American Institute of Mathematical Sciences

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December  2020, 13(6): 1219-1242. doi: 10.3934/krm.2020044

An anisotropic interaction model with collision avoidance

Claudia Totzeck ,

University of Mannheim, School of Business Informatics and Mathematics, 68159 Mannheim, Germany

* Corresponding author: Claudia Totzeck

Received  December 2019 Revised  July 2020 Published  December 2020 Early access  September 2020

In this article an anisotropic interaction model avoiding collisions is proposed. Starting point is a general isotropic interacting particle system, as used for swarming or follower-leader dynamics. An anisotropy is induced by rotation of the force vector resulting from the interaction of two agents. In this way the anisotropy is leading to a smooth evasion behaviour. In fact, the proposed model generalizes the standard models, and compensates their drawback of not being able to avoid collisions. Moreover, the model allows for formal passage to the limit 'number of particles to infinity', leading to a mesoscopic description in the mean-field sense. Possible applications are autonomous traffic, swarming or pedestrian motion. Here, we focus on the latter, as the model is validated numerically using two scenarios in pedestrian dynamics. The first one investigates the pattern formation in a channel, where two groups of pedestrians are walking in opposite directions. The second experiment considers a crossing with one group walking from left to right and the other one from bottom to top. The well-known pattern of lanes in the channel and travelling waves at the crossing can be reproduced with the help of this anisotropic model at both, the microscopic and the mesoscopic level. In addition, the 'right-before-left' and 'left-before-right' rule appear intrinsically for different anisotropy parameters.

Keywords: Mathematical modeling, anisotropic interaction, particle systems, mean-field limit, numerical simulations, pedestrian dynamics.
Mathematics Subject Classification: Primary: 93A16, 82C22, 35Q83; Secondary: 37M05.
Citation: Claudia Totzeck. An anisotropic interaction model with collision avoidance. Kinetic & Related Models, 2020, 13 (6) : 1219-1242. doi: 10.3934/krm.2020044
References:
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show all references

References:
[1]

G. Albi, M. Bongini, E. Cristiani and D. Kalise, Invisible control of self-organizing agents leaving unknown environments, SIAM J. Appl. Math., 76 (2016), 1683-1710.  doi: 10.1137/15M1017016.  Google Scholar

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C. Appert-Rolland, J. Cividini, H. J. Hilhorst and P. Degond, Pedestrian flows: From individuals to crowds, Transportation Research Procedia, 2 (2014), 468-476.  doi: 10.1016/j.trpro.2014.09.062.  Google Scholar

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R. Bailo, J. A. Carrillo and P. Degond, Pedestrian models based on rational behaviour, in Crowd Dynamics, Volume 1: Theory, Models, and Safety Problems (eds. L. Gibelli and N. Bellomo), Springer Internat. Publishing, (2018), 259–292.  Google Scholar

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N. Bellomo, P. Degond and E. Tadmor, Active Particles, Volume 1, Birkhäuser, Basel, 2017. doi: 10.1007/978-3-319-49996-3.  Google Scholar

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N. W. F. Bode and E. A. Codling, Statistical models for pedestrian behaviour in front of bottlenecks, in Traffic and Granular Flow'15 (eds. V. Knoop and W. Daamen) Springer, Cham (2016), 81–88. doi: 10.1007/978-3-319-33482-0_11.  Google Scholar

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N. W. F. Bode and E. Ronchi, Statistical model fitting and model selection in pedestrian dynamics research, Collective Dynamics, 4 (2019), 1-32.  doi: 10.17815/CD.2019.20.  Google Scholar

[8]

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R. Borsche, A. Klar, S. Kühn and A. Meurer, Coupling traffic flow networks to pedestrian motion, Math. Mod. Meth. Appl. Sci., 24 (2014), 359-380.  doi: 10.1142/S0218202513400113.  Google Scholar

[10]

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[11]

M. Burger, B. Düring, L. M. Kreusser, P. A. Markowich and C.-B. Schönlieb, Pattern formation of a nonlocal, anisotropic interaction model, Math. Mod. Meth. Appl. Sci., 28 (2018), 409-451.  doi: 10.1142/S0218202518500112.  Google Scholar

[12]

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[13]

