喷涂机器人喷枪轨迹设计与优化研究
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摘要
本文研究了喷涂机器人喷枪轨迹的设计与优化,该设计与优化的目的是要确定一条使得工件表面涂层厚度差异达到最小的喷枪最优轨迹。全文主要由以下几个部分组成：
1.首先阐述了喷涂机器人喷枪轨迹优化的研究背景、意义、发展历程、喷涂机器人离线编程的主要结构、轨迹优化研究概况与现阶段存在的问题以及本文所作的主要工作。
2.根据已有的实验数据,通过理论分析建立了涂层生长速率新模型,对其进行了曲线仿真拟合。以工作面上任意一点涂层厚度和工作面涂层平均厚度的方差最小作为衡量涂层均匀性的标准,对沿指定空间路径的喷枪轨迹优化问题进行研究,由于涂层生长新模型应用在平面喷枪轨迹优化时较复杂,提出了运用最小二乘法将该方程拟合成为简单的β分布方程进行求解。
3.研究了小曲率复杂曲面上喷枪轨迹设计与优化问题。由于现有涂层生长速率函数表达式过于复杂,故采用实验方法建立了表达式较为简单的涂层生长速率模型。对复杂曲面进行喷涂作业时,在小曲率复杂曲面的分片原则下,对曲面进行分片,并在每片上进行轨迹设计。为保证曲面上涂层厚度的均匀性,要对面片交界处喷枪轨迹进行优化,针对多面片组合优化时出现涂层干涉的问题,提出变喷枪高度的方法进行优化。
4.研究了大曲率柱面上喷涂轨迹设计与优化问题。在现有涂层生长速率模型的基础上,运用自由曲面上涂层厚度建模原理推导出圆柱面上的3D涂层厚度模型。提出了将柱面通过最小二乘圆弧逼近的思想,将柱面分片成若干圆柱面片的组合,柱面上的喷涂轨迹优化问题可近似看做圆柱面来处理,并在每个圆柱面片上建立喷枪轨迹优化目标函数,讨论了两圆柱面片交界处的涂层厚度计算方法及喷枪轨迹优化问题。
本文对主要的设计方案进行了仿真计算。仿真结果表明,本文提出的数学模型以及各种喷枪轨迹设计、优化方法均可获得良好的效果。
This thesis is undertaken on the design and optimization of spray-gun trajectory for spray-painting robot.The objective of the design and optimization is to determine a trajectory that results in minimal variation in accumulated film thickness on the surface of a workpiece.
1.Firstly,the general situation and background about paining robot research and the developing about research are discussed. And then the frame of off-line programming system and the major problems of research are given. Finally some problems,which are studied in this thesis are illtroduced.
2. According to the experimental data, founding the new model of paint desposition and simulating the peripheral curve of distribution off film base on theoretical analysis. The paint thickness at any point deviation from the average paint thickness required minimum, the variation in paint thickness can be as a standard of coating uniformity. Research on the problem of trajectory optimization along the designated path. Because it is more complex to use this equation when the trajectory optimization of in-plane spray painting is carried on, a method of fitting this equation to a simpleβ-distribution-equation is come up by the least square method.
3. Trajectory design and optimization for painting robot of complex surface with small curvature is discussed. Because the current paint deposition rate function is too complicated, the paint deposition rate function on a plane according to the experiment data is provided. A complex surface with small curvature is divided into several patches under the principle of sharding and trajectory optimization for each patch is performed. In order to satisfy the material uniformity requirements, trajectory at the junction of patches need optimized. For the interferential problem of coating distribution appeared in the progress of the integrated spraying trajectory optimization on numerous patches, the method of changing the spray gun's height and optimizing the distance of trajectory is advanced.
4. Trajectory design and optimization for painting robot of cylinder with large curvature is discussed. On the basis of current paint deposition rate function, the modeling principle of coating thickness on free surface is used, and the spray gun 3D model was deduced in cylndrical surface. The method of least squares approximation for cylindrical surface was proposed, cylindrical surface fractionize into the combination of cylndrical surface, the trajectory optimization problem in cylindrical surface can be approximated as cylndrical surface to deal with,and trajectory optimization objective function was established in each cylndrical surface, the calculation method of coating thickness and optimization algorithms are developed to integrate the trajectories on the intersecting area of two patch.
In this thesis, simulations are made for major design schemes.The results of simulations have shown that the model and the tool planning algorithms are feasible.
