Time-delayed follow-the-leader model for pedestrians walking in line

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September 2015, 10(3): 579-608. doi: 10.3934/nhm.2015.10.579

Time-delayed follow-the-leader model for pedestrians walking in line

Jérôme Fehrenbach ^1, , Jacek Narski ^1, , Jiale Hua ^2, , Samuel Lemercier ^3, , Asja Jelić ^4, , Cécile Appert-Rolland ^5, , Stéphane Donikian ^6, , Julien Pettré ^3, and Pierre Degond ^7,

1.	Université de Toulouse; UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse; F-31062 Toulouse, France, France
2.	Donghua University, No. 2999 North Renmin Road, Songjiang, Shanghai 201620, China
3.	INRIA Rennes - Bretagne Atlantique, Campus de Beaulieu, 35042 Rennes, France
4.	Istituto Sistemi Complessi, Consiglio Nazionale delle Ricerche, UOS Sapienza, 00185 Rome, Italy
5.	Laboratoire de Physique Théorique, Université Paris Sud, btiment 210, 91405 Orsay cedex
6.	Golaem S.A.S., Bâtiment Germanium, 80 avenue des Buttes de Coësmes, 35 700 Rennes, France
7.	Imperial College London, South Kensington Campus, London SW7 2AZ

Received October 2014 Revised January 2015 Published July 2015

Abstract
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We use the results of a pedestrian tracking experiment to identify a follow-the-leader model for pedestrians walking-in-line. We demonstrate the existence of a time-delay between a subject's response and the predecessor's corresponding behavior. This time-delay induces an instability which can be damped out by a suitable relaxation. By comparisons with the experimental data, we show that the model reproduces well the emergence of large-scale structures such as congestions waves. The resulting model can be used either for modeling pedestrian queuing behavior or can be incorporated into bi-dimensional models of pedestrian traffic.

Keywords: following behavior, jam., individual tracking, Motion capture, relaxation, individual-based model.

Mathematics Subject Classification: Primary: 90B06, 90B20; Secondary: 91C99, 65L9.

Citation: Jérôme Fehrenbach, Jacek Narski, Jiale Hua, Samuel Lemercier, Asja Jelić, Cécile Appert-Rolland, Stéphane Donikian, Julien Pettré, Pierre Degond. Time-delayed follow-the-leader model for pedestrians walking in line. Networks and Heterogeneous Media, 2015, 10 (3) : 579-608. doi: 10.3934/nhm.2015.10.579

References:

