任意阶参数连续的形状可调过渡曲线
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摘要
目的针对现有研究未能给出可以使过渡曲线在端点处与被过渡曲线之间达到C~k(k为任意自然数)连续的多项式势函数统一表达式的问题展开研究,以期用简单有效的方式解决这一问题。方法从过渡曲线的方程出发,借助莱布尼兹公式得出其k阶导矢表达式,根据预设的连续性目标,反推出可使过渡曲线在端点处达到C~k连续的势函数需满足的基本条件。由这些基本条件中所含条件的数量,以及对势函数、过渡曲线的其他期望所对应的条件个数的总量,确定多项式势函数的次数,将势函数表达成相同次数的Bernstein基函数的线性组合,组合系数待定。由势函数需满足的基本条件、其他预期条件,以及Bernstein基函数在端点处的函数值、导数值等信息,得出关于待定系数的方程组。解该方程组,得出满足所有预期目标,并含一个自由参数的多项式势函数的统一表达式。结果势函数中存在两个参数k和λ,k用于控制过渡曲线与被过渡曲线在端点处的连续阶,在k取定以后,λ可用于控制过渡曲线与被过渡曲线的贴近程度。势函数具有对称性、中点性、有界性,分析了当k固定时,势函数关于变量t和参数λ的单调性,分析了使势函数图形只存在唯一拐点时,自由参数的取值范围。由该势函数构造的过渡曲线,取一般参数时在端点处可达C~k连续,取特殊参数时可达C~(k+1)连续。分析了过渡曲线的形状特征,当k取定时,λ的值越大,过渡曲线越贴近被过渡曲线。结论实验数据验证了理论分析结果的正确性,同时直观显示了所给方法的有效性。
Objective The existing research has failed to provide a general expression of the polynomial potential function,which can enable the transition curve to reach Ck(where k is an arbitrary natural number) continuity at endpoints. This research aims to solve this problem in a simple and effective manner. Method First,by using the equation of the transition curve,the kth derivative of the transition curve is obtained with the help of the Leibniz formula. According to the predetermined continuity goal,the basic conditions,which should be met by the potential function to enable the transition curve to reach Ckcontinuity at endpoints,are deduced. Second,according to the total number of conditions contained in the basic conditions and those corresponding to other expectations of the potential function and transition curve,the degree of the polynomial potential function is determined. The potential function is expressed as a linear combination of the Bernstein basis functions with the same degree,and the combination coefficients are established. Finally,according to the basic and otherexpected conditions to be satisfied by the potential function,as well as the function and derivative values of the Bernstein basis functions at endpoints,an equation set is obtained for the undetermined coefficients. To solve the equation set,the unified expression of the polynomial potential function,which satisfies all expected goals and contains a free parameter,should be obtained. Result Two parameters exist in the potential function,namely,k and λ. Parameter k is used to control the continuity order between the transition and initial curves at the endpoints. After k is determined,parameter λ can be used to control the degree of proximity between the transition and initial curves. The potential function has symmetry,midpoint property,and boundedness. The monotonicity of potential function with respect to the variable t and the parameter λis analyzed when k is fixed. The value range of the free parameter,which depicts the curve of the potential function and has a unique inflection point,is analyzed. For the general parameter values,the transition curve that is constructed by the potential function can reach Ckcontinuity at the endpoints. For the special parameter values,the transition curve can Ck + 1 reach continuity. The shape characteristics of the transition curve are further analyzed. When the value of k is set,the greater the value of λ and the closer the transition to the initial curve. Conclusion The numerical examples verify the correctness of the theoretical analytical results and the effectiveness of the proposed method.
