Stability Analysis of Pedestrian Flow in 2D OV Model with Asymmetric Interaction
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Stability Analysis of Pedestrian Flow in 2D OV Model with
Asymmetric Interaction
Tian Huan-huan Xue Yu
August 13, 2013
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Outline
• Review the 1D OV Model
• Review the 2D OV Model
• The 2D OV Model with the asymmetric
interaction
• Linear stability analysis
• Conclusions
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1. The 1D OV Model
------the coordinate of the nth vehicle
-----the acceleration
-----------the driver’s sensitivity
-----the optimal velocity (OV) function
[1] BANDO M, HASEBE K, NAKAYAMA A, et al. Phys. Rev. E, 1995, 51:1035–1042.
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• -----the position of jth pedestrian• -----the desired velocity• -----the interaction between pedestrians
2. The 2D OV Model (TOVM)
[2] SUGIYAMA Y, NAKAYAMA A, HASEBE K. PED’01, 155-160[3] NAKAYAMA A, HASEBE K, SUGIYAMA Y. Physical Review E, 2005, 71:036121.
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• The strength of the interaction is determined by the distance (between jth and kth pedestrians) and the angle (between and );
• indicated that the pedestrian is more sensitive to pedestrians in front than those behind.
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• The first derivative of the function is centered on the inflectant point; That’s to say, the process of the acceleration and deceleration is symmetrical in the pedestrian flow, so the interaction between pedestrians is symmetrical.
• In reality, the response to the acceleration and deceleration is different. Especially in a high-density situation, in order to avoid collision and pushing, pedestrians are willing to slow down, which is similar with drivers’ behavior.
The interaction is
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3. The asymmetric interaction
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4. Stability analysis
(1)
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The homogeneous solution of equation (1) is
is a constant vector ; is a constant velocity .
Consider a small perturbation as follows:
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The linearized equations of equation(1) are
(3)
(2)
where
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Suppose that the small wave propagates at the angle with the x axis.
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• The longitudinal mode
The linearized equations (2) and (3) are
The 2D wave is classified into two types of modes: longitudinal mode and transverse mode.
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• The transverse mode
The linearized equations (2) and (3) are
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The phase diagram
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The phase diagram of ATOVM
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RegionIs the homogeneous flow stable?
The transverse modes along the x axis
The longitudinal modes along the x axis
A Y Y
B N Y
C Y N
D unknown
E N Y
F N N
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4. Conclusions• The asymmetrical interaction between
pedestrians is considered in the 2D OV model.• The stability of homogenous flow is
investigated with linear stability analysis. • The phase diagram is obtained.• The critical curve of longitudinal mode move
leftward along r-axis and the regions below the curves of longitudinal mode becomes smaller. The critical curve of transverse mode move rightward along r-axis.
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• The phase diagram is obtained. There are six regions above the critical curves in the new model. The region A in the original model is divided into two regions (A and E) in new model, the region C in the original model is divided into two regions (C and F) in new model.
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