Force-based models of pedestrian dynamics

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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

Abstract
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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 and Heterogeneous Media, 2011, 6 (3) : 425-442. doi: 10.3934/nhm.2011.6.425

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), 842-855. doi: 10.1016/j.trc.2010.01.005.
[2]	J. V. Berg, M. Lin and D. Manocha, Reciprocal velocity obstacles for real-time multi-agent navigation, in "Robotics and Automation," IEEE International Conference on Robotics and Automation Pasadena, CA, USA, 2008.
[3]	P. Bourke, Minimum distance between a point and a line, accessed 24, December 2010. Available from: http://paulbourke.net/geometry/pointline/.
[4]	M. Chraibi and A. Seyfried, Pedestrian dynamics with event-driven simulation, in [27], 713-718.
[5]	M. Chraibi, A. Seyfried and A. Schadschneider, Generalized centrifugal force model for pedestrian dynamics, Phys. Rev. E, 82 (2010), 046111. doi: 10.1103/PhysRevE.82.046111.
[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, IEEE Computer Society, Los Alamitos, CA, USA, (2009), 583-586.
[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.
[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), 459-465. doi: 10.1016/j.firesaf.2007.12.005.
[9]	E. R. Galea, ed., "Pedestrian and Evacuation Dynamics 2003," CMS Press, London, 2003.
[10]	C. Gloor, L. Mauron and K. A. Nagel, "Pedestrian Simulation for Hiking in the Alps," Proceedings of Swiss Transport Research Conference (STRC), Monte Verita, 2003.
[11]	, GNU General public license, http://www.gnu.org/licenses/gpl.html.
[12]	D. Helbing, Collective phenomena and states in traffic and self-driven many-particle systems, Computational Materials Science, 30 (2004), 180-187. doi: 10.1016/j.commatsci.2004.01.026.
[13]	D. Helbing and P. Molnár, Social force model for pedestrian dynamics, Phys. Rev. E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282.
[14]	D. Helbing, I. J. Farkas, P. Molnár and T. Viscek, Simulation of pedestrian crowds in normal and evacuation situations, in [44], 21-58.
[15]	M. Höcker, V. Berkhahn, A. Kneidl, A. Borrmann and W. Klein, "Graph-based Approaches for Simulating Pedestrian Dynamics in Building Models," 8^th European Conference on Product & Process Modelling (ECPPM), University College Cork, Cork, Ireland, 2010.
[16]	S. Holl and A. Seyfried, Hermes - an evacuation assistant for mass events, inSiDe, 7 (2009), 60-61.
[17]	S. P. Hoogendoorn, "Walking Behavior in Bottlenecks and its Implications for Capacity," TRB 2004 Annual Meeting, 2004.
[18]	S. P. Hoogendoorn and W. Daamen, A novel calibration approach of microscopic pedestrian models, in "Pedestrian Behavior" (ed. H. Timmermans), Emerald, 2009, p. 195.
[19]	S. P. Hoogendoorn and W. Daamen, Pedestrian behavior at bottlenecks, Transportation Science, 39 (2005), 147-159. doi: 10.1287/trsc.1040.0102.
[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, 2003.
[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], 89-100.
[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], 253.
[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), 271-288. doi: 10.1142/S0219525907001355.
[24]	A. Kirchner, "Modellierung und Statistische Physik biologischer und sozialer Systeme," Ph.D thesis, Universität zu Köln, Germany, 2003.
[25]	A. Kirchner and A. Schadschneider, Simulation of evacuation processes using a bionics-inspired cellular automaton model for pedestrian dynamics, Physica A, 312 (2002), 260-276. doi: 10.1016/S0378-4371(02)00857-9.
[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), 056122. doi: 10.1103/PhysRevE.67.056122.
[27]	W. Klingsch, C. Rogsch, A. Schadschneider and M. Schreckenberg, eds., "Pedestrian and Evacuation Dynamics 2008," Springer-Verlag, Berlin, Heidelberg, 2010.
[28]	T. Kretz, A. Grünebohm and M. Schreckenberg, Experimental study of pedestrian flow through a bottleneck, J. Stat. Mech., 10 (2006), P10014.
[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 "International Symposium of Transport Simulation" (ed. M Sarvi), Monash University, Melbourne, Australia, 2008.
[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), 339-352. doi: 10.1177/0037549705052772.
[31]	K. Lewin, "Field Theory in Social Science," Greenwood Press Publishers, 1951.
[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, arXiv:0911.4350.
[33]	R. Löhner, On the modelling of pedestrian motion, Applied Mathematical Modelling, 34 (2010), 366-382. doi: 10.1016/j.apm.2009.04.017.
[34]	K. Müller, "Zur Gestaltung und Bemessung von Fluchtwegen für die Evakuierung von Personen aus Bauwerken auf der Grundlage von Modellversuchen," Ph.D thesis, Magdeburg, Germany, 1981.