1.首先阐述了喷涂机器人喷枪轨迹优化的研究背景、意义、发展历程、喷涂机器人离线编程的主要结构、轨迹优化研究概况与现阶段存在的问题以及本文所作的主要工作。
2.根据已有的实验数据,通过理论分析建立了涂层生长速率新模型,对其进行了曲线仿真拟合。以工作面上任意一点涂层厚度和工作面涂层平均厚度的方差最小作为衡量涂层均匀性的标准,对沿指定空间路径的喷枪轨迹优化问题进行研究,由于涂层生长新模型应用在平面喷枪轨迹优化时较复杂,提出了运用最小二乘法将该方程拟合成为简单的β分布方程进行求解。
3.研究了小曲率复杂曲面上喷枪轨迹设计与优化问题。由于现有涂层生长速率函数表达式过于复杂,故采用实验方法建立了表达式较为简单的涂层生长速率模型。对复杂曲面进行喷涂作业时,在小曲率复杂曲面的分片原则下,对曲面进行分片,并在每片上进行轨迹设计。为保证曲面上涂层厚度的均匀性,要对面片交界处喷枪轨迹进行优化,针对多面片组合优化时出现涂层干涉的问题,提出变喷枪高度的方法进行优化。
4.研究了大曲率柱面上喷涂轨迹设计与优化问题。在现有涂层生长速率模型的基础上,运用自由曲面上涂层厚度建模原理推导出圆柱面上的3D涂层厚度模型。提出了将柱面通过最小二乘圆弧逼近的思想,将柱面分片成若干圆柱面片的组合,柱面上的喷涂轨迹优化问题可近似看做圆柱面来处理,并在每个圆柱面片上建立喷枪轨迹优化目标函数,讨论了两圆柱面片交界处的涂层厚度计算方法及喷枪轨迹优化问题。
本文对主要的设计方案进行了仿真计算。仿真结果表明,本文提出的数学模型以及各种喷枪轨迹设计、优化方法均可获得良好的效果。
This thesis is undertaken on the design and optimization of spray-gun trajectory for spray-painting robot.The objective of the design and optimization is to determine a trajectory that results in minimal variation in accumulated film thickness on the surface of a workpiece.
1.Firstly,the general situation and background about paining robot research and the developing about research are discussed. And then the frame of off-line programming system and the major problems of research are given. Finally some problems,which are studied in this thesis are illtroduced.
2. According to the experimental data, founding the new model of paint desposition and simulating the peripheral curve of distribution off film base on theoretical analysis. The paint thickness at any point deviation from the average paint thickness required minimum, the variation in paint thickness can be as a standard of coating uniformity. Research on the problem of trajectory optimization along the designated path. Because it is more complex to use this equation when the trajectory optimization of in-plane spray painting is carried on, a method of fitting this equation to a simpleβ-distribution-equation is come up by the least square method.
3. Trajectory design and optimization for painting robot of complex surface with small curvature is discussed. Because the current paint deposition rate function is too complicated, the paint deposition rate function on a plane according to the experiment data is provided. A complex surface with small curvature is divided into several patches under the principle of sharding and trajectory optimization for each patch is performed. In order to satisfy the material uniformity requirements, trajectory at the junction of patches need optimized. For the interferential problem of coating distribution appeared in the progress of the integrated spraying trajectory optimization on numerous patches, the method of changing the spray gun's height and optimizing the distance of trajectory is advanced.
4. Trajectory design and optimization for painting robot of cylinder with large curvature is discussed. On the basis of current paint deposition rate function, the modeling principle of coating thickness on free surface is used, and the spray gun 3D model was deduced in cylndrical surface. The method of least squares approximation for cylindrical surface was proposed, cylindrical surface fractionize into the combination of cylndrical surface, the trajectory optimization problem in cylindrical surface can be approximated as cylndrical surface to deal with,and trajectory optimization objective function was established in each cylndrical surface, the calculation method of coating thickness and optimization algorithms are developed to integrate the trajectories on the intersecting area of two patch.
In this thesis, simulations are made for major design schemes.The results of simulations have shown that the model and the tool planning algorithms are feasible.
