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Pedestrian Models Based on Rational Behaviour CROWDS: Models and Control Centre International de Rencontres Mathématiques Rafael Bailo , José A. Carrillo, Pierre Degond 3 rd June 2019 Department of Mathematics, Imperial College London 1

Background

Multi-Agent Systems • N discrete, indistinguishable agents. • Agents characterised by position-velocity pair ( x i , v i ) . • Simple interaction rules — attraction, repulsion, alignment. 2

Multi-Agent Systems • CGI flocking — Reynolds (1987). • Pedestrian dynamics — Helbing and Molnár (1995). • Milling in fish — D’Orsogna et al. (2006). • Opinion dynamics — Cucker and Smale (2007a,b). • Predator-prey dynamics — Chen and Kolokolnikov (2014). 3

A Model with Rational Behaviour

Two-Step Dynamics Originally from Degond et al. (2013); Moussaïd et al. (2011): I. Perception Phase: visual stimuli informs agents of their environment. II. Decision Phase: agents react to information by adjusting their paths. 4

Perception Phase — Heuristics v j x j v i x i 5

Distance between i and j : � 2 . � � d 2 i , j ( t ) = � x j + v j t − x i − v i t Time to interaction of τ i , j : � � � � x j − x i v j − v i · � � d i , j τ i , j = arg min = − . � 2 � � � v j − v i t ∈ R Point of closest approach p i , j : � � � � � p i , j − p j , i � = min � x i ( t ) − x j ( t ) and p i , j = x i + v i τ i , j . � t ∈ R 6

Distance to interaction D i , j : � � � � x j − x i v j − v i · � � D i , j = � p i , j − x i � = τ i , j � v i � = − � v i � . � 2 � � � v j − v i Distance of closest approach C i , j : � 1 � �� 2 �� � � x j − x i v j − v i 2 · � 2 − � � � � C i , j = � p i , j − p j , i � x j − x i � = . � 2 � � � v j − v i Note τ i , j ≡ τ j , i and C i , j ≡ C j , i . 7

v j x j (visual horizon) 0 ≤ D i , j ≤ L Distance to Interaction v i p i , j x i p j , i (personal space) 0 ≤ C i , j ≤ R Distance of Closest Approach 8

v j x j (visual horizon) 0 ≤ D i , j ≤ L Distance to Interaction v i p i , j x i p j , i (personal space) 0 ≤ C i , j ≤ R Distance of Closest Approach 9

Perception Phase — Assumptions on the Heuristics Agent i reacts to j if: • τ i , j > 0 and D i , j > 0. • D i , j < L , the visual horizon. • C i , j < R , the personal space. • Cone of vision: � � x j − x i · v i � � v i � > cos( ϑ/ 2 ) . � � � x j − x i 10

Perception Phase — Global Heuristics Global distance to interaction D i : � � D i = min D i , j for perceived j . j Global distance of closest approach C i : C i = C i , j for j that minimises D i , j . 11

Decision Phase • Construct a decision function Φ i from the heuristics. • Choose the optimal velocity at every step: x n + 1 = x n i + v n v n Φ n i ∆ t , i ( v ) . i = arg min i v 12

Possible paths 13

Decision Phase — Gradient Formulation Alternatively, descend the gradient of Φ : dx i dv i dt = v i , dt = −∇ v Φ i ( v i ) . 14

Decision Function Φ i ( v ) = k i � 2 . 2 � D i v − Lv ∗ 15

Towards a High Density Model

Modified Decision Function Φ S , i ( v ) = k b + k s i � 2 � 2 � v � 2 − � v ∗ i � 2 � 2 R 2 � D i C i v − LRv ∗ . 2 � �� � � �� � Speed control Collision avoidance 16

Environmental Coercion Include high-density environmental effects: dx i dv i dt = v i , dt = −∇ v Φ S , i ( v i ) + ǫ i . 17

I. Repulsion as Anticipation: x j − x i � �� � �� ǫ f , i ( x i ) = f � x j − x i � , � � � x j − x i j � = i dr ( r ) , ( r ) = D exp {− ar 2 } where f ( r ) = dV . r p II. Friction and the Fundamental Diagram: ǫ µ, i ( x i , v i ) = − µ ( ρ i ) v i , Cone of vision where µ ( ρ ) ∝ ρ max / ( ρ max − ρ ) . 18

Simulations rationalbehaviour.rafaelbailo.com 19

Outlook • Empirical calibration of the models. • Data-driven fundamental diagram. • Live simulations and optimal control of crowds. • Mesoscopic and macroscopic models. 20

Pedestrian Models Based on Rational Behaviour CROWDS: Models and Control Centre International de Rencontres Mathématiques Rafael Bailo , José A. Carrillo, Pierre Degond 3 rd June 2019 Department of Mathematics, Imperial College London 21

References i References Chen, Y. and T. Kolokolnikov 2014. A minimal model of predator-swarm interactions. Journal of The Royal Society Interface , 11(94):20131208–20131208. Cucker, F. and S. Smale 2007a. Emergent Behavior in Flocks. IEEE Transactions on Automatic Control , 52(5):852–862. 22

References ii Cucker, F. and S. Smale 2007b. On the mathematics of emergence. Japanese Journal of Mathematics , 2(1):197–227. Degond, P., C. Appert-Rolland, M. Moussaïd, J. Pettré, and G. Theraulaz 2013. A Hierarchy of Heuristic-Based Models of Crowd Dynamics. Journal of Statistical Physics , 152(6):1033–1068. D’Orsogna, M. R., Y. L. Chuang, A. L. Bertozzi, and L. S. Chayes 2006. Self-Propelled Particles with Soft-Core Interactions: Patterns, Stability, and Collapse. Physical Review Letters , 96(10):104302. Helbing, D. and P. Molnár 1995. Social force model for pedestrian dynamics. Physical Review E , 51(5):4282–4286. 23

References iii Moussaïd, M., D. Helbing, and G. Theraulaz 2011. How simple rules determine pedestrian behavior and crowd disasters. Proceedings of the National Academy of Sciences , 108(17):6884–6888. Reynolds, C. W. 1987. Flocks, herds and schools: A distributed behavioral model. ACM SIGGRAPH Computer Graphics , 21(4):25–34. 24