[1]	C. Appert-Rolland, P. Degond and S. Motsch, Two-way multi-lane traffic model for pedestrians in corridors, Netw. Heter. Media., 6 (2011), 351-381. doi: 10.3934/nhm.2011.6.351.
[2]	A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278. doi: 10.1137/S0036139900380955.
[3]	R. Bellman and K. Cooke, Differential-Difference Equations, Academic Press, New-York, 1963.
[4]	N. Bellomo and C. Dogbé, On the modelling crowd dynamics from scaling to hyperbolic macroscopic models, Math. Models Methods Appl. Sci., 18 (2008), 1317-1345. doi: 10.1142/S0218202508003054.
[5]	N. Bellomo and C. Dogbé, On the modeling of traffic and crowds: A survey of models, speculations and perspectives, SIAM Review, 53 (2011), 409-463. doi: 10.1137/090746677.
[6]	S. Berres, R. Ruiz-Baier, H. Schwandt and E. M. Tory, An adaptive finite-volume method for a model of two-phase pedestrian flow, Netw. Heter. Media., 6 (2011), 401-423. doi: 10.3934/nhm.2011.6.401.
[7]	C. Burstedde, K. Klauck , A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A, 295 (2001), 507-525. doi: 10.1016/S0378-4371(01)00141-8.
[8]	R. E. Chandler, R. Herman and E. W. Montroll, Traffic dynamics: Studies in car following, Operations Res., 6 (1958), 165-184. doi: 10.1287/opre.6.2.165.
[9]	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.
[10]	R. M. Colombo and M. D. Rosini, Pedestrian flows and nonclassical shocks, Math. Methods Appl. Sci., 28 (2005), 1553-1567. doi: 10.1002/mma.624.
[11]	V. Coscia and C. Canavesio, First-order macroscopic modelling of human crowd dynamics, Math. Models Methods Appl. Sci., 18 (2008), 1217-1247. doi: 10.1142/S0218202508003017.
[12]	D. C. Gazis, R. Herman and R. Rothery, Nonlinear follow-the-leader models of traffic flow, Operations Res., 9 (1961), 545-567. doi: 10.1287/opre.9.4.545.
[13]	S. J. Guy, J. Chhugani, C. Kim, N. Satish, M. C. Lin, D. Manocha and P. Dubey, Clearpath: Highly parallel collision avoidance for multi-agent simulation, in ACM SIGGRAPH/Eurographics Symposium on Computer Animation, 2009, 177-187. doi: 10.1145/1599470.1599494.
[14]	P. Degond, C. Appert-Rolland, M. Moussaid, J. Pettre and G. Theraulaz, A hierarchy of heuristic-based models of crowd dynamics, J. Stat. Phys., 152 (2013), 1033-1068. doi: 10.1007/s10955-013-0805-x.
[15]	P. Degond, C. Appert-Rolland, J. Pettre and G. Theraulaz, Vision-based macroscopic pedestrian models, Kinet. Relat. Models, 6 (2013), 809-839. doi: 10.3934/krm.2013.6.809.
[16]	P. Degond and J. Hua, Self-Organized Hydrodynamics with congestion and path formation in crowds, J. Comput. Phys., 237 (2013), 299-319. doi: 10.1016/j.jcp.2012.11.033.
[17]	M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram, On the Hughes' model for pedestrian flow: The one-dimensional case, J. Diff. Eq., 250 (2011), 1334-1362. doi: 10.1016/j.jde.2010.10.015.
[18]	D. Helbing, A mathematical model for the behavior of pedestrians, Behavioral Science, 36 (1991), 298-310. doi: 10.1002/bs.3830360405.
[19]	D. Helbing, A fluid dynamic model for the movement of pedestrians, Complex Systems, 6 (1992), 391-415.
[20]	D. Helbing and P. Molnàr, Social force model for pedestrian dynamics, Phys. Rev. E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282.
[21]	D. Helbing and P. Molnàr, Self-organization phenomena in pedestrian crowds, in Self-Organization of Complex Structures: From Individual to Collective Dynamics (ed. F. Schweitzer), Gordon and Breach, London, 1997, 569-577.
[22]	L. F. Henderson, On the fluid mechanics of human crowd motion, Transp. Res., 8 (1974), 509-515. doi: 10.1016/0041-1647(74)90027-6.
[23]	S. Hoogendoorn and P. H. L. Bovy, Simulation of pedestrian flows by optimal control and differential games, Optimal Control Appl. Methods, 24 (2003), 153-172. doi: 10.1002/oca.727.
[24]	W. H. Huang, B. R. Fajen, J. R. Fink and W. H. Warren, Visual navigation and obstacle avoidance using a steering potential function, Robotic and Autonomous Systems, 54 (2006), 288-299. doi: 10.1016/j.robot.2005.11.004.
[25]	L. Huang, S. C. Wong, M. Zhang, C.-W. Shu and W. H. K. Lam, Revisiting Hughes' dynamic continuum model for pedestrian flow and the development of an efficient solution algorithm, Transp. Res. B, 43 (2009), 127-141. doi: 10.1016/j.trb.2008.06.003.