M. Burger, R. Pinnau, C. Totzeck, O. Tse and A. Roth, Instantaneous Control of interacting particle systems in the mean-field limit, J. Comp. Phys., 405 (2020), 109181. doi: 10.1016/j.jcp.2019.109181.  Google Scholar

[14]

J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, in Collective dynamics from bacteria to crowds (eds. A. Muntean, F. Toschi), Springer, (2014), 1–46. doi: 10.1007/978-3-7091-1785-9_1.  Google Scholar

[15]

J. A. Carrillo, Y.-P. Choi, C. Totzeck and O. Tse, An analytical framework for consensus-based global optimization method, Math. Mod. Meth. Appl. Sci., 28 (2018), 1037-1066.  doi: 10.1142/S0218202518500276.  Google Scholar

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J. A. Carrillo, B. Düring, L. M. Kreusser and C.-B. Schönlieb, Stability analysis of line patterns of an anisotropic interaction model, SIAM J. Appl. Dyn. Sys., 18 (2019), 1798-1845.  doi: 10.1137/18M1181638.  Google Scholar

[17]

J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences (eds. G. Naldi, L. Pareschi, G. Toscani), Birkhäuser Boston, (2010), 297–336. doi: 10.1007/978-0-8176-4946-3_12.  Google Scholar

[18]

J. A. Carrillo, A. Klar and A. Roth, Single to double mill small noise transition via semi-lagrangian finite volume methods, Comm. Math. Sci., 14 (2016), 1111-1136.  doi: 10.4310/CMS.2016.v14.n4.a12.  Google Scholar

[19]

J. A. Carrillo, S. Martin and M.-T. Wolfram, An improved version of the Hughes model for pedestrian flow, Math. Models Meth. Appl. Sci., 26 (2016), 671-697.  doi: 10.1142/S0218202516500147.  Google Scholar

[20]

J. Cividini, C. Appert-Rolland and H. J. Hilhorst, Diagonal patterns and chevron effect in intersecting traffic flows, EPL, 102 (2013), 20002. doi: 10.1209/0295-5075/102/20002.  Google Scholar

[21]

E. Cristiani and D. Peri, Handling obstacles in pedestrian simulations: Models and optimization, Appl. Math. Mod., 45 (2017), 285-302.  doi: 10.1016/j.apm.2016.12.020.  Google Scholar

[22]

E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, Springer International Publishing, Switzerland, 2014. doi: 10.1007/978-3-319-06620-2.  Google Scholar

[23]

E. Cristiani, F. S. Priuli and A. Tosin, Modeling rationality to control self-organization of crowds: An environmental approach, SIAM J. Appl. Math., 75 (2015), 605-629.  doi: 10.1137/140962413.  Google Scholar

[24]

P. Degond, C. Appert-Rolland, M. Moussaid, J. Pettré and G. Theraulaz, A hierarchy of heuristic-based models of crowd dynamics, J. Statistical Physics, 152 (2013), 1033-1068.  doi: 10.1007/s10955-013-0805-x.  Google Scholar

[25]

P. Degond, C. Appert-Rolland, J. Pettreé and G. Theraulaz, Vision-based macroscopic pedestrian models, Kin. Rel. Models, 6 (2013), 809-839.  doi: 10.3934/krm.2013.6.809.  Google Scholar

[26]

P. Degond, A. Frouvelle and S. Merino-Aceituno, A new flocking model through body attitude coordination, Math. Mod. Meth. Appl. Sci., 27 (2017), 1005-1049.  doi: 10.1142/S0218202517400085.  Google Scholar

[27]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Mod. Meth. Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005.  Google Scholar

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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. Differential Equations, 250 (2011), 1334-1362.  doi: 10.1016/j.jde.2010.10.015.  Google Scholar

[29]

M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006), 104302. doi: 10.1103/PhysRevLett.96.104302.  Google Scholar

[30]

B. Düring, C. Gottschlich, S. Huckemann, L. M. Kreusser and C. -B.Schönlieb, An anisotropic interaction model for simulating fingerprints, J. Math. Biol., 78 (2019), 2171-2206.  doi: 10.1007/s00285-019-01338-3.  Google Scholar

[31]

