American Institute of Mathematical Sciences

Advanced
September  2011, 6(3): 425-442. doi: 10.3934/nhm.2011.6.425

Force-based models of pedestrian dynamics

Mohcine Chraibi 1, , Ulrich Kemloh 1, , Andreas Schadschneider 2, and  Armin Seyfried 1,

1. 

Jülich Supercomputing Centre, Forschungszentrum Jülich, 52425 Jülich, Germany, Germany, Germany
2. 

Institute for Theoretical Physics, University of Cologne, 50937 Köln, Germany

Received  December 2010 Revised  May 2011 Published  August 2011

Force-based models describe the interactions of pedestrians in terms of physical and social forces. We discuss some intrinsic problems of this approach, like penetration of particles, unrealistic oscillations and velocities as well as conceptual problems related to violations of Newton's laws. We then present the generalized centrifugal force model which solves some of these problems. Furthermore we discuss the problem of choosing a realistic driving force to an exit. We illustrate this problem by investigating the behaviour of pedestrians at bottlenecks.
Keywords: flow, Pedestrian dynamics, bottleneck, force-based models..
Mathematics Subject Classification: Primary: 82C22, 97M99; Secondary: 990B2.
Citation: Mohcine Chraibi, Ulrich Kemloh, Andreas Schadschneider, Armin Seyfried. Force-based models of pedestrian dynamics. Networks & Heterogeneous Media, 2011, 6 (3) : 425-442. doi: 10.3934/nhm.2011.6.425
References:
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M. Chraibi, A. Seyfried and A. Schadschneider, Generalized centrifugal force model for pedestrian dynamics,, Phys. Rev. E, 82 (2010).  doi: 10.1103/PhysRevE.82.046111.  Google Scholar

[6]

M. Chraibi, A. Seyfried, A. Schadschneider and W. Mackens, "Quantitative Description of Pedestrian Dynamics with a Force-Based Model,", IEEE/WIC/ACM International Joint Conference on Web Intelligence and Intelligent Agent Technology, 3 (2009), 583.   Google Scholar

[7]

M. Chraibi, A. Seyfried, A. Schadschneider and W. Mackens, "Quantitative Verification of a Force-Based Model for Pedestrian Dynamics,", Traffic and Granular Flow '09, (2009).   Google Scholar

[8]

Z. Fang, J. P. Yuan, Y. C. Wang and S. M. Lo, Survey of pedestrian movement and development of a crowd dynamics model,, Fire Safety Journal, 43 (2008), 459.  doi: 10.1016/j.firesaf.2007.12.005.  Google Scholar

[9]

E. R. Galea, ed., "Pedestrian and Evacuation Dynamics 2003,", CMS Press, (2003).   Google Scholar

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C. Gloor, L. Mauron and K. A. Nagel, "Pedestrian Simulation for Hiking in the Alps,", Proceedings of Swiss Transport Research Conference (STRC), (2003).   Google Scholar

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, GNU General public license,, , ().   Google Scholar

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[13]

D. Helbing and P. Molnár, Social force model for pedestrian dynamics,, Phys. Rev. E, 51 (1995), 4282.  doi: 10.1103/PhysRevE.51.4282.  Google Scholar

[14]

D. Helbing, I. J. Farkas, P. Molnár and T. Viscek, Simulation of pedestrian crowds in normal and evacuation situations,, in [44], (): 21.   Google Scholar

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M. Höcker, V. Berkhahn, A. Kneidl, A. Borrmann and W. Klein, "Graph-based Approaches for Simulating Pedestrian Dynamics in Building Models,", 8th European Conference on Product & Process Modelling (ECPPM), (2010).   Google Scholar

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S. P. Hoogendoorn, "Walking Behavior in Bottlenecks and its Implications for Capacity,", TRB 2004 Annual Meeting, (2004).   Google Scholar

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S. P. Hoogendoorn and W. Daamen, A novel calibration approach of microscopic pedestrian models,, in, (2009).   Google Scholar

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S. P. Hoogendoorn, W. Daamen and P. H. L. Bovy, Microscopic pedestrian traffic data collection and analysis by walking experiments: Behaviour at bottlenecks,, in [9], (): 89.   Google Scholar

