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Spatiotemporal evolution model of vehicular movement Behavior under path constraints

Pan Deng Zheng Ying-Ping

Spatiotemporal evolution model of vehicular movement Behavior under path constraints

Pan Deng, Zheng Ying-Ping
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  • The path of vehicle movement in a complex geographical environment has the 3D space feature, which is evidently the constraint for vehicular movement behaviors and cannot be described in one-dimensional or two-dimensional space. But the path of vehicle movement in a complex geographical environment can be abstracted into a space curve. By introducing theories in differential geometry, we can build a Serret-Frenet frame moving along this space curve with the geometric invariants of arc length, curvature, and torsion. And then, we can give a mathematical description to the dynamic behavior of the Serret-Frenet frame with the time-varying property at an arbitrary point of space curve. Finally, the spatiotemporal evolution model of the vehicle movement behaviors under the path constraint conditions is established and is rigidly proven in mathematics to be suitable for the longitudinal movement and uniform circular motion of a vehicle (in Serret-Frenet frame). It will lay the theoretical foundation for the future study of vehicular movement behaviors on the transport line in a complex geographical environment, including vehicular microscopic behaviors such as the vehicle following operation, lane changing, as well as the vehicular macroscopic behavior in traffic flows.
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61174183).
    [1]

    Tomer E, Safonov L, Havlin S 2000 Phys. Rev. Lett. 84 382

    [2]

    Koutsopoulos H N, Farah H 2012 Transp. Res. Part. B 46 563

    [3]

    Naito Y, Nagatani T 2011 Phys. Lett. A 375 1319

    [4]

    Naito Y, Nagatani T 2012 Physica A 391 1626

    [5]

    Zheng Z D 2014 Transp. Res. Part B 60 16

    [6]

    Lee H Y, Lee H W, Kim D 1998 Phys. Rev. Lett. 81 1130

    [7]

    Mitarai N, Nakanishi H 2000 Phys. Rev. Lett. 85 1766

    [8]

    Peng G H, Sun D H, He H P 2009 Chin. Phys. B 18 468

    [9]

    Zheng L, Ma S F, Zhong S Q 2011 Chin. Phys. B 20 088701

    [10]

    Peng G H, Cai X H, Liu C Q 2011 Phys. Lett. A 375 3973

    [11]

    Sugiyama N, Nagatani T 2012 Phys. Lett. A 376 1803

    [12]

    Tang T Q, Wang Y P, Yang X B, Wu Y H 2012 Nonlinear Dynam. 70 1397

    [13]

    Jetto K, Ez-Zahraouy H, Benyoussef A 2012 Chin. Phys. B 21 118901

    [14]

    Gupta A K, Sharma S 2012 Chin. Phys. B 21 015201

    [15]

    Kamal M A S, Imura J, Hayakawa T, Ohata A, Aihara K 2014 IEEE Trans. Intell. Transp. Syst. 15 878

    [16]

    Ploeg J, Shukla D P, van de Wouw N, Nijmeijer H 2014 IEEE Trans. Intell. Transp. Syst. 15 854

    [17]

    Davis L C 2014 Physica A 405 128

    [18]

    Laval J A, Leclercq L 2013 Transp. Res. Part B 52 17

    [19]

    Ren D B, Zhang J Y, Zhang J M, Cui S M 2011 Sci. China Ser. E 54 630

    [20]

    Zhang L D, Jia L, Zhu W X 2012 Acta Phys. Sin. 61 074501 (in Chinese) [张立东, 贾磊, 朱文兴 2012 物理学报 61 074501]

    [21]

    Guo L, Huang X H, Ge P S 2013 Journal of Jilin University (Engineering and Technology Edition) 43 323 (in Chinese) [郭烈, 黄晓慧, 葛平淑2013 吉林大学学报 (工学版) 43 323]

    [22]

    Ren D B, Zhang J M, Wang C 2014 Acta Phys. Sin. 63 078902 (in Chinese) [任殿波, 张京明, 王聪 2014 物理学报 63 078902]

    [23]

    He Z C, Sun W B, Zhang L C, Xu F F, Zhuang L J 2013 Acta Phys. Sin. 62 168901 (in Chinese) [何兆成, 孙文博, 张力成, 许菲菲, 庄立坚 2013 物理学报 62 168901]

    [24]

    Saffarian M, de Winter J C F, Happee R 2013 IEEE Trans. Human-Mach. Syst. 43 8

    [25]

    Wang J Q, Zhang L, Zhang D Z, Li K Q 2013 IEEE Trans. Intell. Transp. Syst. 14 1

    [26]

