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 & 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), 606. doi: 10.1109/ITSC.2006.1706808. Google Scholar

[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. doi: 10.3934/nhm.2011.6.351. Google Scholar

[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, (2010), 273. doi: 10.1007/978-0-8176-4946-3_11. Google Scholar

[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. doi: 10.1142/S0218202508003054. Google Scholar

[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. doi: 10.1137/090746677. Google Scholar

[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. doi: 10.3934/nhm.2011.6.401. Google Scholar

[7]

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

[8]

F. Bouchut, On zero pressure gas dynamics,, in Advances in Kinetic Theory and Computing, (1994), 171. Google Scholar

[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. doi: 10.3934/krm.2011.4.1025. Google Scholar

[10]

A. Chertock, A. Kurganov, A. Polizzi and I. Timofeyev, Pedestrian Flow Models with Slowdown Interactions,, Math. Models Methods Appl. Sci., (). Google Scholar

[11]

R. M. Colombo and M. D. Rosini, Pedestrian flows and nonclassical shocks,, Math. Methods Appl. Sci., 28 (2005), 1553. doi: 10.1002/mma.624. Google Scholar

[12]

V. Coscia and C. Canavesio, First-order macroscopic modelling of human crowd dynamics,, Math. Models Methods Appl. Sci., 18 (2008), 1217. doi: 10.1142/S0218202508003017. Google Scholar

[13]

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

[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. Google Scholar

[15]

P. Degond, Macroscopic limits of the Boltzmann equation: A review,, in Modeling and Model. Simul. Sci. Eng. Technol., (2004), 3. Google Scholar

[16]

P. Degond, C. Appert-Rolland, J. Pettre and G. Theraulaz, A macroscopic crowd model based on behavioral heuristics,, J. Stat. Phys, (). Google Scholar

[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. doi: 10.1007/s00332-012-9157-y. Google Scholar

[18]

P. Degond and J. Hua, Self-Organized Hydrodynamics with congestion and path formation in crowds,, J. Comput. Phys., 237 (2013), 299. doi: 10.1016/j.jcp.2012.11.033. Google Scholar

[19]

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

[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., (). Google Scholar

[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. doi: 10.1016/j.jde.2010.10.015. Google Scholar

[22]

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

[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. doi: 10.1145/1599470.1599494. Google Scholar

[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). doi: 10.1103/PhysRevE.85.016110. Google Scholar

[25]

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

[26]

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

[27]

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

[28]

D. Helbing and P. Molnàr, Self-organization phenomena in pedestrian crowds,, in Self-Organization of Complex Structures: From Individual to Collective Dynamics, (1997), 569. Google Scholar

[29]

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

[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. doi: 10.1002/oca.727. Google Scholar

[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. doi: 10.1016/j.trb.2008.06.003. Google Scholar

[32]

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

[33]

R. L. Hughes, The flow of human crowds, Annual review of fluid mechanics,, Ann. Rev. Fluid Mech., 35 (2003), 169. doi: 10.1146/annurev.fluid.35.101101.161136. Google Scholar

[34]

E. P. Hsu, Stochastic Analysis on Manifolds,, Graduate Series in Mathematics, (2002). Google Scholar

[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. doi: 10.1016/j.physa.2010.05.003. Google Scholar

[36]

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

[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. Google Scholar

[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. Google Scholar

[39]

B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion models,, Netw. Heterog. Media, 6 (2011), 485. doi: 10.3934/nhm.2011.6.485. Google Scholar

[40]

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

[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., (). Google Scholar

[42]

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

[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. Google Scholar

[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). Google Scholar

[45]

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

[46]

H. J. Payne, Models of Freeway Traffic and Control,, Simulation Councils Inc., (1971). Google Scholar

[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. doi: 10.1145/1599470.1599495. Google Scholar

[48]

B. Piccoli and A. Tosin, Pedestrian flows in bounded domains with obstacles,, Contin. Mech. Thermodyn., 21 (2009), 85. doi: 10.1007/s00161-009-0100-x. Google Scholar

[49]

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

[50]

C. W. Reynolds, Steering behaviors for autonomous characters,, in Proceedings of Game Developers Conference 1999, (1999), 763. Google Scholar

[51]

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

[52]

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

[53]

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

[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. Google Scholar

[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. doi: 10.1103/PhysRevLett.75.1226. Google Scholar

