 2008.
[25] Bedi S, Mann S, Menzel C. Flank milling with flat end milling cutters.Computer-Aided Design, 2003, 35(3):293–300.
[26] NREC C.MAX-PAC, 2011, http://www.conceptsnrec.com.
[27] Senatore J., Monies F., Redonnet J. M., Rubio W. Analysis of improved positioning in five-axis ruled surface milling using envelope surface. Computer Aided Design, 2005, 37(10): 989-998.
[28] Liu X W. Five-axis NC cylindrical milling of sculptured surfaces. Computer-Aided Design, 1995, 27(12): 887–894.
[29] Lee Y. S., Koc B. Ellipse-offset approach and inclined zig-zag method for multi-axis roughing of ruled surface pockets. Computer-Aided Design, 1998, 30(12): 957-971.
[30] Rubio W, Lagarrigue P, Dessein G, Pastor F. Calculation of tool paths for a torus mill on free-form surfaces on five-axis machines with detection and elimination of interference. The International Journal of Advanced Manufacturing Technology, 1998, 14(1): 13–20.
[31] Redonnet J M, Rubio W, Dessein G. Side milling of ruled surfaces: Optimum positioning of the milling cutter and calculation of interference. The International Journal of Advanced Manufacturing Technology, 1998, 14(7): 459–465.
[32] Monies F, Redonnet J M, Rubio W, et al. Improved position of a conical mill for machining ruled surfaces: application to turbine blades. Proceedings of IMechE, Part B: Journal of Engineering Manufacture, 2000, 214(B): 625–634.
[33] Monies F, Rubio W, Redonnet J M, et al. Comparative study of interference caused by different position settings of a conical milling cutter on a ruled surface. Proceedings of IMechE, Part B: Journal of Engineering Manufacture, 2001, 215(9): 1305–1317.
[34] Monies F, Felices J N, Rubio W, et al. Five-axis NC milling of ruled surfaces: optimal geometry of a conical tool. International Journal of Production Research, 2002, 40(12): 2901–2922.
[35] Tsay D M, Her M J. Accurate 5-axis machining of twisted ruled surfaces. Trans. of ASME, Journal of Manufacturing Science & Engineering, 2001, 123(4): 731–738.
[36]于源,赖天琴,员敏,王小椿.基于特征的直纹面5轴侧铣精加工刀位计算方法.机械工程学报, 2002, 38(6): 130-133.
[37]陈晧晖,刘华明,孙春华.发动机叶轮侧铣数控加工方法及误差计算.机械工程学报, 2003, 39(7): 143-145.
[38] Menzel C, Bedi S, Mann S. Triple tangent flank milling of ruled surfaces. Computer-Aided Design, 2004, 36(3):289–296.
[39] Li C G, Mann S, Bedi S. Error measurements for flank milling. Computer-Aided Design, 2003, 37:1459–1468.
[40] Li C G, Bedi S, Mann S. Flank milling of ruled surface with conical tools - an optimization approach. The International Journal of Advanced Manufacturing Technology, 2006, 29(11-12): 1115–1124.
[41]吴宝海,王尚锦.基于正向杜邦指标线的五坐标侧铣加工.机械工程学报, 2006, 42(11): 192-196.
[42] Senatore J., Monies F., Landon Y., Rubio W. Optimising positioning of the axis of a milling cutter on an offset surface by geometric error minimisation. International Journal of Advanced Manufacturing Technology, 2008, 37(9-10): 861-871.
[43] Senatore J., Landon Y., Rubio W. Analytical estimation of error in flank milling of ruled surfaces. Computer Aided Design, 2008, 40(5): 595-603.
[44] Ding Y., Zhu L., Ding H. Semidefinite programming for Chebyshev fitting of spatial straight line with applications to cutter location planning and tolerance evaluation. Precision Engineering, 2007, 31(4): 364-368.
[45] Ding Y, Zhu LM, Ding H. On a novel approach to planning cylindrical cutter location for flank milling of ruled surfaces. International Journal of Production Research, 2009, 47(12): 3289–3305.
[46]赖天琴,李锋,孔德治等.计算机辅助加工直纹面的新方法.西安交通大学学报, 1996, 30(2): 61-68.
[47] Chu C. H., Chen J. T. Tool path planning for five-axis flank milling with developable surface approximation. International Journal of Advanced Manufacturing Technology, 2006, 29(7-8): 707-713.
[48] Chu C H, Huang WN, Hsu YY. Machining accuracy improvement in five-axis flank milling of ruled surfaces. International Journal of Machine Tools & Manufacture,2008,48:914-921.
[49] Elber G., Fish R. 5-axis freeform surface milling using piecewise ruled surfaceapproximation. Journal of Manufacturing Science and Engineering, 1997, 119(3): 383-387.
