Mathematical Models for Pedestrian Dynamics

Transcription

1 Mathematical Models for Pedestrian Dynamics Jen Worthy May 11, 2018 Cleveland State University MTH 497: Senior Project Dr. Shawn D. Ryan Assistant Professor Department of Mathematics

2 1 Contents 1 Introduction 3 2 Pedestrian Dynamics The Social Force Model Specifications Crowd Dynamics Densities Self-Organization Evacuation Dynamics Model for Panicking Pedestrians Collective Phenomena Modified Model for Pedestrian Dynamics Mathematical Modeling Evacuation with Modified Model Results Further Uses 16 7 Conclusion 16 8 References 16

3 2 Abstract People spend much of the day walking to and from di erent places. But have you ever thought about how and why pedestrians move the way they do? Pedestrian, Crowd, and Evacuation Dynamics, a paper written by Dirk Helbing and Anders Johansson, describes the e orts made to model the movement of pedestrians. Models of pedestrian dynamics attempt to display the behaviors of both individual pedestrians and their interactions in crowds. From these interactions, some patterns of self-organization can be seen. The paper by Helbing and Johansson focuses on dynamics in panic and evacuation situations. For the purposes of this work, we will focus on pedestrian dynamics in some simple scenarios, as well as in an evacuation situation with and without added panic. The goal of this paper is to summarize the findings of and explore the di erent models for pedestrian crowd dynamics, and to provide information on how these models could be better modified to depict di erent scenarios.

4 3 1 Introduction An essential focus of the field of pedestrian dynamics is how coordination patterns are created from individual interactions. Models assuming automatic responses to interactions pedestrians may encounter can accurately produce models of how pedestrians move. These automatic responses imply patterns of motion are formed simply by interactions and not by direct a direct conscious intervention. Some of pedestrians reactions to each other can be assumed to be a result of a learned behavior that optimizes response in terms of minimizing collisions and time. An example of this is the choice of Americans to walk on the right side since we drive on the right side of the road. The development of these models is not solely mathematical. They involve tra c scientists, psychologists, sociologists, biologists, physicists, computer scientists, and other experts. The original models were made by observing the movement of pedestrians over many years, and then replicating what was observed. None of these models took the patterns of self-organization into consideration, but they clearly show signs that self-organization does occur. L. F. Henderson was one of the first to propose a model that produced spatio-temporal patterns of motion. He proposed this model by comparing pedestrian crowd movement to the behavior of gases and/or fluids. Because of this, agent-based models are the main focus of pedestrian researchers and of this work. 2 Pedestrian Dynamics 2.1 The Social Force Model The Social Force model is the most well-known of the agent-based models. It accurately models usual pedestrian tra c; cases where people automatically evade each other when met by other pedestrians. This automatic behavior can be seen as a learning process based o of trial and error, which can be simulated through evolutionary algorithms. These algorithms take the model and add a repulsive force to each pedestrian. This prevents them from walking on top of or through each other. Another element of the model is the use of vectors to show that behavior of pedestrians in certain situations can be described using forces. These vectors determine the amount and direction of pedestrian behavior and behavioral changes that may occur. For these simulations, a certain percentage of how many pedestrians turn right or left is estimated and uncertainty of individual behavior is averaged out. We will use the following components in pedestrian models: t =time v (t) = velocity of pedestrian of

5 4 f (t) = sum of social forces influencing pedestrian (t) = individual fluctuations (randomness) r (t) = change in location of pedestrian = relaxation time v 0 = isolated movement (normal speed) e 0 = desired direction 1 (v 0 e 0 v ) = self-propulsion f (t),f i (t) = repulsive forces avoiding (other pedestrians) and i (obstacles) Social force model derived from Newton s Laws: v (t) = dr (t) dt (1) dv dt = f (t)+ (t) (2) f (t) = 1 (v 0 e 0 v )+ X (6= ) f (t)+ X i f i (t) (3) 2.2 Specifications Specification of the Social Force Model: Simplified Interaction: f (t) =f(d (t)) (4) Circular Specification: f(d )=A e d /B d kd k (5) Elliptical Specification: f (d )=Ae b /B kd k + kd y k 1 2b 2 d kd k + d y kd y k (6) In the simplified interaction force, the radius of reaction to a given situation is limited to d = r r (where r is the radius of the pedestrian), which is the distance from pedestrian to pedestrian. In the circular specification,

