= Social Forces =

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[[NoteBox(note,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 **[wiki:doc/tec/mas 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 ([#helbing1995 Helbing, 1995]) and an extension of it for close-rage collision prediction and avoidance ([#karamouzas2014 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,
{{{
#!Latex
\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
{{{
#!Latex
\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.\\\\
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
{{{
#!Latex
\begin{equation*}
\vec{F}_{\alpha B} = - \nabla_{\vec{r}_{\alpha B}} U(|\vec{r}_{\alpha B}|),
\end{equation*}
}}}
with the repulsive potential
{{{
#!Latex
\begin{equation*}
U = (|\vec{r}_{\alpha B}|) = U_0 \cdot e^{-|\vec{r}_{\alpha B}|/R_{B}}.
\end{equation*}
}}}
In the MAS, //U,,0,,// is [wiki:/doc/app/agtpar#repuls_wall repuls_wall] and //R,,B,,// is [wiki:/doc/app/agtpar#sigma_rep_wall sigma_rep_wall].\\\\
The repulsion by another **pedestrian //β//** is defined as
{{{
#!Latex
\begin{equation*}
\vec{F}_{\alpha \beta} = - \nabla_{\vec{r}_{\alpha \beta}} V(|\vec{r}_{\alpha \beta}|),
\end{equation*}
}}}
with the repulsive potential
{{{
#!Latex
\begin{equation*}
V = (|\vec{r}_{\alpha \beta}|) = V_0 \cdot e^{-|\vec{r}_{\alpha \beta}|/R_{\beta}}.
\end{equation*}
}}}
In the MAS, //V,,0,,// is [wiki:/doc/app/agtpar#repuls_agent repuls_agent] and //R,,β,,// is [wiki:/doc/app/agtpar#sigma_rep_agent 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 [wiki:/doc/app/agtpar#radius_agent radius_agent] are intersecting). Otherwise, a collision avoidance force according to ([#karamouzas2014 Karamouzas et. al, 2014]) is calculated:
{{{
#!Latex
\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, τ,,0,, = 3 s and k is a constant to sets the units with a value of 1.5 m^2^ kg.\\\\

The **acceleration force** 
{{{
#!Latex
\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 target. Here, τ,,α,, is a relaxation time ([wiki:/doc/app/agtpar#tau_accel_agent tau_accel_agent]) that describes how quickly the pedestrian approaches v,,0,, its desired walking speed with the direction to its current target, e.\\\\



== References ==
* [=#helbing1995] '''Helbing, D., Molnar, P.''' (1995). Social force model for pedestrian dynamics. Physical review E, 51(5), 4282. [https://doi.org/10.1103/PhysRevE.51.4282 doi]
* [=#karamouzas2014] '''Karamouzas, I., Skinner, B., Guy, S.J.''' 2014. Universal Power Law Governing Pedestrian Interactions. Pyhsical Review Letters, 113, 238701. [https://doi.org/10.1103/PhysRevLett.113.238701 doi]