Objective The existing research has failed to provide a general expression of the polynomial potential function,which can enable the transition curve to reach Ck(where k is an arbitrary natural number) continuity at endpoints. This research aims to solve this problem in a simple and effective manner. Method First,by using the equation of the transition curve,the kth derivative of the transition curve is obtained with the help of the Leibniz formula. According to the predetermined continuity goal,the basic conditions,which should be met by the potential function to enable the transition curve to reach Ckcontinuity at endpoints,are deduced. Second,according to the total number of conditions contained in the basic conditions and those corresponding to other expectations of the potential function and transition curve,the degree of the polynomial potential function is determined. The potential function is expressed as a linear combination of the Bernstein basis functions with the same degree,and the combination coefficients are established. Finally,according to the basic and otherexpected conditions to be satisfied by the potential function,as well as the function and derivative values of the Bernstein basis functions at endpoints,an equation set is obtained for the undetermined coefficients. To solve the equation set,the unified expression of the polynomial potential function,which satisfies all expected goals and contains a free parameter,should be obtained. Result Two parameters exist in the potential function,namely,k and λ. Parameter k is used to control the continuity order between the transition and initial curves at the endpoints. After k is determined,parameter λ can be used to control the degree of proximity between the transition and initial curves. The potential function has symmetry,midpoint property,and boundedness. The monotonicity of potential function with respect to the variable t and the parameter λis analyzed when k is fixed. The value range of the free parameter,which depicts the curve of the potential function and has a unique inflection point,is analyzed. For the general parameter values,the transition curve that is constructed by the potential function can reach Ckcontinuity at the endpoints. For the special parameter values,the transition curve can Ck + 1 reach continuity. The shape characteristics of the transition curve are further analyzed. When the value of k is set,the greater the value of λ and the closer the transition to the initial curve. Conclusion The numerical examples verify the correctness of the theoretical analytical results and the effectiveness of the proposed method.
引文
[1]Li L,Yang L.Design of transition curve of profiled chamber flow sensor considering slides with arc ends[J].Journal of Zhejiang University-Science A(Applied Physics&Engineering),2018,19(2):137-147.[DOI:10.1631/jzus.A1600666]
[2]Zhang X F,Huang K,Chen Q.Set up of equation on dedendum transion curve and research of 3D accurate modeling for involute gear[J].Modular Machine Tool&Automatic Manufacturing Technique,2008,(2):1-3,7.[张训福,黄康,陈奇.渐开线齿轮齿根过渡曲线方程的建立及三维精确建模[J].组合机床与自动化加工技术,2008,(2):1-3,7.][DOI:10.3969/j.issn.1001-2265.2008.02.001]
[3]Zhao S,Bi Q Z,Wang Y H,et al.A data compression algorithm based on G2continuous Bézier curves for tool paths[J].Journal of Shanghai Jiaotong University,2014,48(5):629-635.[赵晟,毕庆贞,王宇晗,等.基于G2连续Bézier曲线的刀具轨迹压缩算法[J].上海交通大学学报,2014,48(5):629-635.][DOI:10.16183/j.cnki.jsjtu.2014.05.009]
[4]Lin L Q,Lu Y.Low noise transitional curves study for internal contour of the stator of rotary vane pump[J].Machine Tool&Hydraulics,2010,38(19):21-24.[林立强,鲁阳.转子式叶片泵的低噪声定子内轮廓过渡曲线研究[J].机床与液压,2010,38(19):21-24.][DOI:10.3969/j.issn.1001-3881.2010.19.007]
[5]Cai H H,Wang G J.Construction of cubic C-Bézier spiral and its application in highway design[J].Journal of Zhejiang University:Engineering Science,2010,44(1):68-74.[蔡华辉,王国瑾.三次C-Bézier螺线构造及其在道路设计中的应用[J].浙江大学学报:工学版,2010,44(1):68-74.][DOI:10.3785/j.issn.1008-973X.2010.01.013]
[6]Di Mascio P,Loprencipe G,Moretti L,et al.Bridge expansion joint in road transition curve:effects assessment on heavy vehicles[J].Applied Sciences,2017,7(6):#599.[DOI:10.3390/app7060599]
[7]Zboinski K,Woznica P.Combined use of dynamical simulation and optimisation to form railway transition curves[J].Vehicle System Dynamics,2018,56(9):1394-1450.[DOI:10.1080/00423114.2017.1421315]
[8]Zhang P,Liang Y Y,Liu H W,et al.Complex orientation smooth transition technique of industrial robot in task space[J].Machine Tool&Hydraulics,2016,44(5):76-79.[张平,梁艳阳,刘宏伟,等.工业机器人任务空间复杂姿态平滑过渡技术[J].机床与液压,2016,44(5):76-79.][DOI:10.3969/j.issn.1001-3881.2016.05.019]
[9]Zboinski K,Woznica P.Optimization of polynomial transition curves from the viewpoint of jerk value[J].Archives of Civil Engineering,2017,63(1):181-199.[DOI:10.1515/ace-2017-0012]
[10]Li Z,Ma L Z,Meek D.Reconstruction of G2transition curve for two separated circular arcs[J].Journal of Computer-Aided Design&Computer Graphics,2006,18(2):265-269.[李重,马利庄,Meek D.平面两圆弧相离情况下G2连续过渡曲线构造[J].计算机辅助设计与图形学学报,2006,18(2):265-269.][DOI:10.3321/j.issn:1003-9775.2006.02.016]
[11]Zheng Z H,Wang G Z.On curvature monotony for a PH cubic curve and constructing transition curve[J].Journal of ComputerAided Design&Computer Graphics,2014,26(8):1219-1224.[郑志浩,汪国昭.三次PH曲线的曲率单调性及过渡曲线构造[J].计算机辅助设计与图形学学报,2014,26(8):1219-1224.]