[50] Han Z., Chuang J.-J., Yang D. C. H. Isophote-based ruled surface approximation of free-form surfaces and its application in NC machining. International Journal of Production Research, 2001, 39(9): 1911-1930.
[51] Tsay DM, Chen HC, Her MJ. A study on five flank machining of centrifugal compressor impellers. Trans. Of ASME, Journal of Engineering for Gas and Power, 2002, 124(1): 177–181.
[52] Liang Q, Wang YZ, Fu HY, et al. Cutting path planning for ruled surface impellers. Chinese journal of aeronautics, 2008,21(5):462–471.
[53] Young HT, Chuang LC, Gerschwiler K, et al. A five-axis rough machining approach for a centrifugal impeller. The International Journal of Advanced Manufacturing Technology, 2004, 23(3–4): 233–239.
[54]陈良骥,王永章.整体叶轮五轴侧铣数控加工方法的研究.计算机集成制造系统. 2007, 13(1):141-146.
[55] Lartigue C, Duc E, Affouard A. Tool path deformation in 5-axis flank milling using envelope surface. Computer-Aided Design, 2003, 35(4):375– 382.
[56] Wu P. H., Li Y. W., Chu C. H. Optimized tool path generation based on dynamic programming for five-axis flank milling of rule surface. International Journal of Machine Tools and Manufacture, 2008, 48(11): 1224-1233.
[57]宫虎,曹利新,刘健.数控侧铣加工非可展直纹面的刀位整体优化原理与方法.机械工程学报, 2005, 41(11): 134-139.
[58] Gong H, Cao LX, Liu J. Improved positioning of cylindrical cutter for flank milling ruled surfaces. Computer-Aided Design, 2005, 37(12): 1205–1213.
[59]宫虎.五坐标数控加工运动几何学基础及刀位规划原理与方法的研究[博士论文].大连:大连理工大学, 2005.
[60] Gong H, Wang N. Optimize tool paths of flank milling with generic cutters based on approximation using the tool envelope surface. Computer-Aided Design, 2010, 41(12):981–989.
[61] Gong H, Wang N. 5-axis flank milling free-form surfaces considering constraints. Computer-Aided Design, 2011, 43(6):563-572.
[62] International Organization for Standardization. ISO/R 1101. Technical Drawings-Geometrical Tolearncing. Geneva: ISO, 1983.
[63] American Society of Mechanical Engineers. ANSI Standard Y14.5. Dimensioning and tolerancing. New York: ASME, 1994.
[64] Ding H, Zhu LM. Global Optimization of Tool Path for Five-axis Flank Milling with a Cylindrical Cutter. Science in China-Ser. E, 2009, 52(8): 2449-2459.
[65]张小明.五轴数控铣削加工几何-动力学建模与工艺参数优化[博士论文].上海:上海交通大学, 2009.
[66] Zhang XM, Zhu LM, Zheng G, Ding H. Tool path optimisation for flank milling ruled surface based on the distance function. International Journal of Production Research, 2010, 48(14): 4233–4251.
[67] Pechard P-Y, Tournier C, Lartigue C. Geometrical deviations versus smoothness in 5-axis high-speed flank milling. International Journal of Machine Tools and Manufacture, 2009, 41(12): 918-929.
[68] Vickers G. W. a. G., K W. Ball-mills versus end-mills for curved surface machining. Trans. of ASME, Journal of Engineer for Industry , 1989, 111: 22-26.
[69] Jensen CG, Anderson DC. Accurate tool placement and orientation for finish surface machining. Journal Design and Manufacture. 1993, 3(4):251–261.
[70] Kruth JP, Klewais P. Optimization and dynamic adaptation of the cutter inclination during five-axis milling of sculptured surfaces. CIRP Annals- Manufacturing Technology, 1994, 43(1): 443–448.
[71] Deng Z, Leu MC, Wang L, etc. Determination of flat-end orientation in 5-axis machining. ASME MED, 1996,4: 73–80.
[72] Lee YS. Admissible tool orientation control of gouging avoidance for 5-axis complex surface machining. Computer-Aided Design, 1997, 29(7): 507–521.
[73] Rao N, Bedi S, Buchal R. Implementation of the principal-axis method for machining of complex surfaces. International Journal of Advanced Manufacturing Technology.1996,11: 249–57.
[74] Bedi S, Gravelle S, Chen YH. Principal curvature alignment technique for machining complex surface. Trans. Of ASME, Journal of Manufacturing Science & Engineering. 1997, 119(4):756–765.
[75] Lo CC. Efficient cutter-path planning for five-axis surface machining with a flat-end cutter. Computer-Aided Design, 1999, 31(9):557–566.