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A. Johansson, D. Helbing and P. K. Shukla, Specification of the social force pedestrian model by evolutionary adjustment to video tracking data,, Advances in Complex Systems, 10 (2007), 271.  doi: 10.1142/S0219525907001355.  Google Scholar

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A. Kirchner and A. Schadschneider, Simulation of evacuation processes using a bionics-inspired cellular automaton model for pedestrian dynamics,, Physica A, 312 (2002), 260.  doi: 10.1016/S0378-4371(02)00857-9.  Google Scholar

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A. Kirchner, K. Nishinari and A. Schadschneider, Friction effects and clogging in a cellular automaton model for pedestrian dynamics,, Phys. Rev. E, 67 (2003).  doi: 10.1103/PhysRevE.67.056122.  Google Scholar

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W. Klingsch, C. Rogsch, A. Schadschneider and M. Schreckenberg, eds., "Pedestrian and Evacuation Dynamics 2008,", Springer-Verlag, (2010).   Google Scholar

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T. Kretz, A. Grünebohm and M. Schreckenberg, Experimental study of pedestrian flow through a bottleneck,, J. Stat. Mech., 10 (2006).   Google Scholar

[29]

T. Kretz, S. Hengst and P. Vortisch, Pedestrian flow at bottlenecks - validation and calibration of Vissim's social force model of pedestrian traffic and its empirical foundations,, in, (2008).   Google Scholar

[30]

T. I. Lakoba, D. J. Kaup and N. M. Finkelstein, Modifications of the Helbing-Molnár-Farkas-Vicsek social force model for pedestrian evolution,, Simulation, 81 (2005), 339.  doi: 10.1177/0037549705052772.  Google Scholar

[31]

K. Lewin, "Field Theory in Social Science,", Greenwood Press Publishers, (1951).   Google Scholar

[32]

J. Liddle, A. Seyfried, T. Rupprecht, W. Klingsch, A. Schadschneider and A. Winkens, "An Experimental Study of Pedestrian Congestions: Influence of Bottleneck Width and Length,", Traffic and Granular Flow '09, (2009).   Google Scholar

[33]

R. Löhner, On the modelling of pedestrian motion,, Applied Mathematical Modelling, 34 (2010), 366.  doi: 10.1016/j.apm.2009.04.017.  Google Scholar

[34]

K. Müller, "Zur Gestaltung und Bemessung von Fluchtwegen für die Evakuierung von Personen aus Bauwerken auf der Grundlage von Modellversuchen,", Ph.D thesis, (1981).   Google Scholar

[35]

D. R. Parisi and C. O. Dorso, Morphological and dynamical aspects of the room evacuation process,, Physica A, 385 (2007), 343.  doi: 10.1016/j.physa.2007.06.033.  Google Scholar

[36]

D. R. Parisi, M. Gilman and H. Moldovan, A modification of the social force model can reproduce experimental data of pedestrian flows in normal conditions,, Physica A, 388 (2009), 3600.  doi: 10.1016/j.physa.2009.05.027.  Google Scholar

[37]

J. Ondřej, J. Pettré, A. Olivier and S. Donikian, "A Synthetic-Vision-Based Steering Approach for Crowd Simulation,", SIGGRAPH '10: ACM SIGGRAPH, (2010).   Google Scholar

[38]

A. Schadschneider, I'm a football fan ... get me out of here,, Physics World, 21 (2010).   Google Scholar

[39]

A. Schadschneider, W. Klingsch, H. Klüpfel, T. Kretz, C. Rogsch and A. Seyfried, Evacuation dynamics: Empirical results, modeling and applications,, Encyclopedia of Complexity and System Science, (2009).   Google Scholar

[40]

A. Schadschneider, H. Klüpfel, T. Kretz, C. Rogsch and A. Seyfried, Fundamentals of pedestrian and evacuation dynamics,, in, (2009), 124.   Google Scholar

[41]

A. Schadschneider and A. Seyfried, Empirical results for pedestrian dynamics and their implications for modelling,, Networks and Heterogeneous Media, 3 (2011), 545.   Google Scholar

[42]

A. Schadschneider and A. Seyfried, "Modeling Pedestrian Dynamics - From Experiment to Theory and Back,", Traffic and Granular Flow '09, (2009).   Google Scholar

[43]

V. Schneider and R. Könnecke., Simulating evacuation processes with ASERI,, in [44], (): 303.   Google Scholar

[44]

