# Pedestrian dynamics

I've written about modeling the movement of cars as a fluid in the past. We could think about pedestrians like this, but usually pedestrians aren't described well by a fluid model. This is because while cars are mostly constrained to move in one direction (in lanes), this is not true of pedestrians. On a sidewalk, people can be walking in opposite directions, and often someone walking in one direction will directly interact (by getting close to) someone walking in the opposite direction. There are some specific scenarios where a fluid model could work, such as a crowd leaving a stadium after the conclusion of a basketball game. In this case, everyone is trying to get away from the stadium so there is some kind of flow. However, this doesn't work generally, so I will consider a different type of model, similar to the one described in [1].

If there are only two directions that people want to travel and they happen to be opposite, then we could model the pedestrians as charged particles. The pedestrians that want to go in opposite directions would be oppositely charged, and the force that keeps the pedestrians on a trajectory could look like an electric field. However, this would mean that people moving in opposite directions would attract each other, which really does not match expectations. This model also fails if there are multiple directions where pedestrians want to go (such as at an intersection), or if the desired directions of the pedestrians are not opposite. While a plasma (which is a collection of charged particles) model may not be the best to describe the scenario, I will borrow some ideas from plasma dynamics in building my model and I will use techniques used to simulate plasmas to simulate pedestrian movement.

There will be a few effects governing the movement of pedestrians. One effect is for the pedestrians to want to go a desired direction at a desired speed. It turns out that most humans walk at a speed of around $v_d=$1.4 m/s (3.1 mph) and if someone is going slower or faster than this, they will tend to go toward this speed. Let me call the desired speed along the desired direction of the pedestrian $\vec{v}_d,$ and the current walking speed of the pedestrian $\vec{v}.$ I will model the approach to the desired direction as a restorative force that looks like

Here, $\tau$ represents how long it takes the pedestrian to get back to their desired position and direction once they are off track. In general, $\tau$ could be different for every pedestrian, but for simplicity I set it as a constant for all pedestrians here, and I will take it to 0.3 s, which is close to the human reaction time. Note that the restorative force is zero when $\vec{v_d}=\vec{v},$ so if a pedestrian is already going in their desired direction at their desired speed, there will be no restorative force and the pedestrian will continue to go at this direction and speed. You may find it odd that my force has units of acceleration. I am thinking about this more as a generalized sense of the term force as in something that causes velocity changes, but it would also be reasonable to assume that I have set the mass of the pedestrians to 1.

Pedestrians will also avoid colliding with each other, which is the other force I include in the model. While [1] assumes an exponential force for the interaction force, I will assume that pedestrians interact via a generalized Coulomb potential. The general results seem to match without too much regard for the exact shape of the force. I define the force between pedestrian $i$ and pedestrian $j$ is

Where $\vec{r}_{ij} = \vec{r}_i - \vec{r}_j$. $\gamma,$ $\epsilon,$ $\alpha,$ and $r_0$ are constants that I will describe below. $r_0$ is an interaction radius that sets the scale for this interaction. This would not necessarily be the same for everyone. For example if someone is texting, their interaction radius $r_0$ is probably much smaller than someone who is paying attention to where they are going. However, for simplicity I take it to be the same for everyone, and I take it to have a value of 1.2 m.

Since pedestrians travel in 2 dimensions, if $\alpha = 1$ and $\epsilon = 0,$ this would be the Coulomb potential, if $\gamma$ were aptly chosen. In this scenario, however, I do not really want the Coulomb potential. The Coulomb potential is quite long range, meaning that particles in a Coulomb potential can influence particles that are quite far away. As the power $\alpha$ in the equation above gets larger, the force becomes more short-range, which seems to better model the interactions of pedestrians. However, this presents another problem in that the force gets extremely large if two pedestrians happen to get really close to one another. To combat this, $\epsilon$ is a small number that "softens" the force such that the force never gets extremely large (which I took to mean $|\vec{F}_{ij}|$ should never be too much bigger than the maximum possible value of $|\vec{F}_{restore}|$). $\gamma$ then decides the relative importance of this interaction force to the restorative force.

I will simulate this model by considering $N$ people are in a long hallway with aspect ratio 1:10, for example at an airport or a train station. This can also be a model for a long, wide sidewalk as even though there are no walls, people are relatively constrained to stay on the sidewalk. I have some people trying to get to one end of the hallway (in the $+\hat{x}$ direction) and some trying to get to the other end (in the $-\hat{x}$ direction). This is an example of an N-body simulation, which is widely used in studying gravitational systems and plasma systems.