[12]Gao H,Shou H H,Miao Y W,et al.The quasi-cubic Bézier spirals with three control points[J].Journal of Image and Graphics,2014,19(11):1677-1683.[高晖,寿华好,缪永伟,等.3个控制顶点的类三次Bézier螺线[J].中国图象图形学报,2014,19(11):1677-1683.][DOI:10.11834/jig.20141116]
[13]Gao H,Shou H H.The construction ofα-curves and study of curvature monotony condition[J].Applied Mathematics A Journal of Chinese Universities,2015,30(3):291-305.[高晖,寿华好.α-曲线的构造及曲率单调条件的研究[J].高校应用数学学报,2015,30(3):291-305.][DOI:10.3969/j.issn.1000-4424.2015.03.005]
[14]Liu Y Y,Wang X H.Construction of planar cubic PH transition curve[J].Journal of Hefei University of Technology:Natural Science,2016,39(9):1288-1291,1296.[刘莹莹,王旭辉.平面三次PH过渡曲线的构造[J].合肥工业大学学报:自然科学版,2016,39(9):1288-1291,1296.][DOI:10.3969/j.issn.1003-5060.2016.09.026]
[15]Ziatdinov R,Yoshida N,Kim T W.Fitting G2multispiral transition curve joining two straight lines[J].Computer-Aided Design,2012,44(6):591-596.[DOI:10.1016/j.cad.2012.01.007]
[16]Zhang H X,Wang G J.Shape blending of curves:research into geometric continuity preserving[J].Applied Mathematics AJournal of Chinese Universities:Series A,2001,16(2):187-194.[张宏鑫,王国瑾.保持几何连续性的曲线形状调配[J].高校应用数学学报:A辑,2001,16(2):187-194.][DOI:10.3969/j.issn.1000-4424.2001.02.011]
[17]Liu H Y,Duan X J,Zhang D M,et al.Constructing of blending trigonometric Bézier-like curve[J].Journal of Zhejiang University:Science Edition,2013,40(1):42-46.[刘华勇,段小娟,张大明,等.基于三角Bézier-like的过渡曲线构造[J].浙江大学学报:理学版,2013,40(1):42-46.][DOI:10.3785/j.issn.1008-9497.2013.01.010]
[18]Li L F,Tan J R,Zhao H X.Construction method of transition curve based on Metaball technique[J].China Mechanical Engineering,2005,16(6):483-486.[李凌丰,谭建荣,赵海霞.基于Metaball的过渡曲线[J].中国机械工程,2005,16(6):483-486.][DOI:10.3321/j.issn:1004-132X.2005.06.004]
[19]Yan L L,Liang J F,Huang T.Bézier curves with shape parameters[J].Journal of Hefei University of Technology:Natural Science,2009,32(11):1783-1788.[严兰兰,梁炯丰,黄涛.带形状参数的Bézier曲线[J].合肥工业大学学报:自然科学版,2009,32(11):1783-1788.][DOI:10.3969/j.issn.1003-5060.2009.11.039]
[20]Li J C,Song L Z,Liu C Z.Construction of transition curve and surface with a parameter in shape blending[J].Journal of Computer-Aided Design&Computer Graphics,2016,28(12):2088-2096.[李军成,宋来忠,刘成志.形状调配中带参数的过渡曲线与曲面构造[J].计算机辅助设计与图形学学报,2016,28(12):2088-2096.][DOI:10.3969/j.issn.1003-9775.2016.12.007]
[21]Gao H,Shou H H.Construction of potential function and transition curve based on Metaball technique[J].Journal of ComputerAided Design&Computer Graphics,2015,27(5):900-906.[高晖,寿华好.势函数的构造及基于Metaball的过渡曲线[J].计算机辅助设计与图形学学报,2015,27(5):900-906.][DOI:10.3969/j.issn.1003-9775.2015.05.016]
[22]Yan L L,Han X L.Automatic continuous composite curve and surface with adjustable shape and smoothness[J].Journal of Computer-Aided Design&Computer Graphics,2014,26(10):1654-1662.[严兰兰,韩旭里.形状及光滑度可调的自动连续组合曲线曲面[J].计算机辅助设计与图形学学报,2014,26(10):1654-1662.]