[35]	D. R. Parisi and C. O. Dorso, Morphological and dynamical aspects of the room evacuation process, Physica A, 385 (2007), 343-355. doi: 10.1016/j.physa.2007.06.033.
[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), 3600-3608. doi: 10.1016/j.physa.2009.05.027.
[37]	J. Ondřej, J. Pettré, A. Olivier and S. Donikian, "A Synthetic-Vision-Based Steering Approach for Crowd Simulation," SIGGRAPH '10: ACM SIGGRAPH, 2010.
[38]	A. Schadschneider, I'm a football fan ... get me out of here, Physics World, 21, July 2010.
[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, 3142.
[40]	A. Schadschneider, H. Klüpfel, T. Kretz, C. Rogsch and A. Seyfried, Fundamentals of pedestrian and evacuation dynamics, in "Multi-Agent Systems for Traffic and Transportation Engineering" (eds. Ana Bazzan and Franziska Klügl), IGI Global, Hershey, Pennsylvania, USA, 2009, chapter 6, 124-154.
[41]	A. Schadschneider and A. Seyfried, Empirical results for pedestrian dynamics and their implications for modelling, Networks and Heterogeneous Media, 3 (2011), 545-560.
[42]	A. Schadschneider and A. Seyfried, "Modeling Pedestrian Dynamics - From Experiment to Theory and Back," Traffic and Granular Flow '09, 2009.
[43]	V. Schneider and R. Könnecke., Simulating evacuation processes with ASERI, in [44], 303-314.
[44]	M. Schreckenberg and S. D. Sharma, eds., "Pedestrian and Evacuation Dynamics 2001," Springer, 2002.
[45]	W. Schroeder, K. Martin and B. Lorense, "Visualization Toolkit: An Object-Oriented Approach to 3D Graphics," 4^th edition, Kitware Inc., 2006.
[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], 145.
[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), 395-406. doi: 10.1287/trsc.1090.0263.
[48]	A. Seyfried, A. Portz and A. Schadschneider, Phase coexistence in congested states of pedestrian dynamics cellular automata, in "Cellular Automata" (eds. S. Bandini, S. Manzoni, H. Umeo and G. Vizzari), LNCS 6350, Springer, 2010, 496-505, arXiv:1006.3546.
[49]	A. Seyfried and A. Schadschneider, Validation of cellular automata models of pedestrian dynamics using controlled large-scale experiments, Cybernetics and Systems, 40 (2009), 367.
[50]	A. Seyfried, B. Steffen and T. Lippert, Basics of modelling the pedestrian flow, Physica A, 368 (2006), 232-238. doi: 10.1016/j.physa.2005.11.052.
[51]	W. Shao and D. Terzopoulos, Autonomous pedestrians, in "Eurographics/ACM SIGGRAPH Symposium on Computer Animation" (eds. K. Anjyo and P. Faloutsos), 2005.
[52]	B. Steffen and A. Seyfried, Modelling of pedestrian movement around $90^\circ$ and $180^\circ$ bends, in "First International Conference on Soft Computing Technology in Civil" (eds. B. H. V. Topping and Y. Tsompanakis), Structural and environmental engineering, Civil-Comp Press, Stirlingshire, UK, 2009.
[53]	B. Steffen and A. Seyfried, The repulsive force in continous space models of pedestrian movement, arXiv:0803.1319v1, 2008.
[54]	A. Steiner, M. Philipp and A. Schmid, "Parameter Estimation for a Pedestrian Simulation Model," Swiss Transport Research Conference, 2007.
[55]	P. A. Thompson and E. W. Marchant, A computer model for the evacuation of large building populations, Fire Safety Journal, 24 (1995), 131-148. doi: 10.1016/0379-7112(95)00019-P.
[56]	P. A. Thompson and E. W. Marchant, Testing and application of the computer model 'SIMULEX', Fire Safety Journal, 24 (1995), 149-166. doi: 10.1016/0379-7112(95)00020-T.
[57]	H. Timmermans, ed., "Pedestrian Behavior," Emerald, 2009.
[58]	N. Waldau, P. Gattermann, H. Knoflacher and M. Schreckenberg, eds., "Pedestrian and Evacuation Dynamics 2005," Springer, 2007.
[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), 036110. doi: 10.1103/PhysRevE.80.036110.
[60]	W. J. Yu, L. Y. Chen, R. Dong and S. Q. Dai, Centrifugal force model for pedestrian dynamics, Phys. Rev. E, 72 (2005), 026112. doi: 10.1103/PhysRevE.72.026112.
[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), 2717-2726. doi: 10.1016/j.physa.2009.03.017.
[62]	X. Zheng and P. Palffy-Muhoray, Distance of closest approach of two arbitrary hard ellipses in two dimensions, Phys. Rev. E, 75 (2007), 061709. doi: 10.1103/PhysRevE.75.061709.

