December  2013, 6(4): 809-839. doi: 10.3934/krm.2013.6.809

Vision-based macroscopic pedestrian models

1. 

Université de Toulouse; UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse, F-31062 Toulouse, France

2. 

Laboratoire de Physique Théorique, Université Paris Sud, btiment 210, 91405 Orsay cedex, France

3. 

INRIA Rennes - Bretagne Atlantique, Campus de Beaulieu, 35042 Rennes, France

4. 

Centre de Recherches sur la Cognition Animale, UMR-CNRS 5169, Université Paul Sabatier, 31062 Toulouse cedex 9, France

Received  August 2013 Revised  September 2013 Published  November 2013

We propose a hierarchy of kinetic and macroscopic models for a system consisting of a large number of interacting pedestrians. The basic interaction rules are derived from [44] where the dangerousness level of an interaction with another pedestrian is measured in terms of the derivative of the bearing angle (angle between the walking direction and the line connecting the two subjects) and of the time-to-interaction (time before reaching the closest distance between the two subjects). A mean-field kinetic model is derived. Then, three different macroscopic continuum models are proposed. The first two ones rely on two different closure assumptions of the kinetic model, respectively based on a monokinetic and a von Mises-Fisher distribution. The third one is derived through a hydrodynamic limit. In each case, we discuss the relevance of the model for practical simulations of pedestrian crowds.
Citation: Pierre Degond, Cécile Appert-Rolland, Julien Pettré, Guy Theraulaz. Vision-based macroscopic pedestrian models. Kinetic and Related Models, 2013, 6 (4) : 809-839. doi: 10.3934/krm.2013.6.809
References:
[1]

S. Al-nasur and P. Kashroo, A microscopic-to-macroscopic crowd dynamic model, in Intelligent Transportation Systems Conference, (2006), ITSC '06. IEEE", 606-611. doi: 10.1109/ITSC.2006.1706808.

[2]

C. Appert-Rolland, P. Degond and S. Motsch, Two-way multi-lane traffic model for pedestrians in corridors, Netw. Heterog. Media, 6 (2011), 351-381. doi: 10.3934/nhm.2011.6.351.

[3]

N. Bellomo and A. Bellouquid, On the modelling of vehicular traffic and crowds by kinetic theory of active particles, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, (eds. G. Naldi et al), Springer, (2010), 273-296. doi: 10.1007/978-0-8176-4946-3_11.

[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. Heterog. Media, 6 (2011), 401-423. doi: 10.3934/nhm.2011.6.401.

[7]

F. Bolley, J. A. Cañizo and J. A. Carrillo, Mean-field limit for the stochastic Vicsek model, Appl. Math. Lett., 25 (2012), 339-343. doi: 10.1016/j.aml.2011.09.011.

[8]

F. Bouchut, On zero pressure gas dynamics, in Advances in Kinetic Theory and Computing, (ed. B. Perthame), World Scientific, (1994), 171-190.

[9]

M. Burger, P. Markowich and J.-F. Pietschmann, Continuous limit of a crowd motion and herding model: analysis and numerical simulations, Kinet. Relat. Models, 4 (2011), 1025-1047. doi: 10.3934/krm.2011.4.1025.

[10]

A. Chertock, A. Kurganov, A. Polizzi and I. Timofeyev, Pedestrian Flow Models with Slowdown Interactions, Math. Models Methods Appl. Sci., to appear.

[11]

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.

[12]

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.

[13]

E. Cristiani, B. Piccoli and A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics, Multiscale Model. Simul., 9 (2011), 155-182. doi: 10.1137/100797515.

[14]

J. E. Cutting, P. M. Vishton and P. A. Braren, How we avoid collisions with stationary and moving objects, Psychological Review, 102 (1995), 627-651.

[15]

P. Degond, Macroscopic limits of the Boltzmann equation: A review, in Modeling and Model. Simul. Sci. Eng. Technol.,ds for Kinetic Equations, Model. Simul. Sci. Eng. Technol., Birkhaüser Boston, Boston, MA, (2004), 3-57.

[16]

P. Degond, C. Appert-Rolland, J. Pettre and G. Theraulaz, A macroscopic crowd model based on behavioral heuristicsJ. Stat. Phys, to appear. arXiv:1304.1927.

[17]

P. Degond, A. Frouvelle and J-G. Liu, Macroscopic limits and phase transition in a system of self-propelled particles, J. Nonlinear Sci., 23 (2013), 427-456. doi: 10.1007/s00332-012-9157-y.