[76] Chiou CJ,Lee YS. A machining potential field approach to tool path generation for multi-axis sculptured surface machining. Computer-Aided Design, 2002, 34: 357–71.
[77] Yoon J-H, Pottmann H, Lee Y-S. Locally optimal cutting positions for 5-axis sculptured surface machining. Computer-Aided Design, 2003, 35: 69–81.
[78]倪炎榕,马登哲,张洪,郭静萍.圆环面刀具五坐标数控加工复杂曲面优化刀位算法.机械工程学报, 2001, 37(2): 87–90.
[79]王小椿,吴序堂,李艳斌.密切曲率法–一种自由曲面加工的新概念.西安交通大学学报, 1992, 26(5): 51–58.
[80] Wang XC, Ghosh SK, Li YB, Wu XT. Curvature catering-a new approach in manufacture of sculptured surfaces (part 1. theorem). Journal of Materials Processing Technology 1993, 38:159-75.
[81]沈兵,高军,吴联银等.基于密切曲率法的加工理论及算法研究.机械, 1998, 26(4):20-24.
[82]张瑞乾,沈兵,王小椿等.用二阶密切曲率法加工自由曲面.机械工程学报, 1998, 34(2):88-92..
[83] Rao A, Sarma R. On local gouging in five-axis sculptured surface machining using flat-end tools. Computer-Aided Design, 2000, 32: 409–420.
[84] Sarma R. Flat-ended tool swept sections for five-axis NC machining of sculptured surfaces. Trans. of the ASME, Journal of Manufacturing Science & Engineering. 2000, 122(1):158–165.
[85]曹利新,宫虎,刘健.曲面接触问题及其等距面接触特性研究.大连理工大学学报, 2007, 47(1):39-44.
[86] Gong H, Cao LX, Liu J. Second order approximation of tool envelope surface for 5-axis machining with single point contact. Computer-Aided Design, 2008, 40 (5): 604–615.
[87] Gong H, Fang FZ, Hu XT, et al. Optimization of tool positions locally based on the BCELTP for 5-axis machining of free-form surfaces. Computer-Aided Design, 2010, 42(6): 558–570.
[88] Zhu LM, Ding H, Xiong YL. Third Order Point Contact Approach for Five-Axis Sculptured Surface Machining Using Non-ball-end Tools -Part I: Third Order Approximation of Tool Envelope Surface. Science in China-Ser. E, 2010, 53(7): 1904-1912.
[89] Zhu LM, Ding H, Xiong YL. Third Order Point Contact Approach for Five-Axis Sculptured Surface Machining Using Non-ball-end Tools -Part II: Tool Positioning Strategy. Science in China-Ser. E, 2010, 53 (8): 2190-2197.
[90] Warkentin.A., Ismail F., Bedi S. Comparison between multi-point and other 5-axis tool positioning strategies. International Journal of Machine Tools and Manufacture, 2000, 40(2): 185-208.
[91] Li H, Feng HY. Efficient five-axis machining of free-form surfaces with constant scallop height tool paths. International Journal of Production Research,2004, 42(12):2403–2417.
[92] Barakchi Fard MJ, Feng HY. Effect of tool tilt angle on machining strip width in five-axis flat-end milling of free-form surfaces. The International Journal of Advanced Manufacturing Technology, 2009, 44(3-4): 211–222.
[93] Barakchi Fard MJ, Feng HY. Effective determination of feed direction and tool orientation in five-axis flat-end milling. Trans. of ASME, Journal of Manufacturing Science & Engineering. 2010, 132(6):061011.
[94] Ho M. C., Hwang Y. R., Hu C. H. Five-axis tool orientation smoothing using quaternion interpolation algorithm. International Journal of Machine Tool and Manufacture, 2003: 1259-1267.
[95] Zhang W., Zhang Y. F., Ge Q. J. Five-axis Tool Path Generation for Sculptured Surface Machining using Rational Bezier Motions of a Flat-end Cutter. Computer-Aided Design and Applications, 2004, 1(1): 251-260.
[96] Xia J., Ge Q. J. On the Exact Representation of the Boundary Surfaces of the Swept Volume of a Cylinder Undergoing Rational Bézier and B-Spline Motions. Journal of Mechanical Design, 2001, 123(2): 261-265.
[97] Xia J., Ge Q. J. An exact representation of effective cutting shapes of 5-axis CNC machining using rational Bezier and B-spline tool motions. IEEE International Conference on Robotics and Automation, 2001.
[98] Koch J. Method of directing the movement of a tool as part of a process to remove material from a block of material. US patent 6632053. OpenMind Sortware Technologies (DE), 2003.
