基于人工智能技术的虚拟智能对象研究
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摘要
虚拟现实技术是当前计算机领域的一个重要分支,它已经广泛的在工业设计与制造、教育与培训、军事模拟与战场仿真、远程医疗与外科手术、娱乐与游戏等领域中使用,为人类社会生产、生活带来了巨大的影响,是一项蓬勃发展而又充满生机的高新技术。本文以广东省科技中心数字乐园D19“虚拟乒乓球”项目为背景,研究了在虚拟现实系统开发当中所需关键的人工智能技术。本文主要研究了在虚拟现实中虚拟智能对象行为决策、状态变迁以及路径规划问题。为在虚拟现实系统开发和研究提供了一些新的可行的思路和方法,取得了一些研究成果。
本文首先介绍了虚拟现实技术基本背景,综述了虚拟现实有关智能的研究现状。
然后,本文重点研究了在虚拟现实系统开发当中,广泛使用的基于概率方法的人工智能技术,即基于贝叶斯理论和Dempster-Shafer理论的虚拟智能对象行为决策方法。并基于这两种概率方法对虚拟现实中的一些问题提出了相应的解决方法,而且还对比了这两种方法的异同。
本文针对虚拟现实技术具有三I特性,Immersion(沉浸感、临场感)、Interaction(交互性)、Imagination(构想、想像力),尤其是交互性——现在各种成功的虚拟现实系统当中也是广泛利用了这一特性,建立并推导了一种基于6自由度坐标模型的虚拟智能对象状态变迁判断方法。有效解决了虚拟智能对象的状态变迁问题,使得虚拟智能对象可以实时地跟踪参与者的行踪,并作出最为合适的判断。最后使用一个典型例子论证了该方法的有效性。
在虚拟现实中的摄像机的路径规划问题,在一个复杂的虚拟现实场景中进行浏览问题,以及在虚拟现实中仿真日常生活当中有生命的生物自如的移动问题,都是虚拟现实人工智能技术研究中的关键。这些问题都可以使用路径规划的方法来解决。传统的路径规划的方法常使用启发方法,如势场和概率路线图,仅仅对静态的障碍物体有效而且多应用在机器人学科当中。本文建立并推导了一种新的路径规划算法,将网格划分应用于路径规划,并将其应用到虚拟现实领域,得到一种时间复杂度为O ( n log n )的高效最优化的路径规划算法,并适用于虚拟现实动态障碍物的情形。最后使用了一个典型场景论证了该方法的有效性。
最后,对全文进行了总结并指出了后续研究工作的方向。
Virtual Reality(VR) technology is a very important branch among computer science. It would play important role in industry design & manufacture, training & education, military simulation, remote medical treatment & surgical operation, game, entertainment and so on. As a flouring and rapid developing high technology, virtual reality has already take great influence in human work and everyday life. Supported by the Guangdong province science center of the digital park D19--"Virtual ping pang", the work of this paper is to do research on artificial intelligence of virtual reality. The problems in this dissertation are that how the state of Virtual Intelligence Entity changes and how the action of Virtual Intelligence Entity do and path planning in the virtual environment. And the new ideas and the new methods are proved in this dissertation for developing a new system of virtual reality and studying. There are some creative fruits which achieved in it.
In this dissertation the background of virtual reality technology, the related theories of artificial intelligence in virtual reality domain are firstly summarized.
The method basing probability deduction for deciding the action of virtual intelligence entity is gained in the second part. This is feasible. And this can also improve the intelligence level of virtual intelligence entity. There are two ways: one is Bayesian theory and the other is Dempster-Shafer theory. At the end of this part the differences are embodied.
Then aiming at the three I which is Immersion、Interaction、Imagination in virtual reality technology, in this dissertation put forward a coordinate model of 6 degrees of freedom. The most important ingredient in the three I is interaction. Interaction is the cause for the successful systems of virtual reality. The model in this dissertation can solve the problem of the states changing of virtual intelligence entity. This can make virtual intelligence entity real time follow the user. At the end of this part, one typical example is used to prove the effect of this means.
In addition, there are some traditional methods for path planning, such as potential field method and probabilistic road map method. Those methods are just only applied in Robots, which are the same with static obstacle. In this dissertation, there is a new path planning method. Make use of mesh partition. Then use this method in virtual environment. The complexity of this method is O ( n log n ).With some tests, the method has preferably efficiency. The most important is that this method can use for dynamic Obstacle in a virtual environment. And the method can solve the difficult problems in virtual reality, such as path planning of camera, browsing in a complex virtual scene and how the virtual intelligence entity moves in a system of virtual reality.