[18]

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.

[19]

P. Degond, J. Hua and L. Navoret, Numerical simulations of the Euler system with congestion constraint, J. Comput. Phys., 230 (2011), 8057-8088. doi: 10.1016/j.jcp.2011.07.010.

[20]

P. Degond, J.-G. Liu and C. Ringhofer, A Nash equilibrium macroscopic closure for kinetic models coupled with Mean-Field Games, submitted. arXiv:1212.6130.

[21]

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.

[22]

G. Grégoire and H. Chaté, Onset of collective and cohesive motion, Phys. Rev. Lett., 92 (2004), 025702.

[23]

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), 77-187. doi: 10.1145/1599470.1599494.

[24]

S. J. Guy, S. Curtis, M. C. Lin and D. Manocha, Least-effort trajectories lead to emergent crowd behaviors, Phys. Rev. E, 85 (2012), 016110. doi: 10.1103/PhysRevE.85.016110.

[25]

D. Helbing, A mathematical model for the behavior of pedestrians, Behavioral Science, 36 (1991), 298-310. doi: 10.1002/bs.3830360405.

[26]

D. Helbing, A fluid dynamic model for the movement of pedestrians, Complex Systems, 6 (1992), 391-415.

[27]

D. Helbing and P. Molnàr, Social force model for pedestrian dynamics, Phys. Rev. E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282.

[28]

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, (1997), 569-577.

[29]

L. F. Henderson, On the fluid mechanics of human crowd motion, Transportation Research, 8 (1974), 509-515. doi: 10.1016/0041-1647(74)90027-6.

[30]

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.

[31]

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.

[32]

R. L. Hughes, A continuum theory for the flow of pedestrians, Transp. Res. B, 36 (2002), 507-535.

[33]

R. L. Hughes, The flow of human crowds, Annual review of fluid mechanics, Ann. Rev. Fluid Mech., Annual Reviews, Palo Alto, CA, 35 (2003), 169-182. doi: 10.1146/annurev.fluid.35.101101.161136.

[34]

E. P. Hsu, Stochastic Analysis on Manifolds, Graduate Series in Mathematics, Vol. 38, American Mathematical Society, Providence, Rhode Island, 2002.

[35]

Y.-q. Jiang, P. Zhang, S. C. Wong and R.-x. Liu, A higher-order macroscopic model for pedestrian flows, Phys. A, 389 (2010), 4623-4635. doi: 10.1016/j.physa.2010.05.003.

[36]

J.-M. Lasry and P.-L. Lions, Mean field games, Japan J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.

[37]

S. Lemercier, A. Jelic, 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.

[38]

M. J. Lighthill and J. B. Whitham, On kinematic waves. I: flow movement in long rivers. II: A theory of traffic flow on long crowded roads, Proc. Roy. Soc. A, 229 (1955), 1749-1766.

[39]

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.

[40]

M. Moussaid, D. Helbing and G. Theraulaz, How simple rules determine pedestrian behavior and crowd disasters, Proc. Nat. Acad. Sci., 108 (2011), 6884-6888. doi: 10.1073/pnas.1016507108.

[41]

S. Motsch, M. Moussaid, E. G. Guillot, S. Lemercier, J. Pettré, G. Theraulaz, C. Appert-Rolland and P. Degond, Dynamics of cluster formation and traffic efficiency in pedestrian crowds, submitted.

[42]

R. Narain, A. Golas, S. Curtis and M. Lin, Aggregate dynamics for dense crowd simulation, ACM Transactions on Graphics (TOG), 28 (2009), 122.

[43]

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.

[44]

J Ondrej, J. Pettré, A. H. Olivier and S. Donikian, A Synthetic-vision based steering approach for crowd simulation, ACM Transactions on Graphics (TOG), 29 (2010), 123.

[45]

S. Paris, J. Pettré and S. Donikian, Pedestrian reactive navigation for crowd simulation: A predictive approach, Computer Graphics Forum, 26 (2007), 665-674. doi: 10.1111/j.1467-8659.2007.01090.x.

[46]

H. J. Payne, Models of Freeway Traffic and Control, Simulation Councils Inc., La Jolla, California, 1971.

[47]

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. doi: 10.1145/1599470.1599495.

[48]

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.

[49]

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, ACM SIGGRAPH Computer Graphics, 21 (1987), 25-34.

[50]

C. W. Reynolds, Steering behaviors for autonomous characters, in Proceedings of Game Developers Conference 1999, San Jose, California, (1999), 763-782.