[23]Li J C,Song L Z.Construction of transition curves based on Metaball technique using rational potential function with a shape parameter[J].Journal of Zhejiang University:Science Edition,2017,44(3):307-313.[李军成,宋来忠.利用带形状参数的有理势函数构造基于Metaball的过渡曲线[J].浙江大学学报:理学版,2017,44(3):307-313.][DOI:10.3785/j.issn.1008-9497.2017.03.011]
[24]Farin G.Curves and Surfaces for CAGD[M].5th ed.San Francisco:Morgan Kaufmann Publishers,2002:66.
[2]Zhang X F,Huang K,Chen Q.Set up of equation on dedendum transion curve and research of 3D accurate modeling for involute gear[J].Modular Machine Tool&Automatic Manufacturing Technique,2008,(2):1-3,7.[张训福,黄康,陈奇.渐开线齿轮齿根过渡曲线方程的建立及三维精确建模[J].组合机床与自动化加工技术,2008,(2):1-3,7.][DOI:10.3969/j.issn.1001-2265.2008.02.001]
[3]Zhao S,Bi Q Z,Wang Y H,et al.A data compression algorithm based on G2continuous Bézier curves for tool paths[J].Journal of Shanghai Jiaotong University,2014,48(5):629-635.[赵晟,毕庆贞,王宇晗,等.基于G2连续Bézier曲线的刀具轨迹压缩算法[J].上海交通大学学报,2014,48(5):629-635.][DOI:10.16183/j.cnki.jsjtu.2014.05.009]
[4]Lin L Q,Lu Y.Low noise transitional curves study for internal contour of the stator of rotary vane pump[J].Machine Tool&Hydraulics,2010,38(19):21-24.[林立强,鲁阳.转子式叶片泵的低噪声定子内轮廓过渡曲线研究[J].机床与液压,2010,38(19):21-24.][DOI:10.3969/j.issn.1001-3881.2010.19.007]
[5]Cai H H,Wang G J.Construction of cubic C-Bézier spiral and its application in highway design[J].Journal of Zhejiang University:Engineering Science,2010,44(1):68-74.[蔡华辉,王国瑾.三次C-Bézier螺线构造及其在道路设计中的应用[J].浙江大学学报:工学版,2010,44(1):68-74.][DOI:10.3785/j.issn.1008-973X.2010.01.013]
[6]Di Mascio P,Loprencipe G,Moretti L,et al.Bridge expansion joint in road transition curve:effects assessment on heavy vehicles[J].Applied Sciences,2017,7(6):#599.[DOI:10.3390/app7060599]
[7]Zboinski K,Woznica P.Combined use of dynamical simulation and optimisation to form railway transition curves[J].Vehicle System Dynamics,2018,56(9):1394-1450.[DOI:10.1080/00423114.2017.1421315]
[8]Zhang P,Liang Y Y,Liu H W,et al.Complex orientation smooth transition technique of industrial robot in task space[J].Machine Tool&Hydraulics,2016,44(5):76-79.[张平,梁艳阳,刘宏伟,等.工业机器人任务空间复杂姿态平滑过渡技术[J].机床与液压,2016,44(5):76-79.][DOI:10.3969/j.issn.1001-3881.2016.05.019]
[9]Zboinski K,Woznica P.Optimization of polynomial transition curves from the viewpoint of jerk value[J].Archives of Civil Engineering,2017,63(1):181-199.[DOI:10.1515/ace-2017-0012]
[10]Li Z,Ma L Z,Meek D.Reconstruction of G2transition curve for two separated circular arcs[J].Journal of Computer-Aided Design&Computer Graphics,2006,18(2):265-269.[李重,马利庄,Meek D.平面两圆弧相离情况下G2连续过渡曲线构造[J].计算机辅助设计与图形学学报,2006,18(2):265-269.][DOI:10.3321/j.issn:1003-9775.2006.02.016]
[11]Zheng Z H,Wang G Z.On curvature monotony for a PH cubic curve and constructing transition curve[J].Journal of ComputerAided Design&Computer Graphics,2014,26(8):1219-1224.[郑志浩,汪国昭.三次PH曲线的曲率单调性及过渡曲线构造[J].计算机辅助设计与图形学学报,2014,26(8):1219-1224.]