[99] Langeron J. M., Duc E., Lartigue C., Bourdet P. A new format for 5-axis tool path computation, using Bspline curves. Computer Aided Design, 2004, 36(12): 1219-1229.
[100] Jun C. S., Cha K., Lee Y. S. Optimizing tool orientations for 5-axis machining by configuration-space search method. Computer-Aided Design, 2003, 35(6): 549-566.
[101] Wang N, Tang K. Automatic generation of gouge-free and angular-velocity-compliant five-axis toolpath. Computer-Aided Design, 2007, 39(10): 841–852.
[102] Wang N, Tang K. Five-axis tool path generation for a flat-end tool based on iso-conic partitioning. Computer-Aided Design, 2008, 40(12): 1067–1079.
[103] Castagnetti C., E. Duc P. R. The Domain of Admissible Orientation concept: A new method for five-axis tool path optimisation. Computer-Aided Design, 2008, 40(9): 938-950.
[104] Bi QZ, Wang YH and Ding H. A GPU-based algorithm for generating collision-free and orientation-smooth five-axis finishing tool paths of a ball-end cutter. International Journal of Production Research, 2010, 48(4):1105-1124.
[105]梅向明,黄敬之.微分几何.第3版.北京:高等教育出版社,2003.
[106] Murray R., Li Z., Sastry S. A mathematical introduction to robotic manipulation. CRC press, 1994.
[107] Zhu LM, Xiong ZH, Ding H, Xiong YL. A distance function based approach for localization and profile error evaluation of complex surface. Trans. Of ASME, Journal of Manufacturing Science & Engineering, 2004, 126(3): 542–554.
[108] LEE K, SEONG JK, KIM KJ, et al. Minimum distance between two sphere-swept surface. Computer-Aided Design, 2007, 39: 452-459.
[109] Hu S M, Li Y F, Ju T, Zhu X. Modifying the shape of NURBS surfaces with geometric constraints. Computer-Aided Design, 2001, 33(12): 903–912.
[110] Park H. Lofted B-spline surface interpolation by linearly constrained energyminimization. Computer-Aided Design, 2003, 35 (14): 1261–1268.
[111] Zhang CM, Zhang PF, Cheng FH. Fairing spline curves and surfaces by minimizing energy. Computer-Aided Design, 2001, 33(13): 913–923.
[112] Pourazady M, Xu X. Direct manipulations of NURBS surfaces subjected to geometric constraints. Computers & Graphics. 2006,30(4):598-609.
[113] Marler R.T., Arora J.S. Survey of multi-objective optimization methods for engineering. Structural and Multidisciplinary Optimization. 2004, 26(6):369-395.
[114]何乃翔,李世铎.共轭曲面的曲率张量与曲率导数张量.机械工程学报, 1979, 15(3-4): 93–108.
[2]丁汉,朱利民.复杂曲面数字化制造的几何学理论和方法.北京:科学出版社. 2011.
[3] Abdel-Malek K,Yeh HJ. Geometric representation of the swept volume using Jacobian rank-deficiency conditions. Computer Aided Design,1997,29:457–468.
[4] Abdel-Malek K, Seaman W, Yeh HJ. NC-verification of up to 5 axis machining processes using manifold stratification. ASME Journal of Manufacturing Science and Engineering, 2000, 122:1–11.
[5] Yang JZ, Abdel-Malek K. Verification of NC machining processes using swept volumes. International Journal of Advanced Manufacturing Technology, 2006, 28(1-2): 82-91.
[6] Blackmore D, Leu MC, Wang KK. Applications of Flows and Envelopes to NC Machining. 1992, Annals of CIRP, 41(1):493-496.
[7] Blackmore D, Leu MC, Wang LP. The sweep-envelope differential equation algorithm and its application to NC-machining verification. Computer Aided Design, 1997, 29(9): 629–637.
[8] Wang L, Leu MC, Blackmore D. Generating sweep solids for NC verification using the SEDE method. Proceedings of the Fourth Symposium on Solid Modeling and Applications. Atlanta, Georgia, 1997, 364-375.
[9] Blackmore D, Samulyak R, Leu MC. Trimming swept volumes. Computer Aided Design, 1999, 31(3): 215–224.
[10] Roth D, Bedi S, Ismail F, et al. Surface swept by a toroidal cutter during 5-axis machining. Computer Aided Design 2001,33:57–63.
[11] Wang WP, Wang KK. Geometric modeling for swept volume of moving solids. IEEE Computer Graphics and Applications, 1986, 6(12):8-17.
[12] Chung YC, Park JW, Shin H, et al. Modeling the surface swept by a generalizedcutter for NC-verification. Computer Aided Design 1998, 30: 587–593.
