Counterflow Extension for the F.A.S.T.-Model
CCounterflow Extension for the F.A.S.T.-Model
Tobias Kretz , and Maike Kaufman , and Michael Schreckenberg PTV AGStumpfstraße 1 – D-76131 Karlsruhe – GermanyE-mail:
[email protected] Robotics Research GroupDepartment of Engineering ScienceUniversity of OxfordParks Road – OX1 3PJ Oxford – UKE-mail:
[email protected] Physics of Transport and TrafficUniversity of Duisburg-EssenLotharstraße 1 – D-47057 Duisburg – GermanyE-mail:
October 30, 2018
Abstract
The F.A.S.T. (Floor field and Agent based Simulation Tool) modelis a microscopic model of pedestrian dynamics [1], which is discretein space and time [2, 3]. It was developed in a number of more orless consecutive steps from a simple CA model [4, 5, 6, 7, 8, 9, 10].This contribution is a summary of a study [11] on an extension ofthe F.A.S.T-model for counterflow situations. The extensions will beexplained and it will be shown that the extended F.A.S.T.-model iscapable of handling various counterflow situations and to reproducethe well known lane formation effect.
Counterflow situations [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22] pose a specialproblem to models of pedestrian dynamic as well as building construction.To avoid deadlocks a certain degree of self-organization is necessary, whichtypically leads to the formation of lanes, where people in the same lane fol-low each other and different lanes have opposing movement directions. CA1 a r X i v : . [ c s . M A ] A p r odels of pedestrian motion are particularly susceptible to deadlocks, if themain orientation axis of a corridor is aligned to one of the discretizationaxes. The reason for this is the given lane structure which leads to manyhead-on collision situation which lack any asymmetry to decide about theside of mutual passing. One modeling ansatz is to let opposing pedestriansexchange their cells with a certain probability [23]. While this method iscapable of reproducing some elements of reality, one still might be unhappy,as in reality people just do not walk through each other. A major tempta-tion in modelling pedestrian counterflow using discrete models is to makeuse of the lanes preset by the CA lattice and align the two opposing groups’movement along one lattice axis. While such simulations can increase theunderstanding of a model or method, they are merely of academic interestand not sufficient for general use in safety, traffic or city engineering. Theother big difficulty is combining counterflow situations with speeds largerthan one cell per round. The method presented here aims at being indepen-dent of the underlying lattice and it includes speeds larger 1. The probability p xy for an agent i to select a cell ( x, y ) as desired cell ismultiplied by an additional factor p fxy :¯ p xy = p xy · p fxy (1) p fxy = exp k f N (cid:88) j ∈N N i σ ( i, j ) | σ ( i, j ) | P jxy (2)where k f is the coupling parameter that determines the strength of the effect. N N i is the set of nearest neighbors of agent i . N is the maximum number ofnearest neighbors which is considered. N N i only consists of agents withina distance r max , which are visible to agent i (i.e. not hidden by walls and inthe field of view π/ i . σ ( i, j ) = (cid:126)v i · (cid:126)v j is the scalar product of the velocities (cid:126)v i and (cid:126)v j of agent i and agent j , so the fraction in equation 2 is +1, if the agents rather moveinto the same direction and −
1, if the rather move into opposite direction.Finally P jxy is the value of the comoving potential induced by agent j atposition ( x, y ). Assuming motion of agent j at position ( x , y ) along thex-axis, P jxy has the following form (for any other direction of motion, the2otential has to be rotated accordingly): P jxy = 2 h (cid:32) v j v maxj + δ (cid:33) (cid:18) − | y − y | a (cid:19) , if | y − y || x − x | ≤ ab (3) P jxy = 2 h (cid:32) v j v maxj + δ (cid:33) (cid:18) − | x − x | b (cid:19) , if | y − y || x − x | > ab (4)where h is the strength (height) of the potential. The content of the firstbrackets models the velocity dependence of the potential, a is the basewidthof the potential orthogonal to the direction of motion, b is the baselengthalongside the direction of motion of agent j .The values N = 12, h = 4 . δ = 0 . a = 2, b = r max = 15, and k f = 0 . N , but also as itwas increased. All simulations were done with v max = 3. Due to largerfluctuations the method works less well for speeds v max ≥ Figure 1 clearly shows, how lanes are formed and by that deadlocks areavoided and figure 2 the effect on the fundamental diagram by variation of k f and N N i is shown.Figure 1: Lane formation in a corridor and on an area. This contribution presented a model extension to the F.A.S.T.-model for thesimulation of counterflow. It was shown that lane formation was reproducedand deadlocks were avoided for speeds up to 3 cells per round and at densitieswell above the deadlock density of the earlier model. For even larger speeds3igure 2: Fundamental diagrams in dependence of k f and N N i .the model needs to be stabilized, i.e. fluctuations must be restricted tosome maximal value. An interesting empirical question would be to find thenumber of maximally considered nearest neighbors for real pedestrians. References [1] A. Schadschneider, W. Klingsch, H. Kl¨upfel, T. Kretz, C. Rogsch, andA. Seyfried. Evacuation Dynamics: Empirical Results, Modeling andApplications. In R.A. Meyers, editor,
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