J. H. M. Evers, R. C. Fetecau and L. Ryzhik, Anisotropic interactions in a first-order aggregation model, Nonlinearity, 28 (2015), 2847. doi: 10.1088/0951-7715/28/8/2847.  Google Scholar

[32]

J. H. M. Evers, R. C. Fetecau and W. Sun, Small intertia regularization of an anisotropic aggregation model, Math. Mod. Meth. Appl. Sci., 27 (2017), 1795-1842.  doi: 10.1142/S0218202517500324.  Google Scholar

[33]

A. Festa, A. Tosin and M.-T. Wolfram, Kinetic description of collision avoidance in pedestrian crowds by sidestepping, Kinet. Relat. Models, 11 (2018), 491-520.  doi: 10.3934/krm.2018022.  Google Scholar

[34]

L. Gibelli and N. Bellomo, Crowd Dynamics, Volume 1, Birkhäuser, Cham, 2018. doi: 10.1007/978-3-030-05129-7.  Google Scholar

[35]

F. Golse, The mean-field limit for the dynamics of large particle systems, Journées équations aux dérivées partielles, 9 (2003), 1–47.  Google Scholar

[36]

S. N. Gomes, A. M. Stuart and M.-T. Wolfram, Parameter estimation for macroscopic pedestrian dynamics models from microscopic data, SIAM J. Appl. Math., 79 (2019), 1475-1500.  doi: 10.1137/18M1215980.  Google Scholar

[37]

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

[38]

