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.

Running simulations

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

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Mathematical Model

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

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Summary of forces

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:

🟦 Self-Motivated Force

Symbol	Name	Equation in Paper	Description
$F_{a_i}$	Self-driving force	(3)	Moves the agent forward at a constant desired speed.

🟥 Interpersonal & Environmental Forces

Symbol	Name	Equation	Description
$F_{b_i}$	Avoidance force	(4)	Repulsive force from nearby individuals.
$F_{c_i}$	Cohesion force	(5)	Attractive force toward group members.
$F_{w_i}$	Wall repulsion	(8)	Repulsion from walls; strength depends on distance and velocity component into the wall.

🟩 Goal-Oriented Forces

Symbol	Name	Equation	Description
$F_{e_{ik}}$	Visible sign attraction	(9)	Attraction toward visible signs within field of view.
$F_{f_{ik}}$	Memorized sign force	(10)	Attraction to previously seen signs even when no longer visible.
$F_{g_i}$	Exit force	(4)	Attraction toward exit if agent is within exit's effective radius.

🟨 External Influence & Noise

Symbol	Name	Equation	Description
$F_{h_i}$	Herding force	(10)	External influence (e.g., crowd herding).
$F_{31}$	Random fluctuation	(11)	Randomized fluctuation force depending on wall proximity.

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Components of the Forces

1. Social Force $F_{11} = F_{ai} + F_{bi} + F_{ci}$

(a) Forward Driving Force:

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

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 $\tau = \frac{m}{\mu}$

(b) Attraction / Repulsion:

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

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

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

(c) Velocity Matching (Alignment):

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

Encourages alignment of motion in local groups.

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

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2. Environmental Force $F_{21}$

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

• 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.

• 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.

Summary of Region-Specific Forces

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

Region	Active Forces
Exit domain (near exit)	F_gi (goal/exit attraction only)
Visible sign domain	F_eik (only if sign is visible)
Memory domain	F_fik (only if sign was memorized)
All other regions	Wall, social, cohesion, fluctuation forces

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.

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3. Random Force $F_{31}$

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.

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Model Parameters

Driving Force

Parameter	Description	Used In
a	Strength of the driving force	F_ai

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Wall Interaction

Parameter	Description	Used In
d	Cutoff distance to consider wall repulsion	F_wi, F_31
w0	Wall repulsion strength when agent moves into the wall	F_wi
w1	Constant wall repulsion strength	F_wi

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Agent Interaction

Distance-Based Repulsion (via c1)

Parameter	Description	Used In
cn0	Minimum (negative) repulsion at zero distance	c1_func → F_bi
cr0	Maximum repulsion plateau value	c1_func → F_bi
beta	Distance at which repulsion crosses from negative to 0	c1_func → F_bi
nu	Distance at which repulsion reaches cr0	c1_func → F_bi
gamma	Start of decay from cr0	c1_func → F_bi
epsilon	Distance where repulsion drops to 0	c1_func → F_bi

Angle-Based Repulsion (via c2)

Parameter	Description	Used In
cphi1	Max repulsion for aligned direction	c2_func → F_bi
cphi2	Plateau value after initial drop	c2_func → F_bi
phi1-phi4	Transition angles for repulsion decay	c2_func → F_bi

Cohesion (via h1, h2)

Parameter	Description	Used In
hr0	Maximum cohesion value	h1_func → F_ci
lam	Constant cohesion up to this distance	h1_func → F_ci
sigma	Distance where cohesion vanishes	h1_func → F_ci
hphi1	Maximum cohesion for angular alignment	h2_func → F_ci
hphi2	Intermediate angular cohesion	h2_func → F_ci

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Signs & Exits

Parameter	Description	Used In
eta_sign	Strength of attraction toward visible signs	F_eik
eta_mem	Strength of attraction toward memorized signs	F_fik
vision_radius	Radius within which signs are visible to the agent	F_eik
fov_angle	Agent’s field of view angle	F_eik
sign_fov	Sign’s own directional “cone of influence”	F_eik
exit_strength	Strength of attraction toward exits	F_gi

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Herding / Panic Influence

Parameter	Description	Used In
x_panic	Position of panic center (dynamic)	F_hi
strength	Herding force magnitude	F_hi
cutoff	Radius beyond which herding force is not applied	F_hi

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

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🎲 Random Force

Parameter	Description	Used In
q1	Random force strength if far from wall or not moving into wall	F_31
q2	Random force strength if moving into wall	F_31

Implementation Notes

• 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

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Future Extensions

• Calibration

Simulation of overtaking a static pedestrian

Screen.Recording.2025-03-28.at.17.32.16.mov

Simulation of the original scenario from the paper

The paper shows simulations of a passage:

Screen.Recording.2025-03-27.at.16.56.19.mov

Simulation of a room with obstacles

• Two different groups with different initial distances among the group members.

Screen.Recording.2025-04-01.at.09.01.04.mov