M. Schreckenberg and S. D. Sharma, eds., "Pedestrian and Evacuation Dynamics 2001,", Springer, (2002).   Google Scholar

[45]

W. Schroeder, K. Martin and B. Lorense, "Visualization Toolkit: An Object-Oriented Approach to 3D Graphics,", 4th edition, (2006).   Google Scholar

[46]

A. Seyfried, M. Boltes, J. Kähler, W. Klingsch, A. Portz, T. Rupprecht, A. Schadschneider, B. Steffen and A. Winkens, Enhanced empirical data for the fundamental diagram and the flow through bottlenecks,, in [27], ().   Google Scholar

[47]

A. Seyfried, O. Passon, B. Steffen, M. Boltes, T. Rupprecht and W. Klingsch, New insights into pedestrian flow through bottlenecks,, Transportation Science, 43 (2009), 395.  doi: 10.1287/trsc.1090.0263.  Google Scholar

[48]

A. Seyfried, A. Portz and A. Schadschneider, Phase coexistence in congested states of pedestrian dynamics cellular automata,, in, (6350), 496.   Google Scholar

[49]

A. Seyfried and A. Schadschneider, Validation of cellular automata models of pedestrian dynamics using controlled large-scale experiments,, Cybernetics and Systems, 40 (2009).   Google Scholar

[50]

A. Seyfried, B. Steffen and T. Lippert, Basics of modelling the pedestrian flow,, Physica A, 368 (2006), 232.  doi: 10.1016/j.physa.2005.11.052.  Google Scholar

[51]

W. Shao and D. Terzopoulos, Autonomous pedestrians,, in, (2005).   Google Scholar

[52]

B. Steffen and A. Seyfried, Modelling of pedestrian movement around $90^\circ$ and $180^\circ$ bends,, in, (2009).   Google Scholar

[53]

B. Steffen and A. Seyfried, The repulsive force in continous space models of pedestrian movement,, , (2008).   Google Scholar

[54]

A. Steiner, M. Philipp and A. Schmid, "Parameter Estimation for a Pedestrian Simulation Model,", Swiss Transport Research Conference, (2007).   Google Scholar

[55]

P. A. Thompson and E. W. Marchant, A computer model for the evacuation of large building populations,, Fire Safety Journal, 24 (1995), 131.  doi: 10.1016/0379-7112(95)00019-P.  Google Scholar

[56]

P. A. Thompson and E. W. Marchant, Testing and application of the computer model 'SIMULEX',, Fire Safety Journal, 24 (1995), 149.  doi: 10.1016/0379-7112(95)00020-T.  Google Scholar

[57]

H. Timmermans, ed., "Pedestrian Behavior,", Emerald, (2009).   Google Scholar

[58]

N. Waldau, P. Gattermann, H. Knoflacher and M. Schreckenberg, eds., "Pedestrian and Evacuation Dynamics 2005,", Springer, (2007).   Google Scholar

[59]

D. Yanagisawa, A. Kimura, A. Tomoeda, N. Ryosuke, Y. Suma, Kazumichi Ohtsuka and Katsuhiro Nishinari, Introduction of frictional and turning function for pedestrian outflow with an obstacle,, Phys. Rev. E, 80 (2009).  doi: 10.1103/PhysRevE.80.036110.  Google Scholar

[60]

W. J. Yu, L. Y. Chen, R. Dong and S. Q. Dai, Centrifugal force model for pedestrian dynamics,, Phys. Rev. E, 72 (2005).  doi: 10.1103/PhysRevE.72.026112.  Google Scholar

[61]

X. Liu, W. Song and J. Zhang, Extraction and quantitative analysis of microscopic evacuation characteristics based on digital image processing,, Physica A, 388 (2009), 2717.  doi: 10.1016/j.physa.2009.03.017.  Google Scholar

[62]

X. Zheng and P. Palffy-Muhoray, Distance of closest approach of two arbitrary hard ellipses in two dimensions,, Phys. Rev. E, 75 (2007).  doi: 10.1103/PhysRevE.75.061709.  Google Scholar

show all references

References:
[1]

M. Asano, T. Iryo and M. Kuwahara, Microscopic pedestrian simulation model combined with a tactical model for route choice behaviour,, Transportation Research Part C: Emerging Technologies, 18 (2010), 842.  doi: 10.1016/j.trc.2010.01.005.  Google Scholar