    Borhaug E, Pavlov A, Pettersen K Y 2008 Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, Dec. 9-11, 2008 p4984

    [27]

    Ghommam J, Mnif F, Benali A 2009 J. Dyn. Syst.-T. Asme. 131 021006

    [28]

    Sasongko R A, Sembiring J, Muhammad H, Mulyanto T 2011 Proceedings of the 8th Asian Control Conference, Kaohsiung, Taiwan, May 15-18, 2011 p1259

    [29]

    Moe S, Caharija W, Pettersen K Y, Schjolberg I 2014 American Control Conference, Portland, Oregon, USA, June 4-6, 2014 p3856

    [30]

    Burger M, Pettersen K Y 2010 The 49th IEEE Conference on Decision and Control, Atlanta, GA, USA, December 15-17, 2010 p7159

    [31]

    Aggoune W, Morarescu I C, Niculescu S I 2011 Mathematical Reports 13 217

    [32]

    Ali Z, Popov A A, Charles G 2013 Vehicle Syst. Dyn. 51 943

    [33]

    Kim Y C, Yun K H, Min K D 2014 Vehicle Syst. Dyn. 52 456

    [34]

    Pan D, Zheng Y 2014 IET Intell. Trans. SY. 8 232

    [35]

    Xiao L Y, Gao F 2010 Vehicle Syst. Dyn. 48 1167

    [36]

    Davis L C 2012 Phys. Lett. A 376 2658

    [37]

    Lee M H, Park H G, Lee, S H, Yoon K S, Lee K S 2013 Int. J. Precis. Eng. Man. 14 373

    [38]

    Bifulco G N, Pariota L, Simonelli F, Di Pace R 2013 Transp. Res. Part C 29 156

    [39]

    Wang M, Daamen W, Hoogendoorn S, van Arem B 2014 IET Intell. Trans. SY. 8 77

    [40]

    Chen W H 2006 Differential Geometry (Beijing: Peking University Press) pp23-32 (in Chinese) [陈维桓2006微分几何(北京: 北京大学出版社)第23–32页]

    [41]

    Su B Q, Hua Y J, Xin Y L 2010 Introduction to Practical Differential Geometry (Beijing: Science Press) pp10-27 (in Chinese) [苏步青, 华宣积, 忻元龙2010实用微分几何引论(北京: 科学出版社)第10–27页]

  • [1]

    Tomer E, Safonov L, Havlin S 2000 Phys. Rev. Lett. 84 382

    [2]

    Koutsopoulos H N, Farah H 2012 Transp. Res. Part. B 46 563

    [3]

    Naito Y, Nagatani T 2011 Phys. Lett. A 375 1319

    [4]

    Naito Y, Nagatani T 2012 Physica A 391 1626

    [5]

    Zheng Z D 2014 Transp. Res. Part B 60 16

    [6]

    Lee H Y, Lee H W, Kim D 1998 Phys. Rev. Lett. 81 1130

    [7]

    Mitarai N, Nakanishi H 2000 Phys. Rev. Lett. 85 1766

    [8]

    Peng G H, Sun D H, He H P 2009 Chin. Phys. B 18 468

    [9]

    Zheng L, Ma S F, Zhong S Q 2011 Chin. Phys. B 20 088701

    [10]

    Peng G H, Cai X H, Liu C Q 2011 Phys. Lett. A 375 3973

    [11]

    Sugiyama N, Nagatani T 2012 Phys. Lett. A 376 1803

    [12]

    Tang T Q, Wang Y P, Yang X B, Wu Y H 2012 Nonlinear Dynam. 70 1397

    [13]

    Jetto K, Ez-Zahraouy H, Benyoussef A 2012 Chin. Phys. B 21 118901

    [14]

    Gupta A K, Sharma S 2012 Chin. Phys. B 21 015201

    [15]

    Kamal M A S, Imura J, Hayakawa T, Ohata A, Aihara K 2014 IEEE Trans. Intell. Transp. Syst. 15 878

    [16]

    Ploeg J, Shukla D P, van de Wouw N, Nijmeijer H 2014 IEEE Trans. Intell. Transp. Syst. 15 854

    [17]

    Davis L C 2014 Physica A 405 128

    [18]

    Laval J A, Leclercq L 2013 Transp. Res. Part B 52 17

    [19]

    Ren D B, Zhang J Y, Zhang J M, Cui S M 2011 Sci. China Ser. E 54 630

    [20]

    Zhang L D, Jia L, Zhu W X 2012 Acta Phys. Sin. 61 074501 (in Chinese) [张立东, 贾磊, 朱文兴 2012 物理学报 61 074501]

    [21]