[12]Gao H,Shou H H,Miao Y W,et al.The quasi-cubic Bézier spirals with three control points[J].Journal of Image and Graphics,2014,19(11):1677-1683.[高晖,寿华好,缪永伟,等.3个控制顶点的类三次Bézier螺线[J].中国图象图形学报,2014,19(11):1677-1683.][DOI:10.11834/jig.20141116]
[13]Gao H,Shou H H.The construction ofα-curves and study of curvature monotony condition[J].Applied Mathematics A Journal of Chinese Universities,2015,30(3):291-305.[高晖,寿华好.α-曲线的构造及曲率单调条件的研究[J].高校应用数学学报,2015,30(3):291-305.][DOI:10.3969/j.issn.1000-4424.2015.03.005]
[14]Liu Y Y,Wang X H.Construction of planar cubic PH transition curve[J].Journal of Hefei University of Technology:Natural Science,2016,39(9):1288-1291,1296.[刘莹莹,王旭辉.平面三次PH过渡曲线的构造[J].合肥工业大学学报:自然科学版,2016,39(9):1288-1291,1296.][DOI:10.3969/j.issn.1003-5060.2016.09.026]
[15]Ziatdinov R,Yoshida N,Kim T W.Fitting G2multispiral transition curve joining two straight lines[J].Computer-Aided Design,2012,44(6):591-596.[DOI:10.1016/j.cad.2012.01.007]
[16]Zhang H X,Wang G J.Shape blending of curves:research into geometric continuity preserving[J].Applied Mathematics AJournal of Chinese Universities:Series A,2001,16(2):187-194.[张宏鑫,王国瑾.保持几何连续性的曲线形状调配[J].高校应用数学学报:A辑,2001,16(2):187-194.][DOI:10.3969/j.issn.1000-4424.2001.02.011]
[17]Liu H Y,Duan X J,Zhang D M,et al.Constructing of blending trigonometric Bézier-like curve[J].Journal of Zhejiang University:Science Edition,2013,40(1):42-46.[刘华勇,段小娟,张大明,等.基于三角Bézier-like的过渡曲线构造[J].浙江大学学报:理学版,2013,40(1):42-46.][DOI:10.3785/j.issn.1008-9497.2013.01.010]
[18]Li L F,Tan J R,Zhao H X.Construction method of transition curve based on Metaball technique[J].China Mechanical Engineering,2005,16(6):483-486.[李凌丰,谭建荣,赵海霞.基于Metaball的过渡曲线[J].中国机械工程,2005,16(6):483-486.][DOI:10.3321/j.issn:1004-132X.2005.06.004]
[19]Yan L L,Liang J F,Huang T.Bézier curves with shape parameters[J].Journal of Hefei University of Technology:Natural Science,2009,32(11):1783-1788.[严兰兰,梁炯丰,黄涛.带形状参数的Bézier曲线[J].合肥工业大学学报:自然科学版,2009,32(11):1783-1788.][DOI:10.3969/j.issn.1003-5060.2009.11.039]
[20]Li J C,Song L Z,Liu C Z.Construction of transition curve and surface with a parameter in shape blending[J].Journal of Computer-Aided Design&Computer Graphics,2016,28(12):2088-2096.[李军成,宋来忠,刘成志.形状调配中带参数的过渡曲线与曲面构造[J].计算机辅助设计与图形学学报,2016,28(12):2088-2096.][DOI:10.3969/j.issn.1003-9775.2016.12.007]
[21]Gao H,Shou H H.Construction of potential function and transition curve based on Metaball technique[J].Journal of ComputerAided Design&Computer Graphics,2015,27(5):900-906.[高晖,寿华好.势函数的构造及基于Metaball的过渡曲线[J].计算机辅助设计与图形学学报,2015,27(5):900-906.][DOI:10.3969/j.issn.1003-9775.2015.05.016]
[22]Yan L L,Han X L.Automatic continuous composite curve and surface with adjustable shape and smoothness[J].Journal of Computer-Aided Design&Computer Graphics,2014,26(10):1654-1662.[严兰兰,韩旭里.形状及光滑度可调的自动连续组合曲线曲面[J].计算机辅助设计与图形学学报,2014,26(10):1654-1662.]
[23]Li J C,Song L Z.Construction of transition curves based on Metaball technique using rational potential function with a shape parameter[J].Journal of Zhejiang University:Science Edition,2017,44(3):307-313.[李军成,宋来忠.利用带形状参数的有理势函数构造基于Metaball的过渡曲线[J].浙江大学学报:理学版,2017,44(3):307-313.][DOI:10.3785/j.issn.1008-9497.2017.03.011]
[24]Farin G.Curves and Surfaces for CAGD[M].5th ed.San Francisco:Morgan Kaufmann Publishers,2002:66.