Social Forces

This page is part of the Multi Agent System (MAS) documentation.
It contains a description of the Social Forces approach for pedestrian interactions used in the MAS.
For an overview of all MAS-related pages, see the MAS main page.

Agent movement and close-range interaction is implemented using a modified Social Force Model. The implementation uses concepts from the original Social Force Model (Helbing, 1995) and an extension of it for close-rage collision prediction and avoidance (Karamouzas et. al, 2014).
The Social Forces approach is based on the idea that pedestrian movement is the result of all forces exerted on the pedestrian by its surroundings and goals. These forces can be either repulsive or attractive. Repulsive forces are associated with obstacles such as buildings, trees or other pedestrians. The current goal of each pedestrian exerts an attractive force on it. The resulting force on a pedestrian α is the sum of all attractive and repulsive forces,

\begin{equation*}
\vec{F_{\alpha}} = \sum_i{\vec{F_i}}.
\end{equation*}

This force determines the acceleration of the pedestrian. Thus, the pedestrian's walking speed is given by

\begin{equation*}
\frac{d\vec{v}_{\alpha}}{dt} = \vec{F}_{\alpha}.
\end{equation*}

In the MAS, a simple Euler-forward method is used for time-integration.

Note: As human reactions take place on a very short time scale and due to the chosen social forces approach, the time step for the agent model must be very short (~0.02 s - 0.04 s recommended). The forces considered here are repulsion by obstacles and other pedestrians as well as the acceleration term driving the pedestrian toward its target.


The repulsion by an obstacle B is defined as

\begin{equation*}
\vec{F}_{\alpha B} = - \nabla_{\vec{r}_{\alpha B}} U(|\vec{r}_{\alpha B}|),
\end{equation*}

with the repulsive potential

\begin{equation*}
U = (|\vec{r}_{\alpha B}|) = U_0 \cdot e^{-|\vec{r}_{\alpha B}|/R_{B}}.
\end{equation*}

In the MAS, U0 is repuls_wall and RB is sigma_rep_wall.


The repulsion by another pedestrian β is defined as

\begin{equation*}
\vec{F}_{\alpha \beta} = - \nabla_{\vec{r}_{\alpha \beta}} V(|\vec{r}_{\alpha \beta}|),
\end{equation*}

with the repulsive potential

\begin{equation*}
V = (|\vec{r}_{\alpha \beta}|) = V_0 \cdot e^{-|\vec{r}_{\alpha \beta}|/R_{\beta}}.
\end{equation*}

In the MAS, V0 is repuls_agent and Rβ is sigma_rep_agent. This agent repulsive force is only used if α and β are already colliding (i.e. two circles with centers at the respective agents' position with radius radius_agent are intersecting). In this way, a quasi-solid core of the agents is realized.
However, the standard Social-Force Model performs poorly regarding collision avoidance since it contains no mechanism to address this issue. This shows itself in frequent collisions between agents.
To remedy this, Karamouzas et. al, 2014 performed statistical analysis on movement tracking data. They found a 'universal power law governing pedestrian interactions' regardless of scenario. This approach performs very well concerning close-range collision avoidance maneuvers of agents in a wide variety of cases. It causes agents to slow down, speed up or slightly alter their path to avoid colliding with each other. The corresponding force is calculated as

\begin{equation*}
\vec{F}_{\alpha} = -\nabla_{\vec{r}_{\alpha \beta}} \left( \frac{k}{\tau^2}e^{-\tau/\tau_0} \right).
\end{equation*}

Here, τ is the time until, given the current trajectory, a collision between the two pedestrians would occur, τ0 = 3 s and k is a constant to sets the units with a value of 1.5 m2 kg.


The acceleration force

\begin{equation*}
\vec{F_a} = \frac{}{}(v_0\vec{e}_{\alpha} - \vec{v}_{\alpha})
\end{equation*}

describes the tendency of a pedestrian to accelerate toward its current target (more on agent pathfinding here). Here, τα is a relaxation time (tau_accel_agent) that describes how quickly the pedestrian approaches v0, its desired walking speed.

References

  • Helbing, D., Molnar, P. (1995). Social force model for pedestrian dynamics. Physical review E, 51(5), 4282. doi
  • Karamouzas, I., Skinner, B., Guy, S.J. 2014. Universal Power Law Governing Pedestrian Interactions. Pyhsical Review Letters, 113, 238701. doi
Last modified 3 years ago Last modified on Feb 26, 2021 3:51:25 PM