[26]	R. L. Hughes, A continuum theory for the flow of pedestrians, Transp. Res. B, 36 (2002), 507-535. doi: 10.1016/S0191-2615(01)00015-7.
[27]	R. L. Hughes, The flow of human crowds, Ann. Rev. Fluid Mech., 35 (2003), 169-182. doi: 10.1146/annurev.fluid.35.101101.161136.
[28]	A. Jelić, C. Appert-Rolland, S. Lemercier and J. Pettré, Properties of pedestrians walking in line - Fundamental diagrams, Phys. Rev. E, 85 (2012), 036111.
[29]	A. Jelić, C. Appert-Rolland, S. Lemercier and J. Pettré, Properties of pedestrians walking in line. II. stepping behavior, Phys. Rev. E, 86 (2012), 046111.
[30]	D. Jezbera, D. Kordek, J. Kříž, Petr Šeba and P. Šroll, Walkers on the circle, J. Stat. Mech. Theory Exp., 2010 (2010), L01001. doi: 10.1088/1742-5468/2010/01/L01001.
[31]	Y.-q. Jiang, P. Zhang, S. C. Wong and R.-x. Liu, A higher-order macroscopic model for pedestrian flows, Physica A, 389 (2010), 4623-4635. doi: 10.1016/j.physa.2010.05.003.
[32]	A. Johansson, Constant-net-time headway as a key mechanism behind pedestrian flow dynamics, Phys. Rev. E, 80 (2009), 026120. doi: 10.1103/PhysRevE.80.026120.
[33]	S. Lemercier, A. Jelić, R. Kulpa, J. Hua, J. Fehrenbach, P. Degond, C. Appert-Rolland, S. Donikian and J. Pettré, Realistic following behaviors for crowd simulation, Computer Graphics Forum, 31 (2012), 489-498. doi: 10.1111/j.1467-8659.2012.03028.x.
[34]	S. Lemercier, M. Moreau, M. Moussaïd, G. Theraulaz, S. Donikian and J. Pettré, Reconstructing motion capture data for human crowd study, in Motion in Games, Lecture Notes in Computer Science, 7060, Springer, Berlin-Heidelberg, 2011, 365-376. doi: 10.1007/978-3-642-25090-3_31.
[35]	B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion models, Netw. Heterog. Media, 6 (2011), 485-519. doi: 10.3934/nhm.2011.6.485.
[36]	M. Moussaïd, E. G. Guillot, M. Moreau, J. Fehrenbach, O. Chabiron, S. Lemercier, J. Pettré, C. Appert-Rolland, P. Degond and G. Theraulaz, Traffic Instabilities in Self-organized Pedestrian Crowds, PLoS Comput. Biol., 8 (2012), e1002442.
[37]	M. Moussaïd, D. Helbing and G. Theraulaz, How simple rules determine pedestrian behavior and crowd disasters, Proc. Nat. Acad. Sci., 108 (2011), 6884-6888.
[38]	K. Nishinari, A. Kirchner, A. Namazi and A. Schadschneider, Extended floor field CA model for evacuation dynamics, IEICE Transp. Inf. & Syst., E87-D (2004), 726-732.
[39]	J. Ondrej, J. Pettré, A. H. Olivier and S. Donikian, A Synthetic-vision based steering approach for crowd simulation, in SIGGRAPH'10, 29 (2010), p123. doi: 10.1145/1833349.1778860.
[40]	S. Paris, J. Pettré and S. Donikian, Pedestrian reactive navigation for crowd simulation: A predictive approach, Eurographics, 26 (2007), 665-674. doi: 10.1111/j.1467-8659.2007.01090.x.
[41]	J. Pettré, J. Ondřej, A.-H. Olivier, A. Cretual and S. Donikian, Experiment-based modeling, simulation and validation of interactions between virtual walkers, in SCA '09: Proceedings of the 2009 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, 2009, 189-198.
[42]	B. Piccoli and A. Tosin, Pedestrian flows in bounded domains with obstacles, Contin. Mech. Thermodyn., 21 (2009), 85-107. doi: 10.1007/s00161-009-0100-x.
[43]	L. Pontrjagin, On the zeros of some elementary transcendental functions, Amer. Math. Soc. Transl. Ser. 2, 1 (1955), 95-110.
[44]	C. W. Reynolds, Steering behaviors for autonomous characters, in Proceedings of Game Developers Conference, San Jose, California, 1999, 763-782.
[45]	A. Seyfried, B. Steffen, W. Klingsch and M. Boltes, The fundamental diagram of pedestrian movement revisited, J. Stat. Mech. Theory Exp., 2005 (2005), P10002. doi: 10.1088/1742-5468/2005/10/P10002.
[46]	A. Seyfried, B. Steffen and T. Lippert, Basics of modelling the pedestrian flow, Phys. A, 368 (2006), 232-238. doi: 10.1016/j.physa.2005.11.052.
[47]	J. van den Berg and H. Overmars, Planning time-minimal safe paths amidst unpredictably moving obstacles, Int. Journal on Robotics Research, 27 (2008), 1274-1294.
[48]	J. Zhang, W. Klingsch, A. Schadschneider and A. Seyfried, Ordering in bidirectional pedestrian flows and its influence on the fundamental diagram, J. Stat. Mech. Theory Exp., 2012 (2012), P02002. doi: 10.1088/1742-5468/2012/02/P02002.