引文
[1]王燚,赵德安,王振滨等.喷漆机器人喷枪最优轨迹规划的研究[J].江苏理工大学学报：自然科学版2001,22(5)：55-59
[2]王振滨,赵德安,李医民等.喷漆机器人离线编程系统探讨[J].江苏理工大学学报：自然科学版,2000,21(5)：75-82
[3]刁训娣,赵德安,李医民等.喷漆机器人喷枪轨迹设计及影响因素研究[J].机械科学与技术,2004(4)：396-398
[4]Klein A.CAD-Based off-line programming of painting robots[J]. Robotica, 1987,5(4):267-271
[5]Suk-Han Suh, Jung-Jae Lee, Yong-Jong Choi, and Sung-Kwon Lee.Prototype Integrated Robotic Painting System:Software and Hardware Development [J]. Manufacturing Systems,1991,12(6)
[6]Suh S H,Woo I K, Noh S K.Development of automatic trajectory planning system (ATPS) for spray painting robots[C].IEEE Intrenational Conference on Robotics and Automation, USA:Sacramento,1991:1948-1955
[7]Antonio J K.Optimal Trajectory Planning Problems for spray coating[C].IEEE International Conference on Robotics and Automation,USA:Atlanta,1993: 2570-2577
[8]Antonio J K.Optimal Trajectory Planning Problems for spray coating[R]. Puedue University,School of Electrical Engineering, Technical RePort No.TR-EE-93-29, SePtember.1993
[9]Ramabhadran R,Antonio J K.Planning Spatial Paths for Automated Spray Coating Applications[C].IEEE International Conference on Robotics and Automation, USA:Minneapolis 1996:1255-1260
[10]Ramabhadran R,Antonio J K.Fast Solution Techniques for a Class of Optimal Trajectory Planning Problems with Applications to Automated Spray Coating [J].IEEE Transactions on Robotics and Antomation,1997,13(4):519-530
[11]Ramabhadran R,Anonio J K.Fast Solution Teehniques for a Class of Optimal Trajeetory Planning Problems with Applications to Automated Spray Coating [R]. Puedue University,School of Electrical Engineering, Technical Report No. TR-EE-95-9, March.1995
[12]Antonio J K, Ramabhadran R, Ling T L.A Framework for Trajeetory Plannlng for Automated Spray Coating[J].International Journal of Robotics and Automation,1997,12(4):124-134
[13]Zaki Aziza,Eskander Mona. Spray painting of a general three-dimensional surface[C].IEEE/RSJ Intrenational Conference on Robotics and Automation, USA:San Francisco,2000:2172-2176
[14]Sheng W H, Xi N, Song M,Chen Y Automated cad-guided robot path planning for spray Painting of compound surfaces[c].IEEE/RSJ International Conferenee On Intelligent Robots And Systems, Japan:Tankamutsa,2000:1918-1923
[15]Veljko potkonjak, Goran S Dordevie, Dordevic, Kostic.Dynamics of anthropomorphic painting robot:Quality analysis and cost reduction [J]. Robotics and Autonomous Systems,2000,32(2):17-38
[16]Chen H P,Sheng W H,Xi N, M.Song.Automated robot trajectory planning for spray painting off free form surfaces in automotive manufacturing[C].IEEE International Conference on Robotics and Automation, USA:Wshington DC, 2002:450-455
[17]Atkar N Prasad,Choset Howie, Rizzi A Alfred.Towards Optimal Coverage of 2-Dimensional Surfaces Embedded in R3:Choice of Start Curve[C].IEEE/RSJ International Conference on Intelligent Robots and Systems, USA:LasVegas, 2003:3581-3587
[18]Sheng W H, Xi N, Chen H P. Suface Partitioning in automated CAD guided tool Planning for additive manufacturing[C].IEEE/RSJ International Conference On Intelligent Robots And Systems, USA:LasVegas,2003:2072-2077
