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The flow of large crowds of pedestrians
Mathematics and Computers in Simulation 53 (2000) 367–370
The flow of large crowds of pedestrians R.L. Hughes Department of Civil and Environmental Engineering, The University of Melbourne, Parkville, Victoria, Australia Accepted 28 July 2000
Abstract Despite popular belief the motion of a crowd is governed by well-defined rules of behaviour. These rules imply a set of coupled, non-linear, partial differential equations for the density and velocity potential for each type of pedestrian in the crowd. As may be expected, the solution of these equations may, in different regions of space, be supercritical or subcritical with the possibility of a shock wave separating the regions. Less predictable is the remarkable finding that these coupled, non-linear, time dependent equations are conformally mappable and this finding enables solutions to be obtained easily for both supercritical and subcritical flows. © 2000 Published by Elsevier Science B.V. on behalf of IMACS. Keywords: Eulerian simulation; Lagrangian simulation; Pedestrians
1. Introduction Developing an understanding of the behaviour of a moving crowd of pedestrians is an important but largely neglected area of science and engineering. To an engineer there are two fundamentally distinct methods of modelling crowd motion. The first method (Lagrangian simulation) involves direct simulation. Each model pedestrian is given distinct properties and is followed throughout the domain. The second method (Eulerian simulation) is to grid the region of interest and study the flow without regard to specific model pedestrians. Individual model pedestrians are not followed, instead the number of model pedestrians in each grid box is studied. Both methods have their place and should be seen as complementing each other. However, neither method can be considered to be scientifically acceptable as both methods require a preliminary specification of the path taken by pedestrians and this is given using the modellers intuition. The present study uses observed rules of pedestrian behaviour, which formulates as hypotherses, to obtain a rational method for determining the path taken by pedestrians and the associated pedestrian density. Here, the determination of the path requires integration of partial differential equations. However, in choosing a path an actual pedestrian generally does not do a calculation based on these or any other equations. An actual pedestrian draws upon experience of similar situations to choose a path. The results of this experience can be simulated by calcualtion. E-mail address: [email protected] (R.L. Hughes). 0378-4754/00/$20.00 © 2000 Published by Elsevier Science B.V. on behalf of IMACS. PII: S 0 3 7 8 - 4 7 5 4 ( 0 0 ) 0 0 2 2 8 - 7
368
R.L. Hughes / Mathematics and Computers in Simulation 53 (2000) 367–370
2. Formulation The present study seeks to rectify the problem of the choice of path by using a continuum model based on well-defined observations of pedestrian behaviour, referred to here as hypotheses. For a single type of pedestrian the hypotheses are as follow Hypothesis 1. It states that the speed, f , at which pedestrians walk is determined solely by the surrounding pedestrian density and the behavioural characteristics of the pedestrians, i.e. the velocity components (u, v) are given by u = f (ρ)φˆ x ,
v = f (ρ)φˆ y
(1)
where φˆ x and φˆ y are the direction cosines of the motion. This hypothesis is standard. Hypothesis 2. It states that pedestrians have a common sense of the task (called potential) that they face to reach their common destination such that any two individuals at different locations having the same potential would see no advantage to either in changing places. There is no perceived advantage to a pedestrian of moving along a line of constant potential. Thus the motion of any pedestrian is in the direction perpendicular to the potential, i.e. in the direction for which the direction cosines are −(∂φ/∂x) φˆ x = p (∂φ/∂x)2 + (∂φ/∂y)2
−(∂φ/∂y) φˆ y = p , (∂φ/∂x)2 + (∂φ/∂y)2
(2)
where φ is the potential. This hypothesis is not appropriate to vehicular traffic but appears to be applicable to pedestrian flows where pedestrians can visually assess the situation. Hypothesis 3. It states that pedestrians seek to minimize their (accurately) estimated travel time, but temper this behaviour to avoid extremely high densities. As two pedestrians on a given potential must both be at the same new potential as each other at some later time (noting time is a measure of potential by Hypothesis 3), the distance between potentials must be proportional to pedestrian speed irrespective of the path followed by a pedestrian. Thus we write p 1 p = u2 + v 2 (∂φ/∂x)2 + (∂φ/∂y)2
(3)
where φ has been scaled appropriately. Eqs. (1)–(3) and the usual equation of continuity combine to form the governing equations for pedestrian flow ∂φ ∂φ ∂ ∂ ∂ρ 2 2 + ρg(ρ)f (ρ) + ρg(ρ)f (ρ) =0 (4) − ∂t ∂x ∂x ∂y ∂y and 1 g(ρ)f (ρ) = p , (∂φ/∂x)2 + (∂φ/∂y)2
(5)
R.L. Hughes / Mathematics and Computers in Simulation 53 (2000) 367–370
369
where the factor g(ρ) accounts for the tempering behaviour at high densities as referred to in Hypothesis 3. This formulation can be easily extended to crowds involving multiple pedestrian types. Observations of pedestrian motion suggest that f (ρ) and g(ρ) can be approximated by A, ρ ≤ ρtrans r A ρtrans , ρtrans < ρ ≤ ρcrit (6) f (ρ) = ρ r ρtrans ρcrit (ρmax − ρ) , ρcrit < ρ ≤ ρmax A ρ 2 (ρmax − ρcrit ) and
ρ ≤ ρcrit 1, g(ρ) = ρ(ρmax − ρcrit ) , ρcrit < ρ ≤ ρmax ρcrit (ρmax − ρ)
(7)