[2]

J. V. Berg, M. Lin and D. Manocha, Reciprocal velocity obstacles for real-time multi-agent navigation,, in, (2008).   Google Scholar

[3]

P. Bourke, Minimum distance between a point and a line, accessed 24, December 2010., Available from: , ().   Google Scholar

[4]

M. Chraibi and A. Seyfried, Pedestrian dynamics with event-driven simulation,, in [27], (): 713.   Google Scholar

[5]

M. Chraibi, A. Seyfried and A. Schadschneider, Generalized centrifugal force model for pedestrian dynamics,, Phys. Rev. E, 82 (2010).  doi: 10.1103/PhysRevE.82.046111.  Google Scholar

[6]

M. Chraibi, A. Seyfried, A. Schadschneider and W. Mackens, "Quantitative Description of Pedestrian Dynamics with a Force-Based Model,", IEEE/WIC/ACM International Joint Conference on Web Intelligence and Intelligent Agent Technology, 3 (2009), 583.   Google Scholar

[7]

M. Chraibi, A. Seyfried, A. Schadschneider and W. Mackens, "Quantitative Verification of a Force-Based Model for Pedestrian Dynamics,", Traffic and Granular Flow '09, (2009).   Google Scholar

[8]

Z. Fang, J. P. Yuan, Y. C. Wang and S. M. Lo, Survey of pedestrian movement and development of a crowd dynamics model,, Fire Safety Journal, 43 (2008), 459.  doi: 10.1016/j.firesaf.2007.12.005.  Google Scholar

[9]

E. R. Galea, ed., "Pedestrian and Evacuation Dynamics 2003,", CMS Press, (2003).   Google Scholar

[10]

C. Gloor, L. Mauron and K. A. Nagel, "Pedestrian Simulation for Hiking in the Alps,", Proceedings of Swiss Transport Research Conference (STRC), (2003).   Google Scholar

[11]

, GNU General public license,, , ().   Google Scholar

[12]

D. Helbing, Collective phenomena and states in traffic and self-driven many-particle systems,, Computational Materials Science, 30 (2004), 180.  doi: 10.1016/j.commatsci.2004.01.026.  Google Scholar

[13]

D. Helbing and P. Molnár, Social force model for pedestrian dynamics,, Phys. Rev. E, 51 (1995), 4282.  doi: 10.1103/PhysRevE.51.4282.  Google Scholar

[14]

D. Helbing, I. J. Farkas, P. Molnár and T. Viscek, Simulation of pedestrian crowds in normal and evacuation situations,, in [44], (): 21.   Google Scholar

[15]

M. Höcker, V. Berkhahn, A. Kneidl, A. Borrmann and W. Klein, "Graph-based Approaches for Simulating Pedestrian Dynamics in Building Models,", 8th European Conference on Product & Process Modelling (ECPPM), (2010).   Google Scholar

[16]

S. Holl and A. Seyfried, Hermes - an evacuation assistant for mass events,, inSiDe, 7 (2009), 60.   Google Scholar

[17]

S. P. Hoogendoorn, "Walking Behavior in Bottlenecks and its Implications for Capacity,", TRB 2004 Annual Meeting, (2004).   Google Scholar

[18]

S. P. Hoogendoorn and W. Daamen, A novel calibration approach of microscopic pedestrian models,, in, (2009).   Google Scholar

[19]

S. P. Hoogendoorn and W. Daamen, Pedestrian behavior at bottlenecks,, Transportation Science, 39 (2005), 147.  doi: 10.1287/trsc.1040.0102.  Google Scholar

[20]

S. P. Hoogendoorn, W. Daamen and P. H. L Bovy, "Extracting Microscopic Pedestrian Characteristics from Video Data,", TRB 2004 Annual Meeting Washington DC: National Academy Press, (2004).   Google Scholar

[21]

S. P. Hoogendoorn, W. Daamen and P. H. L. Bovy, Microscopic pedestrian traffic data collection and analysis by walking experiments: Behaviour at bottlenecks,, in [9], (): 89.   Google Scholar

[22]

S. P. Hoogendoorn, W. Daamen and R. Landman, Microscopic calibration and validation of pedestrian models - Cross-comparison of models using experimental data,, in [58], ().   Google Scholar

[23]