Finally, this thesis makes a summarization of the entire work and points out the future research direction.
本文首先介绍了虚拟现实技术基本背景,综述了虚拟现实有关智能的研究现状。
然后,本文重点研究了在虚拟现实系统开发当中,广泛使用的基于概率方法的人工智能技术,即基于贝叶斯理论和Dempster-Shafer理论的虚拟智能对象行为决策方法。并基于这两种概率方法对虚拟现实中的一些问题提出了相应的解决方法,而且还对比了这两种方法的异同。
本文针对虚拟现实技术具有三I特性,Immersion(沉浸感、临场感)、Interaction(交互性)、Imagination(构想、想像力),尤其是交互性——现在各种成功的虚拟现实系统当中也是广泛利用了这一特性,建立并推导了一种基于6自由度坐标模型的虚拟智能对象状态变迁判断方法。有效解决了虚拟智能对象的状态变迁问题,使得虚拟智能对象可以实时地跟踪参与者的行踪,并作出最为合适的判断。最后使用一个典型例子论证了该方法的有效性。
在虚拟现实中的摄像机的路径规划问题,在一个复杂的虚拟现实场景中进行浏览问题,以及在虚拟现实中仿真日常生活当中有生命的生物自如的移动问题,都是虚拟现实人工智能技术研究中的关键。这些问题都可以使用路径规划的方法来解决。传统的路径规划的方法常使用启发方法,如势场和概率路线图,仅仅对静态的障碍物体有效而且多应用在机器人学科当中。本文建立并推导了一种新的路径规划算法,将网格划分应用于路径规划,并将其应用到虚拟现实领域,得到一种时间复杂度为O ( n log n )的高效最优化的路径规划算法,并适用于虚拟现实动态障碍物的情形。最后使用了一个典型场景论证了该方法的有效性。
最后,对全文进行了总结并指出了后续研究工作的方向。
Virtual Reality(VR) technology is a very important branch among computer science. It would play important role in industry design & manufacture, training & education, military simulation, remote medical treatment & surgical operation, game, entertainment and so on. As a flouring and rapid developing high technology, virtual reality has already take great influence in human work and everyday life. Supported by the Guangdong province science center of the digital park D19--"Virtual ping pang", the work of this paper is to do research on artificial intelligence of virtual reality. The problems in this dissertation are that how the state of Virtual Intelligence Entity changes and how the action of Virtual Intelligence Entity do and path planning in the virtual environment. And the new ideas and the new methods are proved in this dissertation for developing a new system of virtual reality and studying. There are some creative fruits which achieved in it.
In this dissertation the background of virtual reality technology, the related theories of artificial intelligence in virtual reality domain are firstly summarized.
The method basing probability deduction for deciding the action of virtual intelligence entity is gained in the second part. This is feasible. And this can also improve the intelligence level of virtual intelligence entity. There are two ways: one is Bayesian theory and the other is Dempster-Shafer theory. At the end of this part the differences are embodied.
Then aiming at the three I which is Immersion、Interaction、Imagination in virtual reality technology, in this dissertation put forward a coordinate model of 6 degrees of freedom. The most important ingredient in the three I is interaction. Interaction is the cause for the successful systems of virtual reality. The model in this dissertation can solve the problem of the states changing of virtual intelligence entity. This can make virtual intelligence entity real time follow the user. At the end of this part, one typical example is used to prove the effect of this means.
In addition, there are some traditional methods for path planning, such as potential field method and probabilistic road map method. Those methods are just only applied in Robots, which are the same with static obstacle. In this dissertation, there is a new path planning method. Make use of mesh partition. Then use this method in virtual environment. The complexity of this method is O ( n log n ).With some tests, the method has preferably efficiency. The most important is that this method can use for dynamic Obstacle in a virtual environment. And the method can solve the difficult problems in virtual reality, such as path planning of camera, browsing in a complex virtual scene and how the virtual intelligence entity moves in a system of virtual reality.
Finally, this thesis makes a summarization of the entire work and points out the future research direction.
引文
[1] Sutherland I E. Sketchpad, A Man-Machine Graphical Communication System.AFIPS Spring Joint Computer Conference, 1963, 23: 329-346.
[2] Sutherland I E.The Ultimate Display.In Processings of IFIP Congress,1965:506-508.
[3] Sutherland I E.A Head-Mounted Three Dimension Display.In Proceeding of the AFIPS Fall Joint Computer Conference, 1968: 757-764.