[56]

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

show all references

References:
[1]

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

[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. doi: 10.3934/nhm.2011.6.351. Google Scholar

[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, (2010), 273. doi: 10.1007/978-0-8176-4946-3_11. Google Scholar

[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. doi: 10.1142/S0218202508003054. Google Scholar

[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. doi: 10.1137/090746677. Google Scholar

[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. doi: 10.3934/nhm.2011.6.401. Google Scholar

[7]

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

[8]

F. Bouchut, On zero pressure gas dynamics,, in Advances in Kinetic Theory and Computing, (1994), 171. Google Scholar

[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. doi: 10.3934/krm.2011.4.1025. Google Scholar

[10]

A. Chertock, A. Kurganov, A. Polizzi and I. Timofeyev, Pedestrian Flow Models with Slowdown Interactions,, Math. Models Methods Appl. Sci., (). Google Scholar

[11]

R. M. Colombo and M. D. Rosini, Pedestrian flows and nonclassical shocks,, Math. Methods Appl. Sci., 28 (2005), 1553. doi: 10.1002/mma.624. Google Scholar

[12]

V. Coscia and C. Canavesio, First-order macroscopic modelling of human crowd dynamics,, Math. Models Methods Appl. Sci., 18 (2008), 1217. doi: 10.1142/S0218202508003017. Google Scholar

[13]

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

[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. Google Scholar

[15]

P. Degond, Macroscopic limits of the Boltzmann equation: A review,, in Modeling and Model. Simul. Sci. Eng. Technol., (2004), 3. Google Scholar

[16]

P. Degond, C. Appert-Rolland, J. Pettre and G. Theraulaz, A macroscopic crowd model based on behavioral heuristics,, J. Stat. Phys, (). Google Scholar

[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. doi: 10.1007/s00332-012-9157-y. Google Scholar

[18]

P. Degond and J. Hua, Self-Organized Hydrodynamics with congestion and path formation in crowds,, J. Comput. Phys., 237 (2013), 299. doi: 10.1016/j.jcp.2012.11.033. Google Scholar

[19]

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

[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., (). Google Scholar

[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. doi: 10.1016/j.jde.2010.10.015. Google Scholar

[22]

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

[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. doi: 10.1145/1599470.1599494. Google Scholar

[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). doi: 10.1103/PhysRevE.85.016110. Google Scholar

[25]

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

[26]

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

[27]

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

[28]

D. Helbing and P. Molnàr, Self-organization phenomena in pedestrian crowds,, in Self-Organization of Complex Structures: From Individual to Collective Dynamics, (1997), 569. Google Scholar

[29]

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

[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. doi: 10.1002/oca.727. Google Scholar

[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. doi: 10.1016/j.trb.2008.06.003. Google Scholar

[32]

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

[33]

R. L. Hughes, The flow of human crowds, Annual review of fluid mechanics,, Ann. Rev. Fluid Mech., 35 (2003), 169. doi: 10.1146/annurev.fluid.35.101101.161136. Google Scholar

[34]

E. P. Hsu, Stochastic Analysis on Manifolds,, Graduate Series in Mathematics, (2002). Google Scholar

[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. doi: 10.1016/j.physa.2010.05.003. Google Scholar

[36]

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

[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. Google Scholar

[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. Google Scholar

[39]

B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion models,, Netw. Heterog. Media, 6 (2011), 485. doi: 10.3934/nhm.2011.6.485. Google Scholar

[40]

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

[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., (). Google Scholar

[42]

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

[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. Google Scholar

[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). Google Scholar

[45]

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

[46]

H. J. Payne, Models of Freeway Traffic and Control,, Simulation Councils Inc., (1971). Google Scholar

[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. doi: 10.1145/1599470.1599495. Google Scholar

[48]

B. Piccoli and A. Tosin, Pedestrian flows in bounded domains with obstacles,, Contin. Mech. Thermodyn., 21 (2009), 85. doi: 10.1007/s00161-009-0100-x. Google Scholar

[49]

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

[50]

C. W. Reynolds, Steering behaviors for autonomous characters,, in Proceedings of Game Developers Conference 1999, (1999), 763. Google Scholar

[51]

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

[52]

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

[53]

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

[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. Google Scholar

[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. doi: 10.1103/PhysRevLett.75.1226. Google Scholar

[56]

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

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