show all references

References:

[1]	C. Appert-Rolland, P. Degond and S. Motsch, Two-way multi-lane traffic model for pedestrians in corridors, Netw. Heter. Media., 6 (2011), 351-381. doi: 10.3934/nhm.2011.6.351.
[2]	A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278. doi: 10.1137/S0036139900380955.
[3]	R. Bellman and K. Cooke, Differential-Difference Equations, Academic Press, New-York, 1963.
[4]	N. Bellomo and C. Dogbé, On the modelling crowd dynamics from scaling to hyperbolic macroscopic models, Math. Models Methods Appl. Sci., 18 (2008), 1317-1345. doi: 10.1142/S0218202508003054.
[5]	N. Bellomo and C. Dogbé, On the modeling of traffic and crowds: A survey of models, speculations and perspectives, SIAM Review, 53 (2011), 409-463. doi: 10.1137/090746677.
[6]	S. Berres, R. Ruiz-Baier, H. Schwandt and E. M. Tory, An adaptive finite-volume method for a model of two-phase pedestrian flow, Netw. Heter. Media., 6 (2011), 401-423. doi: 10.3934/nhm.2011.6.401.
[7]	C. Burstedde, K. Klauck , A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A, 295 (2001), 507-525. doi: 10.1016/S0378-4371(01)00141-8.
[8]	R. E. Chandler, R. Herman and E. W. Montroll, Traffic dynamics: Studies in car following, Operations Res., 6 (1958), 165-184. doi: 10.1287/opre.6.2.165.
[9]	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.
[10]	R. M. Colombo and M. D. Rosini, Pedestrian flows and nonclassical shocks, Math. Methods Appl. Sci., 28 (2005), 1553-1567. doi: 10.1002/mma.624.
[11]	V. Coscia and C. Canavesio, First-order macroscopic modelling of human crowd dynamics, Math. Models Methods Appl. Sci., 18 (2008), 1217-1247. doi: 10.1142/S0218202508003017.
[12]	D. C. Gazis, R. Herman and R. Rothery, Nonlinear follow-the-leader models of traffic flow, Operations Res., 9 (1961), 545-567. doi: 10.1287/opre.9.4.545.
[13]	S. J. Guy, J. Chhugani, C. Kim, N. Satish, M. C. Lin, D. Manocha and P. Dubey, Clearpath: Highly parallel collision avoidance for multi-agent simulation, in ACM SIGGRAPH/Eurographics Symposium on Computer Animation, 2009, 177-187. doi: 10.1145/1599470.1599494.
[14]	P. Degond, C. Appert-Rolland, M. Moussaid, J. Pettre and G. Theraulaz, A hierarchy of heuristic-based models of crowd dynamics, J. Stat. Phys., 152 (2013), 1033-1068. doi: 10.1007/s10955-013-0805-x.
[15]	P. Degond, C. Appert-Rolland, J. Pettre and G. Theraulaz, Vision-based macroscopic pedestrian models, Kinet. Relat. Models, 6 (2013), 809-839. doi: 10.3934/krm.2013.6.809.
[16]	P. Degond and J. Hua, Self-Organized Hydrodynamics with congestion and path formation in crowds, J. Comput. Phys., 237 (2013), 299-319. doi: 10.1016/j.jcp.2012.11.033.
[17]	M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram, On the Hughes' model for pedestrian flow: The one-dimensional case, J. Diff. Eq., 250 (2011), 1334-1362. doi: 10.1016/j.jde.2010.10.015.