[4] Sutherland I E.Computer Displays.Scientific American, 1970, 222(6): 56-81.
[5] McGreevy M W.The Presence of Field Geologists in Mars-like Terrain.Teleoperators and Virtual Environments, 1993, 1(4): 375-403.
[6] Fisher S S, McGreevy M W.,Humphries J.,et al.Virtual Environment Display System.Processdings of ACM Workshop on Interactive 3-D Graphics, 1986: 231-240.
[7] Fisher S S, McGreevy M W., Humphries J., et al.Virtual Workstation:A Multimodal, Stereoscopic Display Environment.In Processdings of the SPIE, 1986, 726: 517-522.
[8] Fisher S S.Telepresence Master Glove Controller for Dextrous Robotic End-Effectors.In Proceedings of the SPIE, 1986, 726: 391-401.
[9] Fisher S S.Virtual Interface Environment.Proceedings of IEEE/AIAA 7th Digital Avionics System, 1986: 346-350.
[10] Fisher S S.,Jacoby R.,Byrson S.,et al.The Ames Virtual Environment Workstation:Implementation Issues and Requirements.Human-Machine Interface for Teleoperators and Virtual Environments, NASA, 1991: 20-24.
[11] Fisher S S.,Amkraut S., Girard M.,et al.Menagerie:Designing a Virtual Experience.SPIE-The International Society for Optical Engineering-Stereoscopic Display and Virtual Reality System:The Engineering Reality of Virtual Reality, 1994, 2177: 397-400.
[12]汪成为,高文,王行仁.灵境(虚拟现实)技术的理论、实现及应用.清华大学出版社/广西科学技术出版社, 1996.
[13]汪成为.灵境(虚拟现实)是建立人机和谐仿真系统的关键技术.系统仿真学报, 1995, 7(4): 1-17.
[14] Nancy Talbert. Toward Human-Centered Systems. IEEE Computer Graphics &Applications, July/August, 1997: 21-28.
[15] Carl Machover, Steve E Tice. Virtual Reality. IEEE Computer Graphics and Applications, January, 1994: 15-16.
[16] Andrew S.Forsberg, Joseph J. LaViola, Lee Markosian, et al. Projects in VR: Seamless Interaction in Virtual Reality. IEEE Computer Graphics and Applications, November/December, 1997: 6-10.
[17] John N. Latta, David J. Oberg. A Conceptual Virtual Reality Model. IEEE Computer Graphics and Applications, January, 1994: 23-29.
[18] James Cremer, Joseph Kearney, Yiannis Papelis. Driving Simulation: Challenges for VR Technology. IEEE Computer Graphic & Application, September,1996: 16-20.
[19] James Cremer. Simulation and Scenario Support for Virtual Environment. Computer & Graphics, 1996, 20(2): 199-206.
[20] Gerard Hegron. Motion Control in Animation, Simulation and Visualization. Computer Graphics Forum, 1989, 8(5): 347-352.
[21] Lawrence Rosenblum, Jim Durbin, Upul Obeysekare, et al. Shipbord VR: From damage control to design. IEEE Computer Graphics and Applications, 1996: 10-13.
[22] Clinton Potter, Rachael Brady, Patrick Moran, et al. EVAC: A Virtual Environment for Control of Remote Imaging Instrumentation. IEEE Computer Graphics and Applications, 1996: 62-66.
[23] William Ribarsky, Jay Bolter, Augusto Op den Bosh, et al. Visualization and Analysis Using Virtual Reality. IEEE Computer Graphics and Applications, 1994: 10-12.
[24] Xiaoyuan Tu, Terzopoulos,D. Artificial Fishes: Physics, Locomotion, Perception,Behavior, Proceedings of Siggraph, 1994, 28(3):43-50.
[25] Xiaoyuan Tu.Artificial Animals for Computer Animation: Biomechanics, Locomotion, Perception and Behavior[Ph.D Thesis]. Toronto:University of Toronto, 1996.
[26] Blumberg, B.M. and Galyean, T.A. Multi-level direction of autonomous creatures for real-time virtual environment. Computer Graphics, In Proceedings of SIGGRAPH 1995: 47-54.
[27]陈泓鹃等.计算机动画的人工生命方法研究.自动化学报, 2003, 29(6): 986-989.
[28] Smith, A.R.Plants,fractals and formal languages. SIGGRAPH, 1984: 1-10.