6 5 the additional parameters A and B correspond to the interaction strength and the interaction range respectively. In the elliptical specification, 2b ensures the symmetricity between pedestrians and, and y =(v v ) t. This specification allows for interactions to depend on both distance and relative velocity, and also does not restrict the repulsive force to being directed from pedestrian to pedestrian. The authors use this information to examine video tracking of pedestrians near escalators to extract trajectories (Figure 1). To start the model, the position, velocity, and acceleration for each pedestrian needs to be calculated and assigned. Next, assign the speed (maximum) vector to each pedestrian. Then assign the desired end point for each pedestrian. To reach good model performance, the pedestrians interactions must be velocity dependent. Figure 1: Right: Video tracking of pedestrians head. Left: Simulation of resulting trajectories (Helbing & Johansson, 2013). 3 Crowd Dynamics 3.1 Densities The social force model has been modified to model more complex situations. To better simulate these situations, it helps to explain how crowd dynamics move. When crowd density is low, pedestrians can move more freely; the motion compares to the behavior of gases or isolated individuals. When crowd density is medium to high, pedestrians have more restricted movements; their movement compares to the motion of liquids. This can be seen when looking at the footprints made by pedestrians in the snow, which is analogous to streamlines of fluids. When crowd density is high, pedestrians are restricted in how and where they can move to. This motion compares to that of granular flows. This can be seen in Figure 2, when the distance between pedestrians is low, then the interaction

7 6 force is high, and when the distance is high, then the force is low. Figure 2: Distance dependence of the interaction force between pedestrians (Helbing & Johansson, 2013). 3.2 Self-Organization The model can be further modified to simulate pedestrians in patterns of self-organization. Patterns of selforganization are unplanned phenomena resulting from interactions between objects and subjects. These patterns occur without prior coordination or communication. For the purposes of this paper, we will focus on observations that are common and occur internationally. The primary patterns seen are lane formation, stripe formation, and oscillatory flows. Figure 3: Left: Lane formation. Top right: Number of pedestrians passing through a bottleneck from both sides. Bottom right: Simulation of two crossing pedestrian streams (Helbing & Johansson, 2013). The most frequently occurring pattern is lane formation. This transpires as pedestrians on oppositely moving

8 7 sides form uniform lanes of walking direction, as seen on the left side of Figure 3. This can be seen during both low and high densities, and when there is not a large enough distance between the individuals to create proper lanes. Lane formation has a weak preference for one side over another, which is consistent with the di ering behavior of pedestrians depending on the country. Since the lanes are so distinctly formed, this behavior can be viewed as a segregation phenomenon. The emergence of lanes enables people to minimize the frequency and strength of avoidance maneuvers, i.e. people evading each other. Because of this, lane formation can described as a pattern of collective intelligence, which is the behavior resulting from interactions rather than from individual reasoning. Furthermore, this pattern occurs automatically without side preference, and does not require pedestrians to be consciously placed into lanes. Another self-organizational pattern, and an extension of the behavior described above, is stripe formation. This pattern occurs at areas of intersection. During initial observance, a pattern is not easily identified, and pedestrians appear to be moving chaotically. Upon closer inspection, multiple collective patterns can be detected. These patterns often show rotary and oscillatory flows since they compete with each other until one has temporary dominance over the others. After some time though, another pattern will emerge more dominant and destroy the motion of the one persisting prior. The phenomenon of stripe formation is present when two di erent flows are able to penetrate one another without requiring for pedestrians to stop or collide. As seen in the bottom right of Figure 3, pedestrians can move forward with the stripes while others move sidewards within the stripes. This allows for the minimization of obstructions and the maximization of average speed. Consequently, as in lane formation, stripe formation can be characterized as an intelligent collective behavior. The final pattern observed is that of oscillatory flows occurring at bottlenecks. This phenomenon takes place at entrances and exits. Once a pedestrian passes through a narrow opening, then others flowing in the same direction can easily follow through. This continues until pedestrians on the moving side of the bottleneck experience less pressure than pedestrians on the other, non-moving, side. This allows for the pressure on the still side to eventually grow great enough to change the flow of direction. This can be seen in the top right of Figure 3. These flow changes occur consistently over time, making them oscillatory. A solution to decreasing the oscillatory flow through a bottleneck is to widen the opening, but this only works to an extent. It can be shown that oscillatory flows will persist through any situation, so the widening of an entrance or exit is not a permanent solution. This is due to the fact that a bigger opening will attract more people to walk through it than a smaller opening would.