where A = 1.4 ms−1 , ρtrans = 0.8 ms−2 , ρcrit = 3.0 ms−2 , and ρmax = 5.0 ms−2 typically. The detailed forms of Eqs. (6) and (7) depend on the properties of the pedestrians being studied. However, the forms given are close to the behaviour observed for most crowds and are mathematically convenient. At low densities, less than ρtrans , the speed of pedestrians is constant. For these densities, the speed is limited by the ability of pedestrians to move there limbs quickly. For densities greater than ρtrans , the speed of pedestrians is limited by fear of collision, and interference between pedestrians occurs. The speed of pedestrians decreases with in increased density until pedlock occurs at a density of ρmax . Flows of pedestrians at conditions near pedlock are often frightening to those involved and hence evasive action is taken by many pedestrians, thereby increasing the function g(ρ) above unity. Consideration of the form of Eq. (6) shows that while f (ρ) is continuous, f 0 (ρ) is not continuous at ρ = ρtrans , or ρ = ρcrit . Avoiding issues associated with the behaviour at ρ = ρcrit , it is clear that (ρf (ρ))0 is positive for ρ < ρcrit and negative for ρ > ρcrit . Thus following [1], disturbances are swept downstream for ρ < ρcrit but propogate upstream for ρ > ρcrit . Critical conditions therefore occur when ρ = ρcrit .
3. A solution method An interesting method of solving Eqs. (4) and (5) is conformal mapping. Despite the time dependent nature of the formulation of Eq. (4) and its non-linearity, Eqs. (4) and (5) are conformally mappable. To understand the application of conformal mapping it is necessary to consider the behaviour for ρ ≤ ρcrit , ρtrans < ρ ≤ ρcrit and ρcrit < ρ ≤ ρmax separately. For ρ ≤ ρcrit , pedestrians walk with a constant speed irrespective of density. They walk in straight lines between their origin and destination, only changing direction when forced to do so by boundary geometry. The path taken is the shortest path. For ρtrans < ρ ≤ ρcrit , Eqs. (4) and (5) are conformally mappable as formulated in terms of φ and ρ. Under conformal mapping φ remains unchanged but ρ scales as the square of the Jacobian of the map. By Eq. (5), the speed of pedestrians, f , also scales but as the inverse of the Jacobian.
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R.L. Hughes / Mathematics and Computers in Simulation 53 (2000) 367–370
For ρcrit < ρ ≤ ρmax , Eqs. (4) and (5) are again conformally mappable. Under conformal mapping φ remains unchanged but (ρmax − ρ) scales as the Jacobian of the map. For these densities, the motion of ‘holes’ in the crowd are of more physical significance than the motion of pedestrians in the crowd and so the scaling of (ρmax − ρ) rather than ρ is not surprising. The application of this solution technique is straightforward. The only difficulty involves the choice of boundary conditions. For example, in studying the motion of pilgrims over the Jamarat bridge near Mecca, the attitude of pilgrims to their objective is critical. Pilgrims are required to stone three pillars in turn. If pilgrims are assumed to set as their objective, the stoning of the next pillar then a crowd of pilgrims is predicted to develop in front of each pillar, with low density on the sides of each pillar. However, if pilgrims were to see the leaving of the whole site as their objective, there would be a low density in front of each pillar and high density at the sides as pilgrims, positioning themselves to move to the next pillar, throw their stones as they walk past and route to the next pillar. In practice the former behaviour is observed. 4. Extensions There are two important extensions to this study that need to be noted. Firstly, it has been assumed that the speed of pedestrians is only a function of the density of pedestrians. Many situations exist where the speed, f , also depends on position because of non-uniformity of the surface on which the pedestrians walk. In such cases Eqs. (4) and (5) are still correct, but they are no longer conformally mappable. Secondly, in many cases more than one type of pedestrian are involved. In such cases Eqs. (4) and (5) hold for each type of pedestrian with two equations for each pedestrian type. Surprisingly, it is well established from observations [2] that the only change in the speed f (ρ) is that the ρ now refers to the total density of pedestrians, not the density of each type of pedestrian. This unexpected behaviour results from the way pedestrian crowds walk through each other. 5. Conclusions Despite popular belief the behaviour of pedestrians is rational and easily formulated mathematically. The resulting equations have been shown to be highly non-linear. However, despite this nonlinearity and possible time dependence, the equations have the remarkable property of being conformally mappable. The greatest difficulty in applying this formulation involves the appropriate choice of boundary conditions to match the psychological state of the pedestrians. As shown earlier, the psychological state of pedestrians can completely change the flow pattern, as illustrated by a case study of flow over the Jamarat bridge. There is great scope for improving the safety of pedestrians at many major events. As the location of pedestrian accidents can often be accurately predicted, those changes required for safety can be implemented inexpensively. Fascinating problems involving the prevention of major accidents await anyone who chooses to persue a study of these crowds. References [1] M.J. Lighthill, G.B. Whitham, On kinematic waves: I flood movement in long rivers; II theory of traffic flow on long crowded roads, Proc. R. Soc. A 229 (1955) 281–345. [2] T. Ando, H. Ota, T. Oki, Forecasting the flow of people, Railway Res. Rev. 45 (1988) 8–14 (in Japanese).