[18]	D. Helbing, A mathematical model for the behavior of pedestrians, Behavioral Science, 36 (1991), 298-310. doi: 10.1002/bs.3830360405.
[19]	D. Helbing, A fluid dynamic model for the movement of pedestrians, Complex Systems, 6 (1992), 391-415.
[20]	D. Helbing and P. Molnàr, Social force model for pedestrian dynamics, Phys. Rev. E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282.
[21]	D. Helbing and P. Molnàr, Self-organization phenomena in pedestrian crowds, in Self-Organization of Complex Structures: From Individual to Collective Dynamics (ed. F. Schweitzer), Gordon and Breach, London, 1997, 569-577.
[22]	L. F. Henderson, On the fluid mechanics of human crowd motion, Transp. Res., 8 (1974), 509-515. doi: 10.1016/0041-1647(74)90027-6.
[23]	S. Hoogendoorn and P. H. L. Bovy, Simulation of pedestrian flows by optimal control and differential games, Optimal Control Appl. Methods, 24 (2003), 153-172. doi: 10.1002/oca.727.
[24]	W. H. Huang, B. R. Fajen, J. R. Fink and W. H. Warren, Visual navigation and obstacle avoidance using a steering potential function, Robotic and Autonomous Systems, 54 (2006), 288-299. doi: 10.1016/j.robot.2005.11.004.
[25]	L. Huang, S. C. Wong, M. Zhang, C.-W. Shu and W. H. K. Lam, Revisiting Hughes' dynamic continuum model for pedestrian flow and the development of an efficient solution algorithm, Transp. Res. B, 43 (2009), 127-141. doi: 10.1016/j.trb.2008.06.003.
[26]	R. L. Hughes, A continuum theory for the flow of pedestrians, Transp. Res. B, 36 (2002), 507-535. doi: 10.1016/S0191-2615(01)00015-7.
[27]	R. L. Hughes, The flow of human crowds, Ann. Rev. Fluid Mech., 35 (2003), 169-182. doi: 10.1146/annurev.fluid.35.101101.161136.
[28]	A. Jelić, C. Appert-Rolland, S. Lemercier and J. Pettré, Properties of pedestrians walking in line - Fundamental diagrams, Phys. Rev. E, 85 (2012), 036111.
[29]	A. Jelić, C. Appert-Rolland, S. Lemercier and J. Pettré, Properties of pedestrians walking in line. II. stepping behavior, Phys. Rev. E, 86 (2012), 046111.
[30]	D. Jezbera, D. Kordek, J. Kříž, Petr Šeba and P. Šroll, Walkers on the circle, J. Stat. Mech. Theory Exp., 2010 (2010), L01001. doi: 10.1088/1742-5468/2010/01/L01001.
[31]	Y.-q. Jiang, P. Zhang, S. C. Wong and R.-x. Liu, A higher-order macroscopic model for pedestrian flows, Physica A, 389 (2010), 4623-4635. doi: 10.1016/j.physa.2010.05.003.
[32]	A. Johansson, Constant-net-time headway as a key mechanism behind pedestrian flow dynamics, Phys. Rev. E, 80 (2009), 026120. doi: 10.1103/PhysRevE.80.026120.
[33]	S. Lemercier, A. Jelić, R. Kulpa, J. Hua, J. Fehrenbach, P. Degond, C. Appert-Rolland, S. Donikian and J. Pettré, Realistic following behaviors for crowd simulation, Computer Graphics Forum, 31 (2012), 489-498. doi: 10.1111/j.1467-8659.2012.03028.x.