[51]

W. Shao and D. Terzopoulos, Autonomous pedestrians, in Proceedings of the 2005 ACM SIGGRAPH/Eurographics symposium on Computer animation, ACM Press, (2005), 19-28.

[52]

A. Treuille, S. Cooper and Z. Popovic, Continuum crowds, ACM Transactions on Graphics (TOG), 25 (2006), 1160-1168.

[53]

J. Van Den Berg, S. Guy, M. Lin and D. Manocha, Reciprocal n-body collision avoidance, in Robotics Research, Springer, 70 (2011), 3-19. doi: 10.1007/978-3-642-19457-3_1.

[54]

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.

[55]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.

[56]

G. S. Watson, Distributions on the circle and sphere, J. Appl. Probab., 19 (1982), 265-280.

show all references

References:
[1]

S. Al-nasur and P. Kashroo, A microscopic-to-macroscopic crowd dynamic model, in Intelligent Transportation Systems Conference, (2006), ITSC '06. IEEE", 606-611. doi: 10.1109/ITSC.2006.1706808.

[2]

C. Appert-Rolland, P. Degond and S. Motsch, Two-way multi-lane traffic model for pedestrians in corridors, Netw. Heterog. Media, 6 (2011), 351-381. doi: 10.3934/nhm.2011.6.351.

[3]

N. Bellomo and A. Bellouquid, On the modelling of vehicular traffic and crowds by kinetic theory of active particles, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, (eds. G. Naldi et al), Springer, (2010), 273-296. doi: 10.1007/978-0-8176-4946-3_11.

[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. Heterog. Media, 6 (2011), 401-423. doi: 10.3934/nhm.2011.6.401.

[7]

F. Bolley, J. A. Cañizo and J. A. Carrillo, Mean-field limit for the stochastic Vicsek model, Appl. Math. Lett., 25 (2012), 339-343. doi: 10.1016/j.aml.2011.09.011.

[8]

F. Bouchut, On zero pressure gas dynamics, in Advances in Kinetic Theory and Computing, (ed. B. Perthame), World Scientific, (1994), 171-190.

[9]

M. Burger, P. Markowich and J.-F. Pietschmann, Continuous limit of a crowd motion and herding model: analysis and numerical simulations, Kinet. Relat. Models, 4 (2011), 1025-1047. doi: 10.3934/krm.2011.4.1025.

[10]

A. Chertock, A. Kurganov, A. Polizzi and I. Timofeyev, Pedestrian Flow Models with Slowdown Interactions, Math. Models Methods Appl. Sci., to appear.

[11]

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.

[12]

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.

[13]

E. Cristiani, B. Piccoli and A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics, Multiscale Model. Simul., 9 (2011), 155-182. doi: 10.1137/100797515.

[14]

J. E. Cutting, P. M. Vishton and P. A. Braren, How we avoid collisions with stationary and moving objects, Psychological Review, 102 (1995), 627-651.

[15]

P. Degond, Macroscopic limits of the Boltzmann equation: A review, in Modeling and Model. Simul. Sci. Eng. Technol.,ds for Kinetic Equations, Model. Simul. Sci. Eng. Technol., Birkhaüser Boston, Boston, MA, (2004), 3-57.

[16]

P. Degond, C. Appert-Rolland, J. Pettre and G. Theraulaz, A macroscopic crowd model based on behavioral heuristicsJ. Stat. Phys, to appear. arXiv:1304.1927.

[17]

P. Degond, A. Frouvelle and J-G. Liu, Macroscopic limits and phase transition in a system of self-propelled particles, J. Nonlinear Sci., 23 (2013), 427-456. doi: 10.1007/s00332-012-9157-y.

[18]

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.

[19]

P. Degond, J. Hua and L. Navoret, Numerical simulations of the Euler system with congestion constraint, J. Comput. Phys., 230 (2011), 8057-8088. doi: 10.1016/j.jcp.2011.07.010.

[20]

P. Degond, J.-G. Liu and C. Ringhofer, A Nash equilibrium macroscopic closure for kinetic models coupled with Mean-Field Games, submitted. arXiv:1212.6130.

[21]

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.

[22]

G. Grégoire and H. Chaté, Onset of collective and cohesive motion, Phys. Rev. Lett., 92 (2004), 025702.

[23]

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), 77-187. doi: 10.1145/1599470.1599494.