[19]Sheng W H, Chen H P, Xi N, Tan J D.Optimal tool path planning for compound Surfaces in spray forming proeesses[C],IEEE International Conference on Robotics and Automation, USA:New Orieans,2004:45-50
[20]Chen H P,Xi N, Sheng W H, Chen Y F.Optimal spray gun trajectory planning With variational distribution for forming proeess[C].IEEE International Conference on Robotics and Automation, USA:New Orleans,2004:51-56
[21]Sheng W H, Chen H P,Xi N.Tool path planning for compound surfaces in spray Forming proeesses[J].Automation Science and Engineering,2005,2(6):240-249
[22]Conner C David,Greenfield Aaron, Atkar N Prasad, Rizzi A Alfred and Choset Howie.Paint Deposition Modeling for Trajectory Planning on Automotive Surfaces[J].IEEE Transactions on Antomation Science and Engineering,2005, 2(4):381-392
[23]Atkar N Prasad, Greenfield Aaron,Conner C David,Choset Howie and Rizzi A Alfred.Hierarchical Segmentation of Suifaces Embedded in R3 for Auto-Body Painting[C].Proceedings of the 2005 IEEE International Conference on Robotics and Automation, Spain:Barcelona,2005:574-579
[24]Girma S. Tewolde, Weihua Sheng. Robot Path Integration in Manufacturing Processes:Genetic Algorithm Versus Ant Colony Optimization[J]. IEEE Transactions on Systems,Man,and Cybernetics-Part A:Systems and Humans, 2008:38 (2):278-287
[25]缪苏毅.基于点云切片技术的喷枪轨迹获取方法研究[D].镇江,江苏大学硕士论文,2009
[26]刁训娣,赵德安,李医民等.喷漆机器人喷枪轨迹离线优化方法[J].农机化研究,2004(1)：93-97
[27]冯川,孙增圻.机器人喷涂过程中的喷炬建模及仿真研究[J].机器人,2003,25(4)：353-358
[28]张永贵,黄玉美,高峰等.喷漆机器人空气喷枪的新模型[J].机械工程学报,2006,42(11)：226-233
[29]张永贵,黄玉美,彭中波等.考虑动力学因素的喷漆机器人喷枪路径优化[J].机械科学与技术,2006,25(8)：993-996
[30]崔朝玉.铸造型筒喷枪的轨迹优化及仿真研究[D].兰州,兰州理工大学硕士学位论文,2008
[31]阙骋.喷漆机器人喷枪轨迹离线编程及仿真技术研究[D].镇江,江苏大学硕士论文,2005
[32]王振滨.喷漆机器人最优轨迹设计与仿真[D].镇江,江苏大学硕士论文,2001
[33]郭科,陈聆,魏友华.最优化方法及其应用[M].北京：高等教育出版社,2007：3-5
[34]郭仁生.基于MATLAB和Pro/ENGINEER优化设计实例解析[M].北京：机械工业出版社,2007：61-68
[35]李红.数值分析[M].武汉：华中科技大学出版社,2003：53-59
[36]杨大地,王开荣.数值分析[M].北京：科学技术出版社,2006：133-142
[37]施光燕,董加礼.最优化方法[M]．北京：高等教育出版社,1999：14
[38]Arikan M.A.Sahir and Tuna Balkan.Proeess Modeling, Simulation, and Paint Thickness Measurement for Robotic Spray Painiing[J].Robotic System,2000, 17(9):479-494
[39]Hyotyiemi H.Minor move-global results:Robot trajeetory planning[C].IEEE Conference on Decision and Control, USA:Honolulu,1990:16-22
[40]Peter Hertling, LarsHog, Rune Larsen, John W.Perram, Henrik Gordon Petersen. Task Curve Planning for Painting Robots—Part Ⅰ:Process Modeling and Calibration [J]. IEEE Transaction on Robotics and Automation,1996,12 (2)
[41]陈传波,陆枫.计算机图形学基础[M].北京：电子工业出版社,2003：222-224
[42]Goodman E D, Hoppensteradt L T W.A method for accurate simulation of robotic spray application using empirical parameterization[C].IEEE International Conference on Robotics and Automation, USA:Sacramento,1991: 1357-1368
[43]Kao J H,Prinz F B.Optimal motion planning for deposition in layered manufacturing[C]. Proceedings of 1998 ASME Design Engineering Technical Conferences, USA:Atlanta,1998:1-10
[44]赵德安,陈伟,汤养.面向复杂曲面的喷涂机器人喷枪轨迹优化[J].江苏大学学报自然科学版,2007,28(5)：425-429
[45]Cook C, Schoenefeld D A, Wainwright R L.Finding rural postman tours[C]. Proceedings of the 1998 ACM symposium on applied computing, USA:Atlanta, 1998:318-326
[46]陈伟,赵德安,汤养.自由曲面喷漆机器人喷枪轨迹优化[J]农业机械学报,2008,39(1)：147-150