[13] Sheltami K, Bedi S, Ismail F. Swept volumes of toroidal cutters using generating curves. International Journal of Machine Tools and Manufacture, 1998, 38:855-70.
[14] Pottmann H, Peternell M. Envelopes-computational theory and applications. Proceedings of Spring Conference on Computer Graphics and its Applications. Budmerice, Slovakia, 2000, 3–23.
[15]方向,彭群生.多轴数控加工中刀具包络面的计算.计算机辅助设计与图形学学报,2000,12(8): 609-613.
[16]严思杰,周云飞,陈学东.五轴NC加工中刀具运动包络面的计算.中国机械工程,2005,16(23):2120-2124.
[17]严思杰,周云飞,陈学东.五轴NC加工干涉检查与避免算法研究.中国机械工程,2006,17(17):1822-1825.
[18] Chiou CJ,Lee YS. Swept surface determination for five-axis numerical control machining. International Journal of Machine Tools & Manufacture,2002,42: 1497-1507.
[19] Chiou CJ. Accurate tool position for five-axis ruled surface machining by swept envelope approach. 2004,Computer-Aided Design,36(10):967–974.
[20] Chiou CJ,Lee YS. Optimal tool orientation for five-axis tool-end machining by swept envelope approach. Trans. Of ASME,Journal of Manufacturing Science & Engineering,2005,127:810-818.
[21] Weinert K,Du Shangjian,Damm P,et al. Swept volume generation for the simulation of machining processes. International Journal of Machine Tools & Manufacture,2004,44:617-628.
[22] Du Shangjian,Surmann T,Webber O,et al. Formulating swept profiles for five-axis tool motions. International Journal of Machine Tools & Manufacture,2005,45: 849-861.
[23] Aras E. Generating Cutter Swept Envelopes in Five-Axis Milling by Two-Parameter Families of Spheres, Computer-Aided Design, 2009, 41(2):95–105.
[24] Aras E. Cutter-workpiece engagement identification in multi-aixs milling [Dissertation]. Vancouver: The University of British Columbia, 2008.
[25] Bedi S, Mann S, Menzel C. Flank milling with flat end milling cutters.Computer-Aided Design, 2003, 35(3):293–300.
[26] NREC C.MAX-PAC, 2011, http://www.conceptsnrec.com.
[27] Senatore J., Monies F., Redonnet J. M., Rubio W. Analysis of improved positioning in five-axis ruled surface milling using envelope surface. Computer Aided Design, 2005, 37(10): 989-998.
[28] Liu X W. Five-axis NC cylindrical milling of sculptured surfaces. Computer-Aided Design, 1995, 27(12): 887–894.
[29] Lee Y. S., Koc B. Ellipse-offset approach and inclined zig-zag method for multi-axis roughing of ruled surface pockets. Computer-Aided Design, 1998, 30(12): 957-971.
[30] Rubio W, Lagarrigue P, Dessein G, Pastor F. Calculation of tool paths for a torus mill on free-form surfaces on five-axis machines with detection and elimination of interference. The International Journal of Advanced Manufacturing Technology, 1998, 14(1): 13–20.
[31] Redonnet J M, Rubio W, Dessein G. Side milling of ruled surfaces: Optimum positioning of the milling cutter and calculation of interference. The International Journal of Advanced Manufacturing Technology, 1998, 14(7): 459–465.
[32] Monies F, Redonnet J M, Rubio W, et al. Improved position of a conical mill for machining ruled surfaces: application to turbine blades. Proceedings of IMechE, Part B: Journal of Engineering Manufacture, 2000, 214(B): 625–634.
[33] Monies F, Rubio W, Redonnet J M, et al. Comparative study of interference caused by different position settings of a conical milling cutter on a ruled surface. Proceedings of IMechE, Part B: Journal of Engineering Manufacture, 2001, 215(9): 1305–1317.
[34] Monies F, Felices J N, Rubio W, et al. Five-axis NC milling of ruled surfaces: optimal geometry of a conical tool. International Journal of Production Research, 2002, 40(12): 2901–2922.
[35] Tsay D M, Her M J. Accurate 5-axis machining of twisted ruled surfaces. Trans. of ASME, Journal of Manufacturing Science & Engineering, 2001, 123(4): 731–738.
[36]于源,赖天琴,员敏,王小椿.基于特征的直纹面5轴侧铣精加工刀位计算方法.机械工程学报, 2002, 38(6): 130-133.
[37]陈晧晖,刘华明,孙春华.发动机叶轮侧铣数控加工方法及误差计算.机械工程学报, 2003, 39(7): 143-145.
[38] Menzel C, Bedi S, Mann S. Triple tangent flank milling of ruled surfaces. Computer-Aided Design, 2004, 36(3):289–296.
