Stability Analysis of Pedestrian Flow in 2D OV Model with Asymmetric Interaction

Stability Analysis of Pedestrian Flow in 2D OV Model with

Asymmetric Interaction

Tian Huan-huan Xue Yu

August 13, 2013

Outline

• Review the 1D OV Model

• Review the 2D OV Model

• The 2D OV Model with the asymmetric

interaction

• Linear stability analysis

• Conclusions

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.

• -----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.

• 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.

• 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

3. The asymmetric interaction

4. Stability analysis

(1)

The homogeneous solution of equation (1) is

is a constant vector ; is a constant velocity .

Consider a small perturbation as follows:

The linearized equations of equation(1) are

(3)

(2)

where

Suppose that the small wave propagates at the angle with the x axis.

• 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.

• The transverse mode

The linearized equations (2) and (3) are

The phase diagram

The phase diagram of ATOVM

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

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.

• 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.