[47]谢冬秀.解析几何[M].北京：科学出版社,2009：117
[2]王振滨,赵德安,李医民等.喷漆机器人离线编程系统探讨[J].江苏理工大学学报：自然科学版,2000,21(5)：75-82
[3]刁训娣,赵德安,李医民等.喷漆机器人喷枪轨迹设计及影响因素研究[J].机械科学与技术,2004(4)：396-398
[4]Klein A.CAD-Based off-line programming of painting robots[J]. Robotica, 1987,5(4):267-271
[5]Suk-Han Suh, Jung-Jae Lee, Yong-Jong Choi, and Sung-Kwon Lee.Prototype Integrated Robotic Painting System:Software and Hardware Development [J]. Manufacturing Systems,1991,12(6)
[6]Suh S H,Woo I K, Noh S K.Development of automatic trajectory planning system (ATPS) for spray painting robots[C].IEEE Intrenational Conference on Robotics and Automation, USA:Sacramento,1991:1948-1955
[7]Antonio J K.Optimal Trajectory Planning Problems for spray coating[C].IEEE International Conference on Robotics and Automation,USA:Atlanta,1993: 2570-2577
[8]Antonio J K.Optimal Trajectory Planning Problems for spray coating[R]. Puedue University,School of Electrical Engineering, Technical RePort No.TR-EE-93-29, SePtember.1993
[9]Ramabhadran R,Antonio J K.Planning Spatial Paths for Automated Spray Coating Applications[C].IEEE International Conference on Robotics and Automation, USA:Minneapolis 1996:1255-1260
[10]Ramabhadran R,Antonio J K.Fast Solution Techniques for a Class of Optimal Trajectory Planning Problems with Applications to Automated Spray Coating [J].IEEE Transactions on Robotics and Antomation,1997,13(4):519-530
[11]Ramabhadran R,Anonio J K.Fast Solution Teehniques for a Class of Optimal Trajeetory Planning Problems with Applications to Automated Spray Coating [R]. Puedue University,School of Electrical Engineering, Technical Report No. TR-EE-95-9, March.1995
[12]Antonio J K, Ramabhadran R, Ling T L.A Framework for Trajeetory Plannlng for Automated Spray Coating[J].International Journal of Robotics and Automation,1997,12(4):124-134
[13]Zaki Aziza,Eskander Mona. Spray painting of a general three-dimensional surface[C].IEEE/RSJ Intrenational Conference on Robotics and Automation, USA:San Francisco,2000:2172-2176
[14]Sheng W H, Xi N, Song M,Chen Y Automated cad-guided robot path planning for spray Painting of compound surfaces[c].IEEE/RSJ International Conferenee On Intelligent Robots And Systems, Japan:Tankamutsa,2000:1918-1923
[15]Veljko potkonjak, Goran S Dordevie, Dordevic, Kostic.Dynamics of anthropomorphic painting robot:Quality analysis and cost reduction [J]. Robotics and Autonomous Systems,2000,32(2):17-38
[16]Chen H P,Sheng W H,Xi N, M.Song.Automated robot trajectory planning for spray painting off free form surfaces in automotive manufacturing[C].IEEE International Conference on Robotics and Automation, USA:Wshington DC, 2002:450-455
[17]Atkar N Prasad,Choset Howie, Rizzi A Alfred.Towards Optimal Coverage of 2-Dimensional Surfaces Embedded in R3:Choice of Start Curve[C].IEEE/RSJ International Conference on Intelligent Robots and Systems, USA:LasVegas, 2003:3581-3587
[18]Sheng W H, Xi N, Chen H P. Suface Partitioning in automated CAD guided tool Planning for additive manufacturing[C].IEEE/RSJ International Conference On Intelligent Robots And Systems, USA:LasVegas,2003:2072-2077