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), 842-855. doi: 10.1016/j.trc.2010.01.005.
[2]	J. V. Berg, M. Lin and D. Manocha, Reciprocal velocity obstacles for real-time multi-agent navigation, in "Robotics and Automation," IEEE International Conference on Robotics and Automation Pasadena, CA, USA, 2008.
[3]	P. Bourke, Minimum distance between a point and a line, accessed 24, December 2010. Available from: http://paulbourke.net/geometry/pointline/.
[4]	M. Chraibi and A. Seyfried, Pedestrian dynamics with event-driven simulation, in [27], 713-718.
[5]	M. Chraibi, A. Seyfried and A. Schadschneider, Generalized centrifugal force model for pedestrian dynamics, Phys. Rev. E, 82 (2010), 046111. doi: 10.1103/PhysRevE.82.046111.
[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, IEEE Computer Society, Los Alamitos, CA, USA, (2009), 583-586.
[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.
[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), 459-465. doi: 10.1016/j.firesaf.2007.12.005.
[9]	E. R. Galea, ed., "Pedestrian and Evacuation Dynamics 2003," CMS Press, London, 2003.
[10]	C. Gloor, L. Mauron and K. A. Nagel, "Pedestrian Simulation for Hiking in the Alps," Proceedings of Swiss Transport Research Conference (STRC), Monte Verita, 2003.
[11]	, GNU General public license, http://www.gnu.org/licenses/gpl.html.
[12]	D. Helbing, Collective phenomena and states in traffic and self-driven many-particle systems, Computational Materials Science, 30 (2004), 180-187. doi: 10.1016/j.commatsci.2004.01.026.
[13]	D. Helbing and P. Molnár, Social force model for pedestrian dynamics, Phys. Rev. E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282.
[14]	D. Helbing, I. J. Farkas, P. Molnár and T. Viscek, Simulation of pedestrian crowds in normal and evacuation situations, in [44], 21-58.
[15]	M. Höcker, V. Berkhahn, A. Kneidl, A. Borrmann and W. Klein, "Graph-based Approaches for Simulating Pedestrian Dynamics in Building Models," 8^th European Conference on Product & Process Modelling (ECPPM), University College Cork, Cork, Ireland, 2010.
[16]	S. Holl and A. Seyfried, Hermes - an evacuation assistant for mass events, inSiDe, 7 (2009), 60-61.
[17]	S. P. Hoogendoorn, "Walking Behavior in Bottlenecks and its Implications for Capacity," TRB 2004 Annual Meeting, 2004.
[18]	S. P. Hoogendoorn and W. Daamen, A novel calibration approach of microscopic pedestrian models, in "Pedestrian Behavior" (ed. H. Timmermans), Emerald, 2009, p. 195.
[19]	S. P. Hoogendoorn and W. Daamen, Pedestrian behavior at bottlenecks, Transportation Science, 39 (2005), 147-159. doi: 10.1287/trsc.1040.0102.
[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, 2003.
[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], 89-100.
[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], 253.
[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), 271-288. doi: 10.1142/S0219525907001355.
[24]	A. Kirchner, "Modellierung und Statistische Physik biologischer und sozialer Systeme," Ph.D thesis, Universität zu Köln, Germany, 2003.
[25]	A. Kirchner and A. Schadschneider, Simulation of evacuation processes using a bionics-inspired cellular automaton model for pedestrian dynamics, Physica A, 312 (2002), 260-276. doi: 10.1016/S0378-4371(02)00857-9.
[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), 056122. doi: 10.1103/PhysRevE.67.056122.
[27]	W. Klingsch, C. Rogsch, A. Schadschneider and M. Schreckenberg, eds., "Pedestrian and Evacuation Dynamics 2008," Springer-Verlag, Berlin, Heidelberg, 2010.
[28]	T. Kretz, A. Grünebohm and M. Schreckenberg, Experimental study of pedestrian flow through a bottleneck, J. Stat. Mech., 10 (2006), P10014.
[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 "International Symposium of Transport Simulation" (ed. M Sarvi), Monash University, Melbourne, Australia, 2008.
[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), 339-352. doi: 10.1177/0037549705052772.
[31]	K. Lewin, "Field Theory in Social Science," Greenwood Press Publishers, 1951.
[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, arXiv:0911.4350.
[33]	R. Löhner, On the modelling of pedestrian motion, Applied Mathematical Modelling, 34 (2010), 366-382. doi: 10.1016/j.apm.2009.04.017.
[34]	K. Müller, "Zur Gestaltung und Bemessung von Fluchtwegen für die Evakuierung von Personen aus Bauwerken auf der Grundlage von Modellversuchen," Ph.D thesis, Magdeburg, Germany, 1981.