[39] Li C G, Mann S, Bedi S. Error measurements for flank milling. Computer-Aided Design, 2003, 37:1459–1468.
[40] Li C G, Bedi S, Mann S. Flank milling of ruled surface with conical tools - an optimization approach. The International Journal of Advanced Manufacturing Technology, 2006, 29(11-12): 1115–1124.
[41]吴宝海,王尚锦.基于正向杜邦指标线的五坐标侧铣加工.机械工程学报, 2006, 42(11): 192-196.
[42] Senatore J., Monies F., Landon Y., Rubio W. Optimising positioning of the axis of a milling cutter on an offset surface by geometric error minimisation. International Journal of Advanced Manufacturing Technology, 2008, 37(9-10): 861-871.
[43] Senatore J., Landon Y., Rubio W. Analytical estimation of error in flank milling of ruled surfaces. Computer Aided Design, 2008, 40(5): 595-603.
[44] Ding Y., Zhu L., Ding H. Semidefinite programming for Chebyshev fitting of spatial straight line with applications to cutter location planning and tolerance evaluation. Precision Engineering, 2007, 31(4): 364-368.
[45] Ding Y, Zhu LM, Ding H. On a novel approach to planning cylindrical cutter location for flank milling of ruled surfaces. International Journal of Production Research, 2009, 47(12): 3289–3305.
[46]赖天琴,李锋,孔德治等.计算机辅助加工直纹面的新方法.西安交通大学学报, 1996, 30(2): 61-68.
[47] Chu C. H., Chen J. T. Tool path planning for five-axis flank milling with developable surface approximation. International Journal of Advanced Manufacturing Technology, 2006, 29(7-8): 707-713.
[48] Chu C H, Huang WN, Hsu YY. Machining accuracy improvement in five-axis flank milling of ruled surfaces. International Journal of Machine Tools & Manufacture,2008,48:914-921.
[49] Elber G., Fish R. 5-axis freeform surface milling using piecewise ruled surfaceapproximation. Journal of Manufacturing Science and Engineering, 1997, 119(3): 383-387.
[50] Han Z., Chuang J.-J., Yang D. C. H. Isophote-based ruled surface approximation of free-form surfaces and its application in NC machining. International Journal of Production Research, 2001, 39(9): 1911-1930.
[51] Tsay DM, Chen HC, Her MJ. A study on five flank machining of centrifugal compressor impellers. Trans. Of ASME, Journal of Engineering for Gas and Power, 2002, 124(1): 177–181.
[52] Liang Q, Wang YZ, Fu HY, et al. Cutting path planning for ruled surface impellers. Chinese journal of aeronautics, 2008,21(5):462–471.
[53] Young HT, Chuang LC, Gerschwiler K, et al. A five-axis rough machining approach for a centrifugal impeller. The International Journal of Advanced Manufacturing Technology, 2004, 23(3–4): 233–239.
[54]陈良骥,王永章.整体叶轮五轴侧铣数控加工方法的研究.计算机集成制造系统. 2007, 13(1):141-146.
[55] Lartigue C, Duc E, Affouard A. Tool path deformation in 5-axis flank milling using envelope surface. Computer-Aided Design, 2003, 35(4):375– 382.
[56] Wu P. H., Li Y. W., Chu C. H. Optimized tool path generation based on dynamic programming for five-axis flank milling of rule surface. International Journal of Machine Tools and Manufacture, 2008, 48(11): 1224-1233.
[57]宫虎,曹利新,刘健.数控侧铣加工非可展直纹面的刀位整体优化原理与方法.机械工程学报, 2005, 41(11): 134-139.
[58] Gong H, Cao LX, Liu J. Improved positioning of cylindrical cutter for flank milling ruled surfaces. Computer-Aided Design, 2005, 37(12): 1205–1213.
[59]宫虎.五坐标数控加工运动几何学基础及刀位规划原理与方法的研究[博士论文].大连:大连理工大学, 2005.
[60] Gong H, Wang N. Optimize tool paths of flank milling with generic cutters based on approximation using the tool envelope surface. Computer-Aided Design, 2010, 41(12):981–989.
[61] Gong H, Wang N. 5-axis flank milling free-form surfaces considering constraints. Computer-Aided Design, 2011, 43(6):563-572.
[62] International Organization for Standardization. ISO/R 1101. Technical Drawings-Geometrical Tolearncing. Geneva: ISO, 1983.
[63] American Society of Mechanical Engineers. ANSI Standard Y14.5. Dimensioning and tolerancing. New York: ASME, 1994.
[64] Ding H, Zhu LM. Global Optimization of Tool Path for Five-axis Flank Milling with a Cylindrical Cutter. Science in China-Ser. E, 2009, 52(8): 2449-2459.