In [1], the walls exerted an exponential force on the pedestrians. I choose a similar model. I set the parameters of the exponential empirically such that the pedestrians keep a reasonable distance from the walls. I set the range of the exponential force to be a tenth of the total width of the corridor. I set the magnitude such that at the maximum, the force due to the wall is the same as the maximum value of $|\vec{F}_{restore}|.$

When a pedestrian reaches the end of the corridor, I record how much time it took for that pedestrian to traverse the corridor. I then regenerate the pedestrian at the other end of the corridor as a new pedestrian. I generate the new pedestrian with a random $y$ coordinate and a random velocity direction, but pointing at least a small bit in the desired direction. The magnitude of the velocity is taken to be $v_d.$ Thus, the simulation is set up such that there will always be $N$ people in the hallway.

A simulation of $N$=100 people in a hallway of dimensions 100 m x 10 m. All pedestrians desire to go to the left of the hallway. The pedestrians relax to a state where they are each about the same distance from each other. It seems that people usually stand closer together on average, so our value of $r_0$ should probably be smaller to match observations.

The first thing I tried was to simply put a few people in the hallway all wanting to go in the same direction, and see what they do. I set the length of the hallway to be 100 m, which made the width of the hallway 10 m. As can be seen above, this isn't too exciting. The pedestrians' paths are mostly unobstructed and they get across the hallway in about 71 s, which is the length of the hallway divided by 1.4 m/s, the comfortable walking speed of the pedestrians. Even in this simple case, though, it is apparent that the pedestrians "relax" into a scenario where the average distance between the pedestrians is roughly the same.

A simulation of $N$=100 people in a hallway of dimensions 50 m x 5 m. Pedestrians are equally likely to want to go left or right. We can see that lanes of people that would like to go in the same direction can form, as was observed in [2]. This effect could be even stronger with an extra "incentive force" for people to be on the right side of the road if they are not already on that side.

The time required to cross the room as a function of density of people in a simulation of $N$=100 people. The y-axis is normalized by the length of the room divided by the desired velocity (1.4 m/s). $p=0.5,$ which means half of the pedestrians desire to go to the left and the other half desire to go to right. I change the density of people by changing the size of the room from 2.5 m x 25 m to 10m x 100 m. As the density is higher, the pedestrians interact more with each other and thus are less likely to be on their desired trajectory.

Next, I looked at the more interesting cases of what happens when there are pedestrians that want to go in different directions. First, I assume that exactly half of the pedestrians would like to go in one direction and half would like to go the other direction. I then varied the length and width of the hallway, keeping the aspect ratio constant, while keeping the number of people in the hallway constant at 100. This has the effect of changing the density of people in the hallway. The y-axis on the graph above is normalized by $L/v_d,$ which is the time a pedestrian with all of his or her velocity in the desired direction would take. This shows that as the density increases, it takes longer (proportionally) for the pedestrians to get across the room. This makes sense as the pedestrians are interacting more often and thus cannot keep going in the desired direction.

A simulation of $N$=100 people in a hallway of dimensions 50 m x 5 m. 90% of pedestrians want to go left, while the other 10% want to go right. The right-going pedestrians undergo many interactions with the left-going pedestrians. In fact, if the left-going pedestrians were denser, this could look like Brownian motion.

The time required to cross the room as a function of $p$, the fraction of $N$=100 people that would like to go left or right. The y-axis is normalized by the length of the room divided by the desired velocity (1.4 m/s). The size of the room is 5 m x 50 m. The blue line is the time to get across for the pedestrians going leftward, and the red line in the time to get across for the pedestrians going rightward. As the fraction of pedestrians going leftward increases, it becomes easier for those pedestrians to get across, but it makes it harder for the pedestrians that would like to go in the opposite direction to get across more slowly.

I then took the number of people in the hallway to be 100 with the length of the hallway being 50 m and the width being 5 m. I observed what happened as I varied the fraction of pedestrians, $p$ that wanted to go in either direction. This effect is shown above. As $p$ is increased, the more dominant pedestrians can get through the corridor more quickly than the less dominant pedestrians. Again, this makes sense as when people go "against the gradient," they have to weave through people to try to get to the other side.

I will note that I have not done this simulation in the most efficient way. For every pedestrian, I calculate the interaction force with all the other pedestrians and add up all the contributions. It turns out one can average or sometimes even ignore the effect of pedestrians far away, which can make the code run about $1/N$ times faster.

The python and gnuplot scripts I used for the simulation and to create the plots are available here.

References:
1. Kwak, J., 2014. Modeling Pedestrian Switching Behavior for Attractions. Transportation Research Procedia. 2. 612-617.
2. Tao, X., 2011. A Macroscopic Approach to the Lane Formation Phenomenon in Pedestrian Counterflow. Chinese Phys. Lett. 28.