[29] P. Prusinkiewicz and A.Lindenmayer.The algorithmic beauty of plants. NewYork : Springer-Verlag, 1990.
[30] P. Prusinkiewicz, A. Lindenmayer, and J.Hanan.Developmental models of herbaceous plants for computer imagery purposes. In Proceedings of SIGGRAPH, 1988: 141-150.
[31] Prusinkiewicz, P.The Algorithmic Beauty of Plants. New York: Springer-Verlag, 1990.
[32] Prusinkiewicz, P. et. al.Synthetic Topiary. In Computer Graphics Proceedings, Annual Conference Series, 1994: 351-358.
[33] Mech, R. & Prusinkiewicz, P.Visual models of plants interacting with their environment[A]. In Computer Graphics Proceedings, Annual Conference Series, 1996:397-410.
[34] Miller, G.S.P. The motion dynamics of snakes and worms. Computer Graphics, 1998, 22(4): 169-177.
[35] Noser H, Thalmann D, Turner R, Animation based on the Interaction of L-systems with Vector Force Fields, Proc. Computer Graphics International 1992: 747-761.
[36] Taylor, T. and Colm Massey.Recent Developments in the Evolution of Morphologiesand Controllers for Physically Simulated Creatures. Artificial Life,2001, 7(1): 77-87.
[37] Pearl Judea. Causality. Cambridge University Press, 2001.
[38] Kohals J and Monney P A. A Mathematical Theory of Hints. Springer, 1995.
[39] Pearl Judea. Probabilistic Reasoning in Intelligent System:Networks of Plausible Interface. Morgan Kaufman, 1987.
[40] Terzopoulos D. Artificial life for computer graphics. Communication of ACM , 1999, 42(8): 33-42.
[41]潘志庚,许威威,张明敏.智能虚拟环境.系统仿真学报, 2001, 11(增刊): 152-155.
[42] Gang Dang, Yunxiang Ling, Huaping Hu, Shiyao Jin. A prototype of layered behavioral description for multi-agent virtual environments. International Journal of Virtual Reality, 2000, 4(4): B1-B5.
[43]李自力,朱光喜,朱耀庭.一种基于会聚双目立体视觉的虚拟智能对象化身模型.电子与信息学报, 2003(6): 763-770.
[44] T.Lozano-Perez. Automatic planning of manipulator transfer movements. IEEE Trans. Syst. ,Man Cybern. , 1981, SMC-11: 681-698.
[45] T.Lozano-Perez. Spatial planning: A configuration space approach. IEEE Trans. Comput. , 1983, C-32: 108-120.
[46] T.Lozano-Perez and M. A. Wesley. An algorithm for planning collsion-free paths among polyhedral obstacles. Commun. ACM. 1979, 22: 560-570.
[47] K. Kedem and M. Sharir. On the union of Jordan regions and collision-free translational motion amidst polyonal obstacles. Discrete Comput. Geom. , 1986,1: 59-71.
[48] B. K. Bhattacharya and J. Zorbas. Solving the two-dimensional findpath problem using a linetriangle reprenstentation of the robot. J. Algorithms, 1988, 9: 449-469.
[49] J. T. Schwartz and M. Sharir. On the“piano movers”problem II: general techniques for computing topological properties of real algebraic manifolds. Adv. Appl. Math.,1983, 4: 298-351.
[50] J. Canny. The Complexity of Robot Motion Planning. MIT Press, Cambridge, 1987.
[51] J. Barraquand and J. C. Latombe. Robot motion planning: A distributed representation approach. Internat. J. Robot. Res. , 1991,10: 628-649.
[52] O, Khatib. Real-time obstacle avoidance for manipulators and mobile robots. Internat. J. Robot. Res. , 1985, 5: 90-98.
[53] R. Volpe and P. Khosla. Artificial potential with elliptical isopotential contours for obstacle avoidance. In Proc. 26th IEEE Conf. on Decision and Control, 1987: 180-185.
[54] M. H. Overmars and P. Svestka. A probabilistic learning approach to motion planning. In Algorithmic Foundations of Robotics, 1995: 19-38
[55] L. Kavraki,P. Svestka, J. C. Latombe,and M.H. Overmars. Probabilistic roadmaps for pathplanning in high dimensional configuration spaces. IEEE Trans. On Robotics and Automation, 1996,12: 566-580.
[56]史红兵,张毅彬,童若锋等.虚拟现实自动漫游的路径规划算法.计算机辅助设计与图形学学报, 2006, 18(4): 592-596.