[65]张小明.五轴数控铣削加工几何-动力学建模与工艺参数优化[博士论文].上海:上海交通大学, 2009.
[66] Zhang XM, Zhu LM, Zheng G, Ding H. Tool path optimisation for flank milling ruled surface based on the distance function. International Journal of Production Research, 2010, 48(14): 4233–4251.
[67] Pechard P-Y, Tournier C, Lartigue C. Geometrical deviations versus smoothness in 5-axis high-speed flank milling. International Journal of Machine Tools and Manufacture, 2009, 41(12): 918-929.
[68] Vickers G. W. a. G., K W. Ball-mills versus end-mills for curved surface machining. Trans. of ASME, Journal of Engineer for Industry , 1989, 111: 22-26.
[69] Jensen CG, Anderson DC. Accurate tool placement and orientation for finish surface machining. Journal Design and Manufacture. 1993, 3(4):251–261.
[70] Kruth JP, Klewais P. Optimization and dynamic adaptation of the cutter inclination during five-axis milling of sculptured surfaces. CIRP Annals- Manufacturing Technology, 1994, 43(1): 443–448.
[71] Deng Z, Leu MC, Wang L, etc. Determination of flat-end orientation in 5-axis machining. ASME MED, 1996,4: 73–80.
[72] Lee YS. Admissible tool orientation control of gouging avoidance for 5-axis complex surface machining. Computer-Aided Design, 1997, 29(7): 507–521.
[73] Rao N, Bedi S, Buchal R. Implementation of the principal-axis method for machining of complex surfaces. International Journal of Advanced Manufacturing Technology.1996,11: 249–57.
[74] Bedi S, Gravelle S, Chen YH. Principal curvature alignment technique for machining complex surface. Trans. Of ASME, Journal of Manufacturing Science & Engineering. 1997, 119(4):756–765.
[75] Lo CC. Efficient cutter-path planning for five-axis surface machining with a flat-end cutter. Computer-Aided Design, 1999, 31(9):557–566.
[76] Chiou CJ,Lee YS. A machining potential field approach to tool path generation for multi-axis sculptured surface machining. Computer-Aided Design, 2002, 34: 357–71.
[77] Yoon J-H, Pottmann H, Lee Y-S. Locally optimal cutting positions for 5-axis sculptured surface machining. Computer-Aided Design, 2003, 35: 69–81.
[78]倪炎榕,马登哲,张洪,郭静萍.圆环面刀具五坐标数控加工复杂曲面优化刀位算法.机械工程学报, 2001, 37(2): 87–90.
[79]王小椿,吴序堂,李艳斌.密切曲率法–一种自由曲面加工的新概念.西安交通大学学报, 1992, 26(5): 51–58.
[80] Wang XC, Ghosh SK, Li YB, Wu XT. Curvature catering-a new approach in manufacture of sculptured surfaces (part 1. theorem). Journal of Materials Processing Technology 1993, 38:159-75.
[81]沈兵,高军,吴联银等.基于密切曲率法的加工理论及算法研究.机械, 1998, 26(4):20-24.
[82]张瑞乾,沈兵,王小椿等.用二阶密切曲率法加工自由曲面.机械工程学报, 1998, 34(2):88-92..
[83] Rao A, Sarma R. On local gouging in five-axis sculptured surface machining using flat-end tools. Computer-Aided Design, 2000, 32: 409–420.
[84] Sarma R. Flat-ended tool swept sections for five-axis NC machining of sculptured surfaces. Trans. of the ASME, Journal of Manufacturing Science & Engineering. 2000, 122(1):158–165.
[85]曹利新,宫虎,刘健.曲面接触问题及其等距面接触特性研究.大连理工大学学报, 2007, 47(1):39-44.
[86] Gong H, Cao LX, Liu J. Second order approximation of tool envelope surface for 5-axis machining with single point contact. Computer-Aided Design, 2008, 40 (5): 604–615.
[87] Gong H, Fang FZ, Hu XT, et al. Optimization of tool positions locally based on the BCELTP for 5-axis machining of free-form surfaces. Computer-Aided Design, 2010, 42(6): 558–570.
[88] Zhu LM, Ding H, Xiong YL. Third Order Point Contact Approach for Five-Axis Sculptured Surface Machining Using Non-ball-end Tools -Part I: Third Order Approximation of Tool Envelope Surface. Science in China-Ser. E, 2010, 53(7): 1904-1912.
[89] Zhu LM, Ding H, Xiong YL. Third Order Point Contact Approach for Five-Axis Sculptured Surface Machining Using Non-ball-end Tools -Part II: Tool Positioning Strategy. Science in China-Ser. E, 2010, 53 (8): 2190-2197.