A. Johansson, D. Helbing and P. K. Shukla, Specification of the social force pedestrian model by evolutionary adjustment to video tracking data,, Advances in Complex Systems, 10 (2007), 271.  doi: 10.1142/S0219525907001355.  Google Scholar

[24]

A. Kirchner, "Modellierung und Statistische Physik biologischer und sozialer Systeme,", Ph.D thesis, (2003).   Google Scholar

[25]

A. Kirchner and A. Schadschneider, Simulation of evacuation processes using a bionics-inspired cellular automaton model for pedestrian dynamics,, Physica A, 312 (2002), 260.  doi: 10.1016/S0378-4371(02)00857-9.  Google Scholar

[26]

A. Kirchner, K. Nishinari and A. Schadschneider, Friction effects and clogging in a cellular automaton model for pedestrian dynamics,, Phys. Rev. E, 67 (2003).  doi: 10.1103/PhysRevE.67.056122.  Google Scholar

[27]

W. Klingsch, C. Rogsch, A. Schadschneider and M. Schreckenberg, eds., "Pedestrian and Evacuation Dynamics 2008,", Springer-Verlag, (2010).   Google Scholar

[28]

T. Kretz, A. Grünebohm and M. Schreckenberg, Experimental study of pedestrian flow through a bottleneck,, J. Stat. Mech., 10 (2006).   Google Scholar

[29]

T. Kretz, S. Hengst and P. Vortisch, Pedestrian flow at bottlenecks - validation and calibration of Vissim's social force model of pedestrian traffic and its empirical foundations,, in, (2008).   Google Scholar

[30]

T. I. Lakoba, D. J. Kaup and N. M. Finkelstein, Modifications of the Helbing-Molnár-Farkas-Vicsek social force model for pedestrian evolution,, Simulation, 81 (2005), 339.  doi: 10.1177/0037549705052772.  Google Scholar

[31]

K. Lewin, "Field Theory in Social Science,", Greenwood Press Publishers, (1951).   Google Scholar

[32]

J. Liddle, A. Seyfried, T. Rupprecht, W. Klingsch, A. Schadschneider and A. Winkens, "An Experimental Study of Pedestrian Congestions: Influence of Bottleneck Width and Length,", Traffic and Granular Flow '09, (2009).   Google Scholar

[33]

R. Löhner, On the modelling of pedestrian motion,, Applied Mathematical Modelling, 34 (2010), 366.  doi: 10.1016/j.apm.2009.04.017.  Google Scholar

[34]

K. Müller, "Zur Gestaltung und Bemessung von Fluchtwegen für die Evakuierung von Personen aus Bauwerken auf der Grundlage von Modellversuchen,", Ph.D thesis, (1981).   Google Scholar

[35]

D. R. Parisi and C. O. Dorso, Morphological and dynamical aspects of the room evacuation process,, Physica A, 385 (2007), 343.  doi: 10.1016/j.physa.2007.06.033.  Google Scholar

[36]

D. R. Parisi, M. Gilman and H. Moldovan, A modification of the social force model can reproduce experimental data of pedestrian flows in normal conditions,, Physica A, 388 (2009), 3600.  doi: 10.1016/j.physa.2009.05.027.  Google Scholar

[37]

J. Ondřej, J. Pettré, A. Olivier and S. Donikian, "A Synthetic-Vision-Based Steering Approach for Crowd Simulation,", SIGGRAPH '10: ACM SIGGRAPH, (2010).   Google Scholar

[38]

A. Schadschneider, I'm a football fan ... get me out of here,, Physics World, 21 (2010).   Google Scholar

[39]

A. Schadschneider, W. Klingsch, H. Klüpfel, T. Kretz, C. Rogsch and A. Seyfried, Evacuation dynamics: Empirical results, modeling and applications,, Encyclopedia of Complexity and System Science, (2009).   Google Scholar

[40]

A. Schadschneider, H. Klüpfel, T. Kretz, C. Rogsch and A. Seyfried, Fundamentals of pedestrian and evacuation dynamics,, in, (2009), 124.   Google Scholar

[41]

A. Schadschneider and A. Seyfried, Empirical results for pedestrian dynamics and their implications for modelling,, Networks and Heterogeneous Media, 3 (2011), 545.   Google Scholar

[42]