[35]	D. R. Parisi and C. O. Dorso, Morphological and dynamical aspects of the room evacuation process, Physica A, 385 (2007), 343-355. doi: 10.1016/j.physa.2007.06.033.
[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), 3600-3608. doi: 10.1016/j.physa.2009.05.027.
[37]	J. Ondřej, J. Pettré, A. Olivier and S. Donikian, "A Synthetic-Vision-Based Steering Approach for Crowd Simulation," SIGGRAPH '10: ACM SIGGRAPH, 2010.
[38]	A. Schadschneider, I'm a football fan ... get me out of here, Physics World, 21, July 2010.
[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, 3142.
[40]	A. Schadschneider, H. Klüpfel, T. Kretz, C. Rogsch and A. Seyfried, Fundamentals of pedestrian and evacuation dynamics, in "Multi-Agent Systems for Traffic and Transportation Engineering" (eds. Ana Bazzan and Franziska Klügl), IGI Global, Hershey, Pennsylvania, USA, 2009, chapter 6, 124-154.
[41]	A. Schadschneider and A. Seyfried, Empirical results for pedestrian dynamics and their implications for modelling, Networks and Heterogeneous Media, 3 (2011), 545-560.
[42]	A. Schadschneider and A. Seyfried, "Modeling Pedestrian Dynamics - From Experiment to Theory and Back," Traffic and Granular Flow '09, 2009.
[43]	V. Schneider and R. Könnecke., Simulating evacuation processes with ASERI, in [44], 303-314.
[44]	M. Schreckenberg and S. D. Sharma, eds., "Pedestrian and Evacuation Dynamics 2001," Springer, 2002.
[45]	W. Schroeder, K. Martin and B. Lorense, "Visualization Toolkit: An Object-Oriented Approach to 3D Graphics," 4^th edition, Kitware Inc., 2006.
[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], 145.
[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), 395-406. doi: 10.1287/trsc.1090.0263.
[48]	A. Seyfried, A. Portz and A. Schadschneider, Phase coexistence in congested states of pedestrian dynamics cellular automata, in "Cellular Automata" (eds. S. Bandini, S. Manzoni, H. Umeo and G. Vizzari), LNCS 6350, Springer, 2010, 496-505, arXiv:1006.3546.
[49]	A. Seyfried and A. Schadschneider, Validation of cellular automata models of pedestrian dynamics using controlled large-scale experiments, Cybernetics and Systems, 40 (2009), 367.
[50]	A. Seyfried, B. Steffen and T. Lippert, Basics of modelling the pedestrian flow, Physica A, 368 (2006), 232-238. doi: 10.1016/j.physa.2005.11.052.
[51]	W. Shao and D. Terzopoulos, Autonomous pedestrians, in "Eurographics/ACM SIGGRAPH Symposium on Computer Animation" (eds. K. Anjyo and P. Faloutsos), 2005.
[52]	B. Steffen and A. Seyfried, Modelling of pedestrian movement around $90^\circ$ and $180^\circ$ bends, in "First International Conference on Soft Computing Technology in Civil" (eds. B. H. V. Topping and Y. Tsompanakis), Structural and environmental engineering, Civil-Comp Press, Stirlingshire, UK, 2009.
[53]	B. Steffen and A. Seyfried, The repulsive force in continous space models of pedestrian movement, arXiv:0803.1319v1, 2008.
[54]	A. Steiner, M. Philipp and A. Schmid, "Parameter Estimation for a Pedestrian Simulation Model," Swiss Transport Research Conference, 2007.
[55]	P. A. Thompson and E. W. Marchant, A computer model for the evacuation of large building populations, Fire Safety Journal, 24 (1995), 131-148. doi: 10.1016/0379-7112(95)00019-P.
[56]	P. A. Thompson and E. W. Marchant, Testing and application of the computer model 'SIMULEX', Fire Safety Journal, 24 (1995), 149-166. doi: 10.1016/0379-7112(95)00020-T.
[57]	H. Timmermans, ed., "Pedestrian Behavior," Emerald, 2009.
[58]	N. Waldau, P. Gattermann, H. Knoflacher and M. Schreckenberg, eds., "Pedestrian and Evacuation Dynamics 2005," Springer, 2007.
[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), 036110. doi: 10.1103/PhysRevE.80.036110.
[60]	W. J. Yu, L. Y. Chen, R. Dong and S. Q. Dai, Centrifugal force model for pedestrian dynamics, Phys. Rev. E, 72 (2005), 026112. doi: 10.1103/PhysRevE.72.026112.
[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), 2717-2726. doi: 10.1016/j.physa.2009.03.017.
[62]	X. Zheng and P. Palffy-Muhoray, Distance of closest approach of two arbitrary hard ellipses in two dimensions, Phys. Rev. E, 75 (2007), 061709. doi: 10.1103/PhysRevE.75.061709.

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