More modifications

Over and above all the modifications made to the social forces model which were outlined above, we also made the following two significant changes to our model.

• A person $X$ can collide with another person $Y$ in two circumstances - either $Y$ is moving towards him or both are moving in the same direction but $Y$ is moving slower than $X$. Both these cases can be characterized as $Y$ moving towards $X$ in the reference frame of $X$. So, instead of experiencing repulsive forces from all people in the 180 degrees field of view in front of him, $X$ is influenced only by people who seem to be moving towards him from his point of view.

• So far, we have estimated the probability of collision only based on the relative positions, without taking into consideration the relative velocities. With the features described until now, there is no way of getting around a head-on collision and also, a person can never overtake another person moving exactly in front of him at a slower pace. To prevent this, we introduce another force which applies on $X$ in a direction perpendicular to his velocity. The probability of $X$ colliding with a person $Y$ is considered to be proportional to the minimum distance possible between $X$ and $Y$, given their current positions and velocities. The force exerted by $Y$ on $X$ is proportional to this probability.