    Guo L, Huang X H, Ge P S 2013 Journal of Jilin University (Engineering and Technology Edition) 43 323 (in Chinese) [郭烈, 黄晓慧, 葛平淑2013 吉林大学学报 (工学版) 43 323]

    [22]

    Ren D B, Zhang J M, Wang C 2014 Acta Phys. Sin. 63 078902 (in Chinese) [任殿波, 张京明, 王聪 2014 物理学报 63 078902]

    [23]

    He Z C, Sun W B, Zhang L C, Xu F F, Zhuang L J 2013 Acta Phys. Sin. 62 168901 (in Chinese) [何兆成, 孙文博, 张力成, 许菲菲, 庄立坚 2013 物理学报 62 168901]

    [24]

    Saffarian M, de Winter J C F, Happee R 2013 IEEE Trans. Human-Mach. Syst. 43 8

    [25]

    Wang J Q, Zhang L, Zhang D Z, Li K Q 2013 IEEE Trans. Intell. Transp. Syst. 14 1

    [26]

    Borhaug E, Pavlov A, Pettersen K Y 2008 Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, Dec. 9-11, 2008 p4984

    [27]

    Ghommam J, Mnif F, Benali A 2009 J. Dyn. Syst.-T. Asme. 131 021006

    [28]

    Sasongko R A, Sembiring J, Muhammad H, Mulyanto T 2011 Proceedings of the 8th Asian Control Conference, Kaohsiung, Taiwan, May 15-18, 2011 p1259

    [29]

    Moe S, Caharija W, Pettersen K Y, Schjolberg I 2014 American Control Conference, Portland, Oregon, USA, June 4-6, 2014 p3856

    [30]

    Burger M, Pettersen K Y 2010 The 49th IEEE Conference on Decision and Control, Atlanta, GA, USA, December 15-17, 2010 p7159

    [31]

    Aggoune W, Morarescu I C, Niculescu S I 2011 Mathematical Reports 13 217

    [32]

    Ali Z, Popov A A, Charles G 2013 Vehicle Syst. Dyn. 51 943

    [33]

    Kim Y C, Yun K H, Min K D 2014 Vehicle Syst. Dyn. 52 456

    [34]

    Pan D, Zheng Y 2014 IET Intell. Trans. SY. 8 232

    [35]

    Xiao L Y, Gao F 2010 Vehicle Syst. Dyn. 48 1167

    [36]

    Davis L C 2012 Phys. Lett. A 376 2658

    [37]

    Lee M H, Park H G, Lee, S H, Yoon K S, Lee K S 2013 Int. J. Precis. Eng. Man. 14 373

    [38]

    Bifulco G N, Pariota L, Simonelli F, Di Pace R 2013 Transp. Res. Part C 29 156

    [39]

    Wang M, Daamen W, Hoogendoorn S, van Arem B 2014 IET Intell. Trans. SY. 8 77

    [40]

    Chen W H 2006 Differential Geometry (Beijing: Peking University Press) pp23-32 (in Chinese) [陈维桓2006微分几何(北京: 北京大学出版社)第23–32页]

    [41]

    Su B Q, Hua Y J, Xin Y L 2010 Introduction to Practical Differential Geometry (Beijing: Science Press) pp10-27 (in Chinese) [苏步青, 华宣积, 忻元龙2010实用微分几何引论(北京: 科学出版社)第10–27页]

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Publishing process
  • Received Date:  28 August 2014
  • Accepted Date:  06 November 2014
  • Published Online:  05 April 2015

Spatiotemporal evolution model of vehicular movement Behavior under path constraints

  • 1. School of Electronic & Information Engineering, Tongji University, Shanghai 201804, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant No. 61174183).

Abstract: The path of vehicle movement in a complex geographical environment has the 3D space feature, which is evidently the constraint for vehicular movement behaviors and cannot be described in one-dimensional or two-dimensional space. But the path of vehicle movement in a complex geographical environment can be abstracted into a space curve. By introducing theories in differential geometry, we can build a Serret-Frenet frame moving along this space curve with the geometric invariants of arc length, curvature, and torsion. And then, we can give a mathematical description to the dynamic behavior of the Serret-Frenet frame with the time-varying property at an arbitrary point of space curve. Finally, the spatiotemporal evolution model of the vehicle movement behaviors under the path constraint conditions is established and is rigidly proven in mathematics to be suitable for the longitudinal movement and uniform circular motion of a vehicle (in Serret-Frenet frame). It will lay the theoretical foundation for the future study of vehicular movement behaviors on the transport line in a complex geographical environment, including vehicular microscopic behaviors such as the vehicle following operation, lane changing, as well as the vehicular macroscopic behavior in traffic flows.

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