OOSCILLATING BEHAVIOR WITHIN THE SOCIAL FORCEMODEL
MOHCINE CHRAIBI
J¨ulich Supercomputing Centre, Forschungszentrum J¨ulich GmbH. 52425 J¨ulich,Germany Introduction
The social force model (SFM) [4] belongs to a class of microscopic force-basedpedestrian model for which the interaction with the neighbors solely depends onthe distance spacings. Yet, distance based models can lead to unrealistic oscillatingbehaviors with collision and negative speed (especially in 1D, see for instance [12]and the harmonic oscillator equation). With oscillation we describe a situationwhere pedestrians perform, in the direction of the intended movement, forward andbackward movement. This behavior is to be distinguished from the oscillationsin the passing direction in bottlenecks [5] and from lateral oscillations along theshoulder direction [6, 9].The problem of oscillations in the movement direction was investigated in the lit-erature and shown to rise in general in simulations with force-based models [1, 7, 3].In [2] the model-induced oscillations were quantified by introducing the “oscillation-proportion” introduced. Furthermore, it was suggested that this phenomena is thedual-problem of another model-induced model, namely the overlapping problem.In this article we show on basis of analytical investigation of a simplified one-dimensional scenario, that only particular values of the parameters allow collision-free and non-oscillating simulation. However, ‘realistic” values for the (physical)parameters of the SFM do not produce such simulations.2.
Local stability analysis
The social force model in one dimension and with a coupled interaction with thepredecessor writes:(1) ¨ x n = − a exp (cid:16) − ∆ x n b (cid:17) + v − ˙ x n τ , with v , a, b, τ >
0. Hereby pedestrians are presented as points and ∆ x n = x n +1 − x n .Equation (1) can be linearly approximated for ∆ x n /b ≈ x n ≈ − a (cid:16) − ∆ x n b (cid:17) + − ˙ x n τ + ˜ v , ˜ v = v τ , E-mail address : [email protected] . Date : November 12, 2018. a r X i v : . [ phy s i c s . s o c - ph ] D ec OSCILLATING BEHAVIOR WITHIN THE SOCIAL FORCE MODEL or(3) ¨ x n + ˙ x n τ − ab ∆ x n + a − ˜ v ≈ . Using the following variable substitution ξ = − ab ∆ x n + a − ˜ v , and assuming that x n +1 is constant we get(4) ba ¨ ξ + baτ ˙ ξ + ξ = 0 , which can be brought in the form of a harmonic oscillator(5) ¨ ξ + r ˙ ξ + ω ξ = 0 , with r = τ and ω = ab .Since the coefficients r and ω are constant, (5) will have a solution of the form ξ = e λt which yields the following characteristic polynomial:(6) λ + rλ + ω = 0 . The roots of (6) are(7) λ , = − r ± (cid:112) r − ω , which gives a general solution of the form ξ = c e λ + c e λ . The system described by (1) does not oscillate if the imaginary part of thesolutions are nil, i.e. if r − ω > ab τ < . Note that the solution does not depend explicitly on v .3. Interpretation
In [13] is has been reported that evaluation of empirical data yields τ = 0 .
61 s.A slightly different value of τ was measured in [10] ( τ = 0 . ± .
05 s). It simulation τ = 0 . v = 1 . τ = 0 . a = 2000 N and b = 0 .
08 m lead to oscillations.See Fig. 1
SCILLATING BEHAVIOR WITHIN THE SOCIAL FORCE MODEL 3 t [s] x n [ m ] Figure 1.
Simulation with a = 2000 N and b = 0 .
08 m.The model [8] with a = 300 N and b = 0 . t [s] x n [ m ] Figure 2.
Simulation with a = 300 N and b = 0 . a = 4 . ± . b = 1 .
25 m (meanvalue) and τ = 0 . ± .
05 s. That means that the system according to Eq. (8) isoscillating.
OSCILLATING BEHAVIOR WITHIN THE SOCIAL FORCE MODEL Conclusion
The presented results show that the social force model has to be extended byusing velocity terms in order to produce realistic collision-free behaviors with rea-sonable values for the parameters.In its original form without velocity dependency (reasonable) values of parame-ters lead inevitably to erroneous oscillating movement.
References [1] M. Chraibi, S. A. Kemloh, U., and A. Schadschneider. Force-based models of pedestriandynamics.
Networks and Heterogeneous Media , 6(3):425–442, 2011.[2] M. Chraibi, A. Seyfried, and A. Schadschneider. Generalized centrifugal force model forpedestrian dynamics.
Physical Review E , 82:046111, 2010.[3] F. Dietrich, G. K¨oster, M. Seitz, and I. Sivers. Bridging the gap: From cellular automatato differential equation models for pedestrian dynamics. In R. Wyrzykowski, J. Dongarra,K. Karczewski, and J. Waniewski, editors,
Parallel Processing and Applied Mathematics ,Lecture Notes in Computer Science, pages 659–668. Springer Berlin Heidelberg, 2014.[4] D. Helbing, I. Farkas, and T. Vicsek. Simulating dynamical features of escape panic.
Nature ,407:487–490, 2000.[5] D. Helbing, P. Molnar, I. J. Farkas, and K. Bolay. Self-organizing pedestrian movement.
Environment and Planning B , 28:361–383, 2001.[6] S. P. Hoogendoorn and W. Daamen. Pedestrian behavior at bottlenecks.
Transport. Sci. ,39(2):147–159, 2005.[7] G. K¨oster, F. Treml, and M. G¨odel. Avoiding numerical pitfalls in social force models.
PhysicalReview E , 87, 2013.[8] T. I. Lakoba, D. J. Kaup, and N. M. Finkelstein. Modifications of the helbing-molnar-farkas-vicsek social force model for pedestrian evolution.
Simulation , 81:339–352, 2005.[9] X. Liu, W. Song, and J. Zhang. Extraction and quantitative analysis of microscopic evacuationcharacteristics based on digital image processing.
Physica A: Statistical Mechanics and itsApplications , 388(13):2717–2726, jul 2009.[10] M. Moussa¨ıd, D. Helbing, S. Garnier, A. Johansson, M. Combe, and G. Theraulaz. Experi-mental study of the behavioural mechanisms underlying self-organization in human crowds.
Proc. R. Soc. B. , 276(1668):2755–2762, 2009.[11] D. R. Parisi, M. Gilman, and H. Moldovan. A modification of the social force model canreproduce experimental data of pedestrian flows in normal conditions.
Physica A: StatisticalMechanics and its Application , 388(17):3600–3608, sep 2009.[12] M. Treiber and A. Kesting.
Traffic Flow Dynamics . Springer, Berlin, 2013. ISBN: 978-3-642-32459-8.[13] T. Werner and D. Helbing. The social force pedestrian model applied to real life scenarios.In E. R. Galea, editor,