9 8 4 Evacuation Dynamics Evacuation dynamics focus on situations that occur at extreme crowd densities. Nearly all evacuations occur in an orderly fashion, but those that don t may result in disasters. While many would call these situations of panic, the definition of that word can become controversial. So for the purposes of this paper, we will define panic as evacuation dynamics with a focus crowd dynamics that occur during high densities with an added psychological stress. 4.1 Model for Panicking Pedestrians The model adds additional physical interaction forces that occur when pedestrians actually touch each other. These forces include a body force, meaning that each body has its own force that expels outward, and a sliding friction force, which makes it more di cult for pedestrians to move. Derived from formulas of granular interactions, we have f ph (t) =k (r d )n + apple (r d ) v t t, (7) where the function (z) is equal to its argument z, ifz 0, otherwise 0, d <r = r + r, k(r d )n is the body force, apple(r d ) v t t is the sliding friction force, and k and apple are large constants. When a pedestrian comes into contact with a boundary or an object, the interaction made is analogous to that touching another pedestrian. This force can be characterized by the following equation: f i = {A exp[(r d i )/B ]+k (r d i )}n i apple (r d i )(v t i )t i, (8) where i is the boundary or object, n i (t) is the direction perpendicular to it, and t i is the directional tangential to it. 4.2 Collective Phenomena Observations of pedestrians in simulated scenarios of panic were modeled with the assumption of the following features: 1. higher level of individual fluctuations (nervousness) 2. higher desired velocity v 0 (want for escape) 3. herding interaction (not knowing what to do)

10 9 The features listed above are responsible for the following outcomes: herding and ignorance of exits, freezing by heating, intermittent flows, faster-is-slower e ect, phantom panic, transition to stop-and-go waves, and transition to crowd turbulence. When people are unsure about how to act in certain situations, they naturally tend to mimic the behavior of those around them, which we will denote as herding behavior. During situations of panic, exits are not easily distinguishable, which causes evacuations to less e cient and less time e ective. As a result, pedestrians will walk straight to where the think the exit is. Many times they will encounter a wall instead. At that point, they move to either the right or left until they reach an exit. If they come across another person(s) before an exit, then they will have to collectively pick one direction to move in. This behavior can also be seen when pedestrians hear something they associate with freedom. The noise will attract all pedestrians to it, whether it leads to an exit or not. Both situations can cause pedestrians to blindly follow each other and ignore the presence of easily accessible exits. This can be seen when looking at the right side of Figure 4. Figure 4: From (a) to (d); Left: The normal evacuation of pedestrians. Right: The evacuation of pedestrians during limited visibility (Helbing & Johansson, 2013). Freezing by heating takes place during very high densities, and when pedestrians are nervous. As people become more nervous, the strength of their individual fluctuations increase (analogous to a rise in temperature). This causes lane formations to be destroyed. Scientifically, when the temperature of a liquid is raised, it becomes