[90] Warkentin.A., Ismail F., Bedi S. Comparison between multi-point and other 5-axis tool positioning strategies. International Journal of Machine Tools and Manufacture, 2000, 40(2): 185-208.
[91] Li H, Feng HY. Efficient five-axis machining of free-form surfaces with constant scallop height tool paths. International Journal of Production Research,2004, 42(12):2403–2417.
[92] Barakchi Fard MJ, Feng HY. Effect of tool tilt angle on machining strip width in five-axis flat-end milling of free-form surfaces. The International Journal of Advanced Manufacturing Technology, 2009, 44(3-4): 211–222.
[93] Barakchi Fard MJ, Feng HY. Effective determination of feed direction and tool orientation in five-axis flat-end milling. Trans. of ASME, Journal of Manufacturing Science & Engineering. 2010, 132(6):061011.
[94] Ho M. C., Hwang Y. R., Hu C. H. Five-axis tool orientation smoothing using quaternion interpolation algorithm. International Journal of Machine Tool and Manufacture, 2003: 1259-1267.
[95] Zhang W., Zhang Y. F., Ge Q. J. Five-axis Tool Path Generation for Sculptured Surface Machining using Rational Bezier Motions of a Flat-end Cutter. Computer-Aided Design and Applications, 2004, 1(1): 251-260.
[96] Xia J., Ge Q. J. On the Exact Representation of the Boundary Surfaces of the Swept Volume of a Cylinder Undergoing Rational Bézier and B-Spline Motions. Journal of Mechanical Design, 2001, 123(2): 261-265.
[97] Xia J., Ge Q. J. An exact representation of effective cutting shapes of 5-axis CNC machining using rational Bezier and B-spline tool motions. IEEE International Conference on Robotics and Automation, 2001.
[98] Koch J. Method of directing the movement of a tool as part of a process to remove material from a block of material. US patent 6632053. OpenMind Sortware Technologies (DE), 2003.
[99] Langeron J. M., Duc E., Lartigue C., Bourdet P. A new format for 5-axis tool path computation, using Bspline curves. Computer Aided Design, 2004, 36(12): 1219-1229.
[100] Jun C. S., Cha K., Lee Y. S. Optimizing tool orientations for 5-axis machining by configuration-space search method. Computer-Aided Design, 2003, 35(6): 549-566.
[101] Wang N, Tang K. Automatic generation of gouge-free and angular-velocity-compliant five-axis toolpath. Computer-Aided Design, 2007, 39(10): 841–852.
[102] Wang N, Tang K. Five-axis tool path generation for a flat-end tool based on iso-conic partitioning. Computer-Aided Design, 2008, 40(12): 1067–1079.
[103] Castagnetti C., E. Duc P. R. The Domain of Admissible Orientation concept: A new method for five-axis tool path optimisation. Computer-Aided Design, 2008, 40(9): 938-950.
[104] Bi QZ, Wang YH and Ding H. A GPU-based algorithm for generating collision-free and orientation-smooth five-axis finishing tool paths of a ball-end cutter. International Journal of Production Research, 2010, 48(4):1105-1124.
[105]梅向明,黄敬之.微分几何.第3版.北京:高等教育出版社,2003.
[106] Murray R., Li Z., Sastry S. A mathematical introduction to robotic manipulation. CRC press, 1994.
[107] Zhu LM, Xiong ZH, Ding H, Xiong YL. A distance function based approach for localization and profile error evaluation of complex surface. Trans. Of ASME, Journal of Manufacturing Science & Engineering, 2004, 126(3): 542–554.
[108] LEE K, SEONG JK, KIM KJ, et al. Minimum distance between two sphere-swept surface. Computer-Aided Design, 2007, 39: 452-459.
[109] Hu S M, Li Y F, Ju T, Zhu X. Modifying the shape of NURBS surfaces with geometric constraints. Computer-Aided Design, 2001, 33(12): 903–912.
[110] Park H. Lofted B-spline surface interpolation by linearly constrained energyminimization. Computer-Aided Design, 2003, 35 (14): 1261–1268.
[111] Zhang CM, Zhang PF, Cheng FH. Fairing spline curves and surfaces by minimizing energy. Computer-Aided Design, 2001, 33(13): 913–923.
[112] Pourazady M, Xu X. Direct manipulations of NURBS surfaces subjected to geometric constraints. Computers & Graphics. 2006,30(4):598-609.
[113] Marler R.T., Arora J.S. Survey of multi-objective optimization methods for engineering. Structural and Multidisciplinary Optimization. 2004, 26(6):369-395.
[114]何乃翔,李世铎.共轭曲面的曲率张量与曲率导数张量.机械工程学报, 1979, 15(3-4): 93–108.