[57]许威威,潘志庚,张明敏.智能虚拟环境中的决策模型及其应用.中国图象图形学报, 2001, 6(5): 497-501.
[58]朱庆保.动态复杂环境下的机器人路径规划蚂蚁预测算法.计算机学报, 2005, 11: 1898-1906.
[2] Sutherland I E.The Ultimate Display.In Processings of IFIP Congress,1965:506-508.
[3] Sutherland I E.A Head-Mounted Three Dimension Display.In Proceeding of the AFIPS Fall Joint Computer Conference, 1968: 757-764.
[4] Sutherland I E.Computer Displays.Scientific American, 1970, 222(6): 56-81.
[5] McGreevy M W.The Presence of Field Geologists in Mars-like Terrain.Teleoperators and Virtual Environments, 1993, 1(4): 375-403.
[6] Fisher S S, McGreevy M W.,Humphries J.,et al.Virtual Environment Display System.Processdings of ACM Workshop on Interactive 3-D Graphics, 1986: 231-240.
[7] Fisher S S, McGreevy M W., Humphries J., et al.Virtual Workstation:A Multimodal, Stereoscopic Display Environment.In Processdings of the SPIE, 1986, 726: 517-522.
[8] Fisher S S.Telepresence Master Glove Controller for Dextrous Robotic End-Effectors.In Proceedings of the SPIE, 1986, 726: 391-401.
[9] Fisher S S.Virtual Interface Environment.Proceedings of IEEE/AIAA 7th Digital Avionics System, 1986: 346-350.
[10] Fisher S S.,Jacoby R.,Byrson S.,et al.The Ames Virtual Environment Workstation:Implementation Issues and Requirements.Human-Machine Interface for Teleoperators and Virtual Environments, NASA, 1991: 20-24.
[11] Fisher S S.,Amkraut S., Girard M.,et al.Menagerie:Designing a Virtual Experience.SPIE-The International Society for Optical Engineering-Stereoscopic Display and Virtual Reality System:The Engineering Reality of Virtual Reality, 1994, 2177: 397-400.
[12]汪成为,高文,王行仁.灵境(虚拟现实)技术的理论、实现及应用.清华大学出版社/广西科学技术出版社, 1996.
[13]汪成为.灵境(虚拟现实)是建立人机和谐仿真系统的关键技术.系统仿真学报, 1995, 7(4): 1-17.
[14] Nancy Talbert. Toward Human-Centered Systems. IEEE Computer Graphics &Applications, July/August, 1997: 21-28.
[15] Carl Machover, Steve E Tice. Virtual Reality. IEEE Computer Graphics and Applications, January, 1994: 15-16.
[16] Andrew S.Forsberg, Joseph J. LaViola, Lee Markosian, et al. Projects in VR: Seamless Interaction in Virtual Reality. IEEE Computer Graphics and Applications, November/December, 1997: 6-10.
[17] John N. Latta, David J. Oberg. A Conceptual Virtual Reality Model. IEEE Computer Graphics and Applications, January, 1994: 23-29.
[18] James Cremer, Joseph Kearney, Yiannis Papelis. Driving Simulation: Challenges for VR Technology. IEEE Computer Graphic & Application, September,1996: 16-20.
[19] James Cremer. Simulation and Scenario Support for Virtual Environment. Computer & Graphics, 1996, 20(2): 199-206.
[20] Gerard Hegron. Motion Control in Animation, Simulation and Visualization. Computer Graphics Forum, 1989, 8(5): 347-352.
[21] Lawrence Rosenblum, Jim Durbin, Upul Obeysekare, et al. Shipbord VR: From damage control to design. IEEE Computer Graphics and Applications, 1996: 10-13.
[22] Clinton Potter, Rachael Brady, Patrick Moran, et al. EVAC: A Virtual Environment for Control of Remote Imaging Instrumentation. IEEE Computer Graphics and Applications, 1996: 62-66.
[23] William Ribarsky, Jay Bolter, Augusto Op den Bosh, et al. Visualization and Analysis Using Virtual Reality. IEEE Computer Graphics and Applications, 1994: 10-12.
[24] Xiaoyuan Tu, Terzopoulos,D. Artificial Fishes: Physics, Locomotion, Perception,Behavior, Proceedings of Siggraph, 1994, 28(3):43-50.
[25] Xiaoyuan Tu.Artificial Animals for Computer Animation: Biomechanics, Locomotion, Perception and Behavior[Ph.D Thesis]. Toronto:University of Toronto, 1996.