A. Schadschneider and A. Seyfried, "Modeling Pedestrian Dynamics - From Experiment to Theory and Back,", Traffic and Granular Flow '09, (2009).   Google Scholar

[43]

V. Schneider and R. Könnecke., Simulating evacuation processes with ASERI,, in [44], (): 303.   Google Scholar

[44]

M. Schreckenberg and S. D. Sharma, eds., "Pedestrian and Evacuation Dynamics 2001,", Springer, (2002).   Google Scholar

[45]

W. Schroeder, K. Martin and B. Lorense, "Visualization Toolkit: An Object-Oriented Approach to 3D Graphics,", 4th edition, (2006).   Google Scholar

[46]

A. Seyfried, M. Boltes, J. Kähler, W. Klingsch, A. Portz, T. Rupprecht, A. Schadschneider, B. Steffen and A. Winkens, Enhanced empirical data for the fundamental diagram and the flow through bottlenecks,, in [27], ().   Google Scholar

[47]

A. Seyfried, O. Passon, B. Steffen, M. Boltes, T. Rupprecht and W. Klingsch, New insights into pedestrian flow through bottlenecks,, Transportation Science, 43 (2009), 395.  doi: 10.1287/trsc.1090.0263.  Google Scholar

[48]

A. Seyfried, A. Portz and A. Schadschneider, Phase coexistence in congested states of pedestrian dynamics cellular automata,, in, (6350), 496.   Google Scholar

[49]

A. Seyfried and A. Schadschneider, Validation of cellular automata models of pedestrian dynamics using controlled large-scale experiments,, Cybernetics and Systems, 40 (2009).   Google Scholar

[50]

A. Seyfried, B. Steffen and T. Lippert, Basics of modelling the pedestrian flow,, Physica A, 368 (2006), 232.  doi: 10.1016/j.physa.2005.11.052.  Google Scholar

[51]

W. Shao and D. Terzopoulos, Autonomous pedestrians,, in, (2005).   Google Scholar

[52]

B. Steffen and A. Seyfried, Modelling of pedestrian movement around $90^\circ$ and $180^\circ$ bends,, in, (2009).   Google Scholar

[53]

B. Steffen and A. Seyfried, The repulsive force in continous space models of pedestrian movement,, , (2008).   Google Scholar

[54]

A. Steiner, M. Philipp and A. Schmid, "Parameter Estimation for a Pedestrian Simulation Model,", Swiss Transport Research Conference, (2007).   Google Scholar

[55]

P. A. Thompson and E. W. Marchant, A computer model for the evacuation of large building populations,, Fire Safety Journal, 24 (1995), 131.  doi: 10.1016/0379-7112(95)00019-P.  Google Scholar

[56]

P. A. Thompson and E. W. Marchant, Testing and application of the computer model 'SIMULEX',, Fire Safety Journal, 24 (1995), 149.  doi: 10.1016/0379-7112(95)00020-T.  Google Scholar

[57]

H. Timmermans, ed., "Pedestrian Behavior,", Emerald, (2009).   Google Scholar

[58]

N. Waldau, P. Gattermann, H. Knoflacher and M. Schreckenberg, eds., "Pedestrian and Evacuation Dynamics 2005,", Springer, (2007).   Google Scholar

[59]

D. Yanagisawa, A. Kimura, A. Tomoeda, N. Ryosuke, Y. Suma, Kazumichi Ohtsuka and Katsuhiro Nishinari, Introduction of frictional and turning function for pedestrian outflow with an obstacle,, Phys. Rev. E, 80 (2009).  doi: 10.1103/PhysRevE.80.036110.  Google Scholar

[60]

W. J. Yu, L. Y. Chen, R. Dong and S. Q. Dai, Centrifugal force model for pedestrian dynamics,, Phys. Rev. E, 72 (2005).  doi: 10.1103/PhysRevE.72.026112.  Google Scholar

[61]

X. Liu, W. Song and J. Zhang, Extraction and quantitative analysis of microscopic evacuation characteristics based on digital image processing,, Physica A, 388 (2009), 2717.  doi: 10.1016/j.physa.2009.03.017.  Google Scholar

[62]

X. Zheng and P. Palffy-Muhoray, Distance of closest approach of two arbitrary hard ellipses in two dimensions,, Phys. Rev. E, 75 (2007).  doi: 10.1103/PhysRevE.75.061709.  Google Scholar

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