[34]	S. Lemercier, M. Moreau, M. Moussaïd, G. Theraulaz, S. Donikian and J. Pettré, Reconstructing motion capture data for human crowd study, in Motion in Games, Lecture Notes in Computer Science, 7060, Springer, Berlin-Heidelberg, 2011, 365-376. doi: 10.1007/978-3-642-25090-3_31.
[35]	B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion models, Netw. Heterog. Media, 6 (2011), 485-519. doi: 10.3934/nhm.2011.6.485.
[36]	M. Moussaïd, E. G. Guillot, M. Moreau, J. Fehrenbach, O. Chabiron, S. Lemercier, J. Pettré, C. Appert-Rolland, P. Degond and G. Theraulaz, Traffic Instabilities in Self-organized Pedestrian Crowds, PLoS Comput. Biol., 8 (2012), e1002442.
[37]	M. Moussaïd, D. Helbing and G. Theraulaz, How simple rules determine pedestrian behavior and crowd disasters, Proc. Nat. Acad. Sci., 108 (2011), 6884-6888.
[38]	K. Nishinari, A. Kirchner, A. Namazi and A. Schadschneider, Extended floor field CA model for evacuation dynamics, IEICE Transp. Inf. & Syst., E87-D (2004), 726-732.
[39]	J. Ondrej, J. Pettré, A. H. Olivier and S. Donikian, A Synthetic-vision based steering approach for crowd simulation, in SIGGRAPH'10, 29 (2010), p123. doi: 10.1145/1833349.1778860.
[40]	S. Paris, J. Pettré and S. Donikian, Pedestrian reactive navigation for crowd simulation: A predictive approach, Eurographics, 26 (2007), 665-674. doi: 10.1111/j.1467-8659.2007.01090.x.
[41]	J. Pettré, J. Ondřej, A.-H. Olivier, A. Cretual and S. Donikian, Experiment-based modeling, simulation and validation of interactions between virtual walkers, in SCA '09: Proceedings of the 2009 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, 2009, 189-198.
[42]	B. Piccoli and A. Tosin, Pedestrian flows in bounded domains with obstacles, Contin. Mech. Thermodyn., 21 (2009), 85-107. doi: 10.1007/s00161-009-0100-x.
[43]	L. Pontrjagin, On the zeros of some elementary transcendental functions, Amer. Math. Soc. Transl. Ser. 2, 1 (1955), 95-110.
[44]	C. W. Reynolds, Steering behaviors for autonomous characters, in Proceedings of Game Developers Conference, San Jose, California, 1999, 763-782.
[45]	A. Seyfried, B. Steffen, W. Klingsch and M. Boltes, The fundamental diagram of pedestrian movement revisited, J. Stat. Mech. Theory Exp., 2005 (2005), P10002. doi: 10.1088/1742-5468/2005/10/P10002.
[46]	A. Seyfried, B. Steffen and T. Lippert, Basics of modelling the pedestrian flow, Phys. A, 368 (2006), 232-238. doi: 10.1016/j.physa.2005.11.052.
[47]	J. van den Berg and H. Overmars, Planning time-minimal safe paths amidst unpredictably moving obstacles, Int. Journal on Robotics Research, 27 (2008), 1274-1294.
[48]	J. Zhang, W. Klingsch, A. Schadschneider and A. Seyfried, Ordering in bidirectional pedestrian flows and its influence on the fundamental diagram, J. Stat. Mech. Theory Exp., 2012 (2012), P02002. doi: 10.1088/1742-5468/2012/02/P02002.

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