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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 3rd June 2019

Department of Mathematics, Imperial College London 1

Background

Multi-Agent Systems

• N discrete, indistinguishable agents.
• Agents characterised by position-velocity pair (xi, vi).
• 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

vi xi vj xj

5

Distance between i and j: d2

i,j (t) =

• xj + vjt − xi − vit
• 2 .

Time to interaction of τi,j: τi,j = arg min

t∈R

• di,j
• = −
• xj − xi
• ·
• vj − vi
• vj − vi
• 2

. Point of closest approach pi,j:

• pi,j − pj,i
• = min

t∈R

• xi(t) − xj(t)
• and

pi,j = xi + viτi,j.

6

Distance to interaction Di,j: Di,j =

• pi,j − xi
• = τi,j vi = −
• xj − xi
• ·
• vj − vi
• vj − vi
• 2

vi . Distance of closest approach Ci,j: Ci,j =

• pi,j − pj,i
• =
• xj − xi
• 2 −
• xj − xi
• ·
• vj − vi

2

• vj − vi
• 2

1

2

. Note τi,j ≡ τj,i and Ci,j ≡ Cj,i.

7

vi xi vj xj pi,j pj,i 0 ≤ Di,j ≤ L Distance to Interaction 0 ≤ Ci,j ≤ R Distance of Closest Approach (visual horizon) (personal space)

8

vi xi vj xj pi,j pj,i 0 ≤ Di,j ≤ L Distance to Interaction 0 ≤ Ci,j ≤ R Distance of Closest Approach (visual horizon) (personal space)

9

Perception Phase — Assumptions on the Heuristics

Agent i reacts to j if:

• τi,j > 0 and Di,j > 0.
• Di,j < L, the visual horizon.
• Ci,j < R, the personal space.
• Cone of vision:
• xj − xi
• · vi
• xj − xi
• vi > cos(ϑ/2).

10

Perception Phase — Global Heuristics

Global distance to interaction Di: Di = min

j

• Di,j
• for perceived j.

Global distance of closest approach Ci: Ci = Ci,j for j that minimises Di,j.

11

Decision Phase

• Construct a decision function Φi from the heuristics.
• Choose the optimal velocity at every step:

xn+1

i

= xn

i + vn i ∆t,

vn

i = arg min v

Φn

i (v) . 12

Possible paths

13

Decision Phase — Gradient Formulation

Alternatively, descend the gradient of Φ: dxi dt = vi, dvi dt = −∇vΦi (vi) .

14

Decision Function Φi (v) = k 2 Div − Lv∗

i 2 .

15

Towards a High Density Model

Modified Decision Function ΦS,i (v) = kb 2R2 DiCiv − LRv∗

i 2

• Collision avoidance

+ ks 2

• v2 − v∗

i 22

• Speed control

.

16

Environmental Coercion

Include high-density environmental effects: dxi dt = vi, dvi dt = −∇vΦS,i (vi) + ǫi.

17

• I. Repulsion as Anticipation:

ǫf,i (xi) =

• j=i

f

• xj − xi
• xj − xi
• xj − xi
• ,

where f (r) = dV

dr (r), (r) = D exp{−ar2} rp

.

• II. Friction and the Fundamental

Diagram: ǫµ,i (xi, vi) = −µ (ρi) vi, where µ (ρ) ∝ ρmax/ (ρmax − ρ). Cone of vision

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 3rd 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.

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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.

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