Hirai and Tarui Crowd Simulation Model

This is a force-based model from 1975 simulating crowd dynamics using social and environmental forces.

K. Hirai and K. Tarui, "A Simulation of the Behavior of a Crowd in Panic", Kobe University, 1975.

grafik

Interactive examples and experiments are provided in the notebooks. These include:

• Basic force tests (e.g., wall force, pairwise interaction)
• Overtaking scenarios
• Original scenario from the paper
• Room evacuation with obstacles

To run the notebooks, install the dependencies and open with Jupyter:

pip install -r requirements.txt
jupyter notebook

The motion of each individual $i$ is governed by the following second-order differential equation:

$$F_{11} + F_{21} + F_{31}=m_i \ddot{x}_i+ \nu_i \dot{x}_i$$

Where:

• $x_i$ is the position vector
• $\dot{x}_i$ is the velocity
• $m_i$ is the mass
• $\nu_i$ is the viscosity (damping) coefficient
• $F_{11}$: social forces
• $F_{21}$: environmental forces
• $F_{31}$: random force due to disturbances

In the Hirai and Tarui model, there are 9 forces explicitly defined, and they are grouped into three conceptual categories. Here's the full list:

$$F_{ai} = a \cdot \hat{x}_i$$

Pushes the individual forward in the direction of motion.

This function alone leads to a constant acceleration. However, with the damping parameter $\nu_i$ the agents accelerate indeed to a constant speed.

The agent's velocity $v(t)$ follows from the equation of motion:

$$ m \frac{d\vec{v}}{dt} = a \frac{\vec{v}}{|\vec{v}|} - \nu \vec{v} $$

Assuming motion in a fixed direction (i.e., constant direction of $\vec{v}$) and starting from $v(0) = 0$, the scalar form becomes:

$$ \frac{dv}{dt} = \frac{a}{m} - \frac{\nu}{m} v $$

This is a linear first-order ODE with solution:

$$ v(t) = \frac{a}{\nu} \left(1 - e^{-\frac{\nu}{m}t} \right) $$

where:

• $a$ is the magnitude of the driving force,
• $\nu$ is the damping coefficient,
• $m$ is the mass of the agent.

Note: The similarity to the traditional driving force in force-based models, e.g. the social force model. Hereby,

$v^0 = \frac{a}{\mu}$ and $\tau = \frac{m}{\mu}$

fig

$$F_{bi} = - \sum_j c(x_i, \dot{x}_i, x_j) \frac{x_j - x_i}{r^{ij}}$$

Attracts agents at a moderate distance, repels those that are too close.

$$c(x_i, \dot{x}_i, x_j) = c_1(r^{ij}) c_2(\theta^{ij})$$

$$F_{ci} = \frac{1}{M} \sum_j h(x_i, \dot{x}_i, x_j)(\dot{x}_j - \dot{x}_i)$$

Encourages alignment of motion in local groups.

$$h(x_i, \dot{x}_i, x_j) = h_1(r^{ij}) h_2(\theta^{ij})$$

fig

$$F_{21} = F_{wi} + \sum_k (F_{eik} + F_{fik}) + F_{gi} + F_{hi}$$

• Wall Repulsion $F_{wi}$:

Repels agents from walls:

Where:

• $\mathbf{e}_w$ is the unit vector pointing away from the wall
• $v_{wi}$ is the component of velocity along $-\mathbf{e}_w$ (i.e., into the wall is positive)
• $d_i$ is the actual distance from the agent to the wall
• $d$ is the cutoff distance: the maximum range in which the wall force acts

This force slows down the agent as they approach the wall, and acts only within a threshold distance.

fig

• Attraction to Signs $F_{eik}$:

$$F_{eik} = \eta \cdot \frac{P_k - x_i}{|P_k - x_i|}$$

Attracts the agent toward visible guiding signs $P_k$.

• Memory-Based Attraction to Signs $F_{fik}$: Same as $F_{eik}$, but persists after the sign is out of sight.

• Exit Attraction $F_{gi}$:

$$F_{gi} = g_i$$ Drives agents toward a known exit nearby.

fig

The model assumes that different forces act in mutually exclusive regions:

Note:
F_gi, F_eik, and F_fik are never active at the same time.
Each agent experiences only one of them depending on their location and memory state.

• Panic Avoidance $F_{hi}$:

$$F_{hi} = h_i$$

Pushes agents away from the panic origin.

Models stochastic disturbances:

$$F_{31} = \begin{cases} q_1 \cdot \text{rand}() & \text{if } d_i > d \\ -q_2 \cdot \text{rand}() & \text{if } d_i \leq d \text{ and } b_{wi} > 0 \\ -q_1 \cdot \text{rand}() & \text{if } d_i \leq d \text{ and } b_{wi} \leq 0 \end{cases} $$

Where rand() is a unit vector in a random direction.

note: cutoff and x_panic were added in this implementation although not mentioned in the original paper.

• Forces are modular Python functions for testing and reuse
• Agents interact with each other and with walls (modeled via Shapely)
• Basic Euler integration is used to update agent positions

• Calibration

The paper shows simulations of a passage:

Screenshot 2025年03月27日 at 16 41 31

• Two different groups with different initial distances among the group members. Screenshot 2025年04月01日 at 08 59 51