[19]Sheng W H, Chen H P, Xi N, Tan J D.Optimal tool path planning for compound Surfaces in spray forming proeesses[C],IEEE International Conference on Robotics and Automation, USA:New Orieans,2004:45-50
[20]Chen H P,Xi N, Sheng W H, Chen Y F.Optimal spray gun trajectory planning With variational distribution for forming proeess[C].IEEE International Conference on Robotics and Automation, USA:New Orleans,2004:51-56
[21]Sheng W H, Chen H P,Xi N.Tool path planning for compound surfaces in spray Forming proeesses[J].Automation Science and Engineering,2005,2(6):240-249
[22]Conner C David,Greenfield Aaron, Atkar N Prasad, Rizzi A Alfred and Choset Howie.Paint Deposition Modeling for Trajectory Planning on Automotive Surfaces[J].IEEE Transactions on Antomation Science and Engineering,2005, 2(4):381-392
[23]Atkar N Prasad, Greenfield Aaron,Conner C David,Choset Howie and Rizzi A Alfred.Hierarchical Segmentation of Suifaces Embedded in R3 for Auto-Body Painting[C].Proceedings of the 2005 IEEE International Conference on Robotics and Automation, Spain:Barcelona,2005:574-579
[24]Girma S. Tewolde, Weihua Sheng. Robot Path Integration in Manufacturing Processes:Genetic Algorithm Versus Ant Colony Optimization[J]. IEEE Transactions on Systems,Man,and Cybernetics-Part A:Systems and Humans, 2008:38 (2):278-287
[25]缪苏毅.基于点云切片技术的喷枪轨迹获取方法研究[D].镇江,江苏大学硕士论文,2009
[26]刁训娣,赵德安,李医民等.喷漆机器人喷枪轨迹离线优化方法[J].农机化研究,2004(1)：93-97
[27]冯川,孙增圻.机器人喷涂过程中的喷炬建模及仿真研究[J].机器人,2003,25(4)：353-358
[28]张永贵,黄玉美,高峰等.喷漆机器人空气喷枪的新模型[J].机械工程学报,2006,42(11)：226-233
[29]张永贵,黄玉美,彭中波等.考虑动力学因素的喷漆机器人喷枪路径优化[J].机械科学与技术,2006,25(8)：993-996
[30]崔朝玉.铸造型筒喷枪的轨迹优化及仿真研究[D].兰州,兰州理工大学硕士学位论文,2008
[31]阙骋.喷漆机器人喷枪轨迹离线编程及仿真技术研究[D].镇江,江苏大学硕士论文,2005
[32]王振滨.喷漆机器人最优轨迹设计与仿真[D].镇江,江苏大学硕士论文,2001
[33]郭科,陈聆,魏友华.最优化方法及其应用[M].北京：高等教育出版社,2007：3-5
[34]郭仁生.基于MATLAB和Pro/ENGINEER优化设计实例解析[M].北京：机械工业出版社,2007：61-68
[35]李红.数值分析[M].武汉：华中科技大学出版社,2003：53-59
[36]杨大地,王开荣.数值分析[M].北京：科学技术出版社,2006：133-142
[37]施光燕,董加礼.最优化方法[M]．北京：高等教育出版社,1999：14
[38]Arikan M.A.Sahir and Tuna Balkan.Proeess Modeling, Simulation, and Paint Thickness Measurement for Robotic Spray Painiing[J].Robotic System,2000, 17(9):479-494
[39]Hyotyiemi H.Minor move-global results:Robot trajeetory planning[C].IEEE Conference on Decision and Control, USA:Honolulu,1990:16-22
[40]Peter Hertling, LarsHog, Rune Larsen, John W.Perram, Henrik Gordon Petersen. Task Curve Planning for Painting Robots—Part Ⅰ:Process Modeling and Calibration [J]. IEEE Transaction on Robotics and Automation,1996,12 (2)
[41]陈传波,陆枫.计算机图形学基础[M].北京：电子工业出版社,2003：222-224
[42]Goodman E D, Hoppensteradt L T W.A method for accurate simulation of robotic spray application using empirical parameterization[C].IEEE International Conference on Robotics and Automation, USA:Sacramento,1991: 1357-1368
[43]Kao J H,Prinz F B.Optimal motion planning for deposition in layered manufacturing[C]. Proceedings of 1998 ASME Design Engineering Technical Conferences, USA:Atlanta,1998:1-10
[44]赵德安,陈伟,汤养.面向复杂曲面的喷涂机器人喷枪轨迹优化[J].江苏大学学报自然科学版,2007,28(5)：425-429
[45]Cook C, Schoenefeld D A, Wainwright R L.Finding rural postman tours[C]. Proceedings of the 1998 ACM symposium on applied computing, USA:Atlanta, 1998:318-326
[46]陈伟,赵德安,汤养.自由曲面喷漆机器人喷枪轨迹优化[J]农业机械学报,2008,39(1)：147-150
[47]谢冬秀.解析几何[M].北京：科学出版社,2009：117