11 10 a gas. In cases of evacuation, the rise in fluctuations does not lead to high density crowds (comparable to fluids) behaving as if they are low density (comparable to gases). Instead, pedestrians become unable to move, analogous to freezing a liquid to make it a solid. Simulations of this show the pedestrians as being completely blocked from moving. In real crowds, pedestrians are able to rotate their bodies allowing others to slide by, so the block does not persist indefinitely. As mentioned before, bottlenecks can cause the oscillatory flows of pedestrians. But when looked at in situations of panic, the flow changes are much more dramatic. In particular, coordination problems can occur when the arrival flow is much higher than the departure. In these situations, people either have a strong desire to move in a certain direction (inquisitiveness), or away from a perceived danger (escape). Both cause people to compete to occupy the same space, which in turn causes more interactions and friction, and eventually causes the movement of the crowd to slow down. This is known as the faster-is-slower e ect. One consequence of this is intermittent flows. Intermittent flows cause openings to become clogged and prevents pedestrians from moving through. Another consequence is the creation of phantom panic. This happens when people who disrupt the flow cause the crowd to stop. Those in the back, who are unaware as to why their movements have slowed, become impatient and pushy, which can cause others around them to panic. Intermittent flows have also been known to transition to stop-and-go waves. These occur when there is a significant drop in the flow rate, and they persist over longer periods of time. An example of this would be road congestion during rush hour. Once the roads become filled with cars, the flow of tra c slows down. Once most of the cars are no longer on the road, or are moving at a normal rate, then we would expect the flows to return to normal, but instead stop-and-go waves still persist. This is why roads may remain congested 30 minutes after rush hour has ended. From this, stop-and-go waves can then transition to crowd turbulence, which stems from a series of instabilities in the flow rate. Unlike stop-and-go waves, turbulence occurs when there are random, irregular flows in all possible directions. This can cause people to be pushed around, and some even fall. Those that fall, become obstacles for the others. In turn, more people can fall by trying to avoid or tripping over those on the ground. Consequently, the involuntary movement of people in a crowd can cause stampedes and other disastrous behaviors.

12 11 5 Modified Model for Pedestrian Dynamics The extensions explored below attempt to confirm the works described above. MATLAB simulations, derived from above formulas, were used to show both simple pedestrian and evacuation dynamics. 5.1 Mathematical Modeling To start, we first analyzed a simplified social force model. Using the same parameters as listed above, all of the following models were derived from this equation for self-propulsion: f (t) = 1 (v 0 e 0 v ), (9) where is relaxation time, v 0 is isolated movement, and v (t) is the velocity of pedestrian of. The first model looks at pedestrian tra c inside of a mall. 100 pedestrians exit escalators and walk into 1 of 3 stores (Figures 5 and 6). The model allows for each pedestrian to walk towards a randomly picked store. The pedestrians cannot walk through or over each other, which is enforced by a repulsion force c rij/ 1,wherec 1 =1is the strength of repulsion, r ij is the position, and =.25 is the e ective repulsion distance. Once they have entered a store, they move o the screen. Once all pedestrians have entered a store, the simulation ends. In the model, the relaxation time = 1, initial velocity v 0 =.25, and initial orientation e 0 is randomly and individually assigned to each pedestrian to determine which store they will enter, and v is the speed at which each pedestrian is able to move once all other parameters have been taken into consideration. Our Model: f (t) = 1 (v 0 e 0 v )+ X (6= ) f (t), (10) where f is the repulsion force. Next, an obstacle (fountain) was added to the previous model (Figure 7). The parameters above all remained the same. The only parameter added was a repulsive force c rij/ 2,wherec 2 = is the strength of repulsion and r ij, are the same as above. This ensured the pedestrians could not walk through or over the fountain. Final Model: f (t) = 1 (v 0 e 0 v )+ X (6= ) f (t)+ X i f i (t), (11) where f i is the obstacle, in this case fountain, repulsion force. The final model examines two situations involving evacuation dynamics.