[26] Blumberg, B.M. and Galyean, T.A. Multi-level direction of autonomous creatures for real-time virtual environment. Computer Graphics, In Proceedings of SIGGRAPH 1995: 47-54.
[27]陈泓鹃等.计算机动画的人工生命方法研究.自动化学报, 2003, 29(6): 986-989.
[28] Smith, A.R.Plants,fractals and formal languages. SIGGRAPH, 1984: 1-10.
[29] P. Prusinkiewicz and A.Lindenmayer.The algorithmic beauty of plants. NewYork : Springer-Verlag, 1990.
[30] P. Prusinkiewicz, A. Lindenmayer, and J.Hanan.Developmental models of herbaceous plants for computer imagery purposes. In Proceedings of SIGGRAPH, 1988: 141-150.
[31] Prusinkiewicz, P.The Algorithmic Beauty of Plants. New York: Springer-Verlag, 1990.
[32] Prusinkiewicz, P. et. al.Synthetic Topiary. In Computer Graphics Proceedings, Annual Conference Series, 1994: 351-358.
[33] Mech, R. & Prusinkiewicz, P.Visual models of plants interacting with their environment[A]. In Computer Graphics Proceedings, Annual Conference Series, 1996:397-410.
[34] Miller, G.S.P. The motion dynamics of snakes and worms. Computer Graphics, 1998, 22(4): 169-177.
[35] Noser H, Thalmann D, Turner R, Animation based on the Interaction of L-systems with Vector Force Fields, Proc. Computer Graphics International 1992: 747-761.
[36] Taylor, T. and Colm Massey.Recent Developments in the Evolution of Morphologiesand Controllers for Physically Simulated Creatures. Artificial Life,2001, 7(1): 77-87.
[37] Pearl Judea. Causality. Cambridge University Press, 2001.
[38] Kohals J and Monney P A. A Mathematical Theory of Hints. Springer, 1995.
[39] Pearl Judea. Probabilistic Reasoning in Intelligent System:Networks of Plausible Interface. Morgan Kaufman, 1987.
[40] Terzopoulos D. Artificial life for computer graphics. Communication of ACM , 1999, 42(8): 33-42.
[41]潘志庚,许威威,张明敏.智能虚拟环境.系统仿真学报, 2001, 11(增刊): 152-155.
[42] Gang Dang, Yunxiang Ling, Huaping Hu, Shiyao Jin. A prototype of layered behavioral description for multi-agent virtual environments. International Journal of Virtual Reality, 2000, 4(4): B1-B5.
[43]李自力,朱光喜,朱耀庭.一种基于会聚双目立体视觉的虚拟智能对象化身模型.电子与信息学报, 2003(6): 763-770.
[44] T.Lozano-Perez. Automatic planning of manipulator transfer movements. IEEE Trans. Syst. ,Man Cybern. , 1981, SMC-11: 681-698.
[45] T.Lozano-Perez. Spatial planning: A configuration space approach. IEEE Trans. Comput. , 1983, C-32: 108-120.
[46] T.Lozano-Perez and M. A. Wesley. An algorithm for planning collsion-free paths among polyhedral obstacles. Commun. ACM. 1979, 22: 560-570.
[47] K. Kedem and M. Sharir. On the union of Jordan regions and collision-free translational motion amidst polyonal obstacles. Discrete Comput. Geom. , 1986,1: 59-71.
[48] B. K. Bhattacharya and J. Zorbas. Solving the two-dimensional findpath problem using a linetriangle reprenstentation of the robot. J. Algorithms, 1988, 9: 449-469.
[49] J. T. Schwartz and M. Sharir. On the“piano movers”problem II: general techniques for computing topological properties of real algebraic manifolds. Adv. Appl. Math.,1983, 4: 298-351.
[50] J. Canny. The Complexity of Robot Motion Planning. MIT Press, Cambridge, 1987.
[51] J. Barraquand and J. C. Latombe. Robot motion planning: A distributed representation approach. Internat. J. Robot. Res. , 1991,10: 628-649.
[52] O, Khatib. Real-time obstacle avoidance for manipulators and mobile robots. Internat. J. Robot. Res. , 1985, 5: 90-98.
[53] R. Volpe and P. Khosla. Artificial potential with elliptical isopotential contours for obstacle avoidance. In Proc. 26th IEEE Conf. on Decision and Control, 1987: 180-185.
[54] M. H. Overmars and P. Svestka. A probabilistic learning approach to motion planning. In Algorithmic Foundations of Robotics, 1995: 19-38
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