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Figure 3.  Illustration of isotropic and anisotropic pairwise interaction induced by different values of $ \lambda $. The initial positions of the pedestrians are highlighted with a point. Top: Two pedestrians are approaching each other. Positive $ \lambda $ leads to right-stepping, negative $ \lambda $ results in left-stepping. The isotropic case corresponds to $ \lambda = 0. $ Bottom: Two pedestrians meet at a crossroad. In the isotropic case, both depart from their desired trajectory. For $ \lambda \ne 0 $ the two return to their desired velocity after avoiding the collision. In case of the crossing with $ \lambda = -0.25 $ the blue pedestrians slows down, while the red pedestrian accelerates to avoid the collision. Then, the pedestrians relax their velocities towards the desired velocity. For $ \lambda = 0.25 $ the two pedestrians change their roles
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Figure 1.  Visualization of the influence of $ \lambda. $ For $ \lambda = 0 $ we are in the case of the grey vector. The interaction force reduces the force resulting for the desired velocity. The particles may stop in front of each other. For $ \lambda > 0 $ we are in the yellow region. The force vector resulting from the interaction is turned. Adding this rotated vector and the vector resulting from the desired velocity, yields the evasive behaviour. The red particle moves to the bottom and the blue particles moves to the top. For $ \lambda < 0 $ the roles of the to particles change. We are then in the blue region and the red particle moves to the top and the blue particle to the bottom
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Figure 3 in the middle. For $ \lambda > 0 $ we are in the yellow region. The force vector resulting from the interaction is turned. Adding this rotated vector and the vector resulting from the desired velocity, yields the evasive behaviour. The red particle slows down and moves to the top before accelerating and adjusting the velocity to get back onto its path and the blue particle does not slow down as much as the red one and moves to the right. For $ \lambda < 0 $ the roles of the to particles change. We are then in the blue region and the red particle moves to the top and the blue particle moves to the right before being push back onto its path. Intrinsically, we see here the 'right-before-left' rule for $ \lambda > 0 $ and 'left-before-right' for $ \lambda <0. $">Figure 2.  Visualization of the influence of $ \lambda $ in a crossing scenario. Top: For $ \lambda = 0 $ we are in the case of the grey vector. The particles push each other diagonally and leave the path given by the desired velocity, see Figure 3 in the middle. For $ \lambda > 0 $ we are in the yellow region. The force vector resulting from the interaction is turned. Adding this rotated vector and the vector resulting from the desired velocity, yields the evasive behaviour. The red particle slows down and moves to the top before accelerating and adjusting the velocity to get back onto its path and the blue particle does not slow down as much as the red one and moves to the right. For $ \lambda < 0 $ the roles of the to particles change. We are then in the blue region and the red particle moves to the top and the blue particle moves to the right before being push back onto its path. Intrinsically, we see here the 'right-before-left' rule for $ \lambda > 0 $ and 'left-before-right' for $ \lambda <0. $
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Figure 4.  Illustration of the reflective boundary conditions. The black particle with velocity depicted with the dashed vector is about to leave the domain in the next time step. Due to the reflecting boundary conditions, it is projected into the domain and the y-component of its velocity is reflected (black vector)
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Figure 5.  Illustration of the boundary setting. The green and yellow parts of the boundary refer to periodic boundary conditions. Walls are indicated by black lines. In case of the crossing the particles that leave the domain through a green boundary are flowing in at the other green boundary and analogous for the yellow boundary parts
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Figure 6.  Left: Illustration of artificial agents with fixed position and artificial velocities modelling a circular obstacle. Right: Influence of a circular obstacle on the trajectories of the pedestrians
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Figure 7.  Lane formation in a channel - particle simulation with only few particles involved, we see a formation of multiple horizontal lanes as stationary state. The parameters are $ N_b = 10 = N_r, t = 150. $ The arrows show the velocities of the particles
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Figure 8.  Lane formation in a channel - particle simulation. The blue pedestrians go from right to left, the red from left to right, i.e. $ u_b = (-0.2,0)^T $ and $ u_r = (0.2,0)^T. $ The arrows show the velocities of the pedestrians. As $ \lambda = 0.25, $ pedestrians prefer to step to the right to avoid a collision. The snapshots are made at the times $ t = 0,\; 50,\; 100,\; 150,\; 200,\; 250. $
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Figure 9.  Lane formation in a channel - mean-field simulation. Initially the red and the blue group are uniformly distributed in the domain. The difference of the densities vanishes as shown in the plot on the top-left. Then we see the formation of diagonal stripes at time t = 50 and finally a formation of lanes as time proceeds. The plots correspond to times t = 0,250,500,750, 1000, 1250
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Figure 10.  Density distribution in a channel - mean-field simulation. The densities $ \rho_i^2, i \in \{r, b\} $ are integrated w.r.t. $ x^1 $ in order to extract the information along the $ x^2 $-axis
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Figure 11.  Lane formation in a channel - mean-field simulation. Initially the red and the blue group are uniformly distributed in the velocity domain $ \Omega_{v_r} $ and $ \Omega_{v_b} $, respectively, as shown in the plot on the top-left. Then the relaxation towards the desired velocities $ u_r = (0.2,0)^T $ and $ u_b = (-0.2,0)^T $ starts, see plot on the top-right. Only small changes in the $ y $-coordinates lead the crowd to the lane configuration as can be seen when comparing the top-left and the bottom plot of the velocity density. The time instances are $ t = 0,\; 250,\; 1500. $
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Figure 12.  Travelling waves at a crossing - particle simulation. The blue go from bottom to top, the red from left to right, i.e. $ u_b = (0,0.2) $ and $ u_r = (0.2,0). $ The arrows show the velocities of the pedestrians. As $ \lambda = 0.25, $ pedestrians prefer to step to the right to avoid a collision. From top-left to bottom-right: $ t = 0 $, $ t = 50 $, $ t = 100 $, $ t = 150 $, $ t = 200 $ and $ t = 250 $. The simulation was done with $ N_b = 150 = N_r. $ At $ t = 250 $ the pattern allows most pedestrians to walk with their desired velocities
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Figure 13.  Travelling waves at a crossing - mean-field simulation. Initially the red and the blue group are uniformly distributed in the spatial domain. The difference of the densities vanishes as shown in the plot on the top-left. At time t = 50 the groups start to separate, see the plot in the top-right. Afterwards the line pattern is forming (t = 150,375 and 750). Finally, we see the stationary configuration of travelling waves at time t = 1500 in the bottom-right plot
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Figure 14.  Travelling waves at a crossing - mean-field simulation. Initially the red and the blue group are uniformly distributed in the velocity domain $ \Omega_v $ as shown in the plot on the top-left. Then the relaxation towards the desired velocities $ u_r = (0.2,0)^T $ and $ u_b = (0, 0.2)^T $ starts, see plot on the top-right. Only small changes in the $ x $- and $ y $-coordinate lead the crowd to the travelling waves configuration as can be seen when comparing the top-left and the bottom plot of the velocity density. The time instances are $ t = 0,250, 1500. $
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