[24]

S. J. Guy, S. Curtis, M. C. Lin and D. Manocha, Least-effort trajectories lead to emergent crowd behaviors, Phys. Rev. E, 85 (2012), 016110. doi: 10.1103/PhysRevE.85.016110.

[25]

D. Helbing, A mathematical model for the behavior of pedestrians, Behavioral Science, 36 (1991), 298-310. doi: 10.1002/bs.3830360405.

[26]

D. Helbing, A fluid dynamic model for the movement of pedestrians, Complex Systems, 6 (1992), 391-415.

[27]

D. Helbing and P. Molnàr, Social force model for pedestrian dynamics, Phys. Rev. E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282.

[28]

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, (1997), 569-577.

[29]

L. F. Henderson, On the fluid mechanics of human crowd motion, Transportation Research, 8 (1974), 509-515. doi: 10.1016/0041-1647(74)90027-6.

[30]

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.

[31]

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.

[32]

R. L. Hughes, A continuum theory for the flow of pedestrians, Transp. Res. B, 36 (2002), 507-535.

[33]

R. L. Hughes, The flow of human crowds, Annual review of fluid mechanics, Ann. Rev. Fluid Mech., Annual Reviews, Palo Alto, CA, 35 (2003), 169-182. doi: 10.1146/annurev.fluid.35.101101.161136.

[34]

E. P. Hsu, Stochastic Analysis on Manifolds, Graduate Series in Mathematics, Vol. 38, American Mathematical Society, Providence, Rhode Island, 2002.

[35]

Y.-q. Jiang, P. Zhang, S. C. Wong and R.-x. Liu, A higher-order macroscopic model for pedestrian flows, Phys. A, 389 (2010), 4623-4635. doi: 10.1016/j.physa.2010.05.003.

[36]

J.-M. Lasry and P.-L. Lions, Mean field games, Japan J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.

[37]

S. Lemercier, A. Jelic, 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.

[38]

M. J. Lighthill and J. B. Whitham, On kinematic waves. I: flow movement in long rivers. II: A theory of traffic flow on long crowded roads, Proc. Roy. Soc. A, 229 (1955), 1749-1766.

[39]

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.

[40]

M. Moussaid, D. Helbing and G. Theraulaz, How simple rules determine pedestrian behavior and crowd disasters, Proc. Nat. Acad. Sci., 108 (2011), 6884-6888. doi: 10.1073/pnas.1016507108.

[41]

S. Motsch, M. Moussaid, E. G. Guillot, S. Lemercier, J. Pettré, G. Theraulaz, C. Appert-Rolland and P. Degond, Dynamics of cluster formation and traffic efficiency in pedestrian crowds, submitted.

[42]

R. Narain, A. Golas, S. Curtis and M. Lin, Aggregate dynamics for dense crowd simulation, ACM Transactions on Graphics (TOG), 28 (2009), 122.

[43]

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.

[44]

J Ondrej, J. Pettré, A. H. Olivier and S. Donikian, A Synthetic-vision based steering approach for crowd simulation, ACM Transactions on Graphics (TOG), 29 (2010), 123.

[45]

S. Paris, J. Pettré and S. Donikian, Pedestrian reactive navigation for crowd simulation: A predictive approach, Computer Graphics Forum, 26 (2007), 665-674. doi: 10.1111/j.1467-8659.2007.01090.x.

[46]

H. J. Payne, Models of Freeway Traffic and Control, Simulation Councils Inc., La Jolla, California, 1971.

[47]

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. doi: 10.1145/1599470.1599495.

[48]

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.

[49]

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, ACM SIGGRAPH Computer Graphics, 21 (1987), 25-34.

[50]

C. W. Reynolds, Steering behaviors for autonomous characters, in Proceedings of Game Developers Conference 1999, San Jose, California, (1999), 763-782.

[51]

W. Shao and D. Terzopoulos, Autonomous pedestrians, in Proceedings of the 2005 ACM SIGGRAPH/Eurographics symposium on Computer animation, ACM Press, (2005), 19-28.

[52]

A. Treuille, S. Cooper and Z. Popovic, Continuum crowds, ACM Transactions on Graphics (TOG), 25 (2006), 1160-1168.

[53]

J. Van Den Berg, S. Guy, M. Lin and D. Manocha, Reciprocal n-body collision avoidance, in Robotics Research, Springer, 70 (2011), 3-19. doi: 10.1007/978-3-642-19457-3_1.

[54]

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.

[55]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.

[56]

G. S. Watson, Distributions on the circle and sphere, J. Appl. Probab., 19 (1982), 265-280.

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