Modifications to the social forces model

As seen in the above video, the naive approach of using a semi-circular social forces model makes the motion of the people extremely jerky, with each person experiencing a lot of vibrations. The principal cause for this is the fact that the field around each person has a uniform $\frac{1}{r^{2}}$ distribution. Due to this, when a person $X$ approaches another person $Y$, he first gets repelled out of $Y$'s region of influence. Then, since the basic tendency of $X$ is to continue moving towards the end it was initially destined to, it enters the region of influence again. This leads to oscillations. This phenomenon will no doubt be observed for any kind of force field which has a finite region of influence. However, the oscillations can be significantly reduced by employing a triangular force field ( $(1 - \frac{r}{R})$, where $R$ is the region of influence) instead of a circular field.

In addition to changing the variation in the force field, we also changed the region of influence from a semi-circle to half an ellipse, wherein the major axis of the ellipse is along the direction of the person's velocity. The major and minor axes of the ellipse were set to $2R$ and $\frac{R}{2}$, respectively, in place of the radius $R$ used earlier. So, for any person with velocity $v$, the force he experiences from a person with position $r$ relative to him is

$$MAX\_NEIGHBOR\_FORCE(1 - \sqrt{(\frac{r \cdot v}{2R})^{2} + (\frac{\sqrt{r \cdot r - r \cdot v}}{\frac{R}{2}})^{2}})$$

The main intuition behind this is that a person $X$ has a higher probability of colliding with another person $Y$ who lies in the direction of $X$'s velocity and so, $Y$ must experience a greater repelling force on $X$.