13 Evacuation with Modified Model In the first simulation pedestrians move from their seats to the exit. This also uses equation 9, and the rows of seats replace the fountain. This can be seen in Figure 8. The second depicts pedestrian movements in a situation of panic (Figure 9). Here, the pedestrians make random movements in the direction of the exit to avoid the fire. Unlike the other simulations, this one does not run until all of the pedestrians have reached the exit. It ends once the fire has reached the exit, leaving some pedestrians still in the room. To simulate pedestrians moving in a panicked manner, the parameter =.4 was added. The only other parameters changed were c 2, which was made to equal 1 in the first simulation lowering the repulsive strength, and 1.5 in the second, and v 0 which was lowered to 0.1 in the second simulation resulting in slower movement speed. This allowed for pedestrians to jump over the seats while they were trying to escape. The inclusion of panic gives the following model: f (t) = 1 (v 0 e 0 v )+ X (6= ) f (t)+ X i f i (t)+", (12) Where " = DdW, D = strength of di usion, and dw = randomness. This allows for noisy motion. The individuals are moving towards the exit, but the panic makes the motion erotic. 5.3 Results The first model derived was used in two di erent simulations. First, when the probability of a pedestrian entering the green store was equal to that of the blue store and the red store. Second, when the probability of a person entering the green store was highest, blue store medium, and red store lowest. The motivation behind this was to see if an external factor, like a big sale, would e ect the speed and e ciency at which pedestrians traveled. One would expect that a large number of people traveling to one location would cause total arrival time for all pedestrians to be longer. To answer this, each simulation was run over a hundred times, then the averages were taken. Overall, there was not a significant di erence in time, so the amount of people traveling to one location does not seem to have a significant e ect on arrival time of all pedestrians.

14 13 (a) (b) (c) (d) (e) (f) Figure 5: Each pedestrian is represented by a circle, each escalator by a black box, and each store by a box colored red, green, or yellow. Pedestrians move from the escalators to a store while avoiding each other. (a) (b) (c) (d) (e) (f) Figure 6: Each pedestrian is represented by a circle, each escalator by a black box, and each store by a box colored red, green, or yellow. Pedestrians move from the escalators to a store while avoiding each other. The second model attempted to answer the question of whether an obstacle, like a fountain, would e ect the

15 14 overall arrival time of pedestrians. This model was also used in two di erent simulations to find out if the number of people entering a store and an obstacle together would e ect total time needed to enter the stores. These simulations used the same store probabilities as those in the first model did respectively. Results from running each simulation several times and taking the averages shows that neither an obstacle nor an unequal distribution of people plus an obstacle e ects the total time by much. Figure 7: Each pedestrian is represented by a circle, each escalator by a black box, each store by a box colored red, green, or yellow, and the obstacle by a large circle (fountain). Pedestrians move from the escalators to a store while avoiding each other and the fountain. Running both the evacuation simulations and comparing their results verified the faster-is-slower e ect described earlier. When panic is added, the pedestrians appear to be moving much faster (even though they have a lower v 0 ), but it takes them much longer to exit the theater.

16 15 (a) (b) (c) (d) (e) (f) Figure 8: Each pedestrian is represented by a circle, rows of seats by rectangles, the screen by a black box, and the exit by a green box. People move from their seats to the exit without experiencing panic. (a) (b) (c) (d) (e) (f) Figure 9: Each pedestrian is represented by a circle, rows of seats by rectangles, the screen by a black box, and the exit by a green box. Pedestrians panic and move from their seats towards the exit to avoid the fire. The fire is represented by a blue circle, and pedestrians touched by the fire turn red.

17 16 6 Further Uses The continual use of these models has the potential to help scientists develop safe and time e cient evacuation procedures. As these models are altered to contain more parameters, they will be capable of simulating more realistic scenarios. These parameters may include number of people, room size, number and size of entrances and exits, number of stairs, escalators, and elevators, number and size of obstacles, amount of security present, possible walking paths, crowd density, and behaviors of the crowd. 7 Conclusion The social force model shows definite signs of collective intelligent behavior in pedestrian dynamics. These behaviors allow for pedestrians to move quickly and smoothly in a variety of di ering situations. These patterns originate from collaborative interactions between pedestrians and occur automatically over time. These models are still being modified and applied for di erent social systems and in di erent settings, specifically those involving evacuation and panic. The overall expectation is that these models will be realistic enough to promote the understanding of collective behaviors. We demonstrated through the simulation of two modified models that we can see collective e ects described in most pedestrian dynamics simulations. 8 References 1. D. Helbing, A. Johansson. Pedestrian, Crowd, and Evacuation Dynamics, Encyclopedia of Complexity and Systems Science 16, pp (Physics and Society, 2013). 2. D. Helbing, P. Molnar, I. J. Farkas, & K. Bolay. Self-Organizing Pedestrian Movement, Environment and Planning B: Planning and Design, 28(3), pp (2001).