The flow of large crowds of pedestrians R.L. Hughes Department of Civil and Environmental Engineering, The University of Melbourne, Parkville, Victoria, Australia Accepted 28 July 2000
Abstract Despite popular belief the motion of a crowd is governed by well-defined rules of behaviour. These rules imply a set of coupled, non-linear, partial differential equations for the density and velocity potential for each type of pedestrian in the crowd. As may be expected, the solution of these equations may, in different regions of space, be supercritical or subcritical with the possibility of a shock wave separating the regions. Less predictable is the remarkable finding that these coupled, non-linear, time dependent equations are conformally mappable and this finding enables solutions to be obtained easily for both supercritical and subcritical flows. © 2000 Published by Elsevier Science B.V. on behalf of IMACS. Keywords: Eulerian simulation; Lagrangian simulation; Pedestrians
1. Introduction Developing an understanding of the behaviour of a moving crowd of pedestrians is an important but largely neglected area of science and engineering. To an engineer there are two fundamentally distinct methods of modelling crowd motion. The first method (Lagrangian simulation) involves direct simulation. Each model pedestrian is given distinct properties and is followed throughout the domain. The second method (Eulerian simulation) is to grid the region of interest and study the flow without regard to specific model pedestrians. Individual model pedestrians are not followed, instead the number of model pedestrians in each grid box is studied. Both methods have their place and should be seen as complementing each other. However, neither method can be considered to be scientifically acceptable as both methods require a preliminary specification of the path taken by pedestrians and this is given using the modellers intuition. The present study uses observed rules of pedestrian behaviour, which formulates as hypotherses, to obtain a rational method for determining the path taken by pedestrians and the associated pedestrian density. Here, the determination of the path requires integration of partial differential equations. However, in choosing a path an actual pedestrian generally does not do a calculation based on these or any other equations. An actual pedestrian draws upon experience of similar situations to choose a path. The results of this experience can be simulated by calcualtion. E-mail address: [email protected] (R.L. Hughes). 0378-4754/00/$20.00 © 2000 Published by Elsevier Science B.V. on behalf of IMACS. PII: S 0 3 7 8 - 4 7 5 4 ( 0 0 ) 0 0 2 2 8 - 7
368
R.L. Hughes / Mathematics and Computers in Simulation 53 (2000) 367–370
2. Formulation The present study seeks to rectify the problem of the choice of path by using a continuum model based on well-defined observations of pedestrian behaviour, referred to here as hypotheses. For a single type of pedestrian the hypotheses are as follow Hypothesis 1. It states that the speed, f , at which pedestrians walk is determined solely by the surrounding pedestrian density and the behavioural characteristics of the pedestrians, i.e. the velocity components (u, v) are given by u = f (ρ)φˆ x ,
v = f (ρ)φˆ y
(1)
where φˆ x and φˆ y are the direction cosines of the motion. This hypothesis is standard. Hypothesis 2. It states that pedestrians have a common sense of the task (called potential) that they face to reach their common destination such that any two individuals at different locations having the same potential would see no advantage to either in changing places. There is no perceived advantage to a pedestrian of moving along a line of constant potential. Thus the motion of any pedestrian is in the direction perpendicular to the potential, i.e. in the direction for which the direction cosines are −(∂φ/∂x) φˆ x = p (∂φ/∂x)2 + (∂φ/∂y)2
−(∂φ/∂y) φˆ y = p , (∂φ/∂x)2 + (∂φ/∂y)2
(2)
where φ is the potential. This hypothesis is not appropriate to vehicular traffic but appears to be applicable to pedestrian flows where pedestrians can visually assess the situation. Hypothesis 3. It states that pedestrians seek to minimize their (accurately) estimated travel time, but temper this behaviour to avoid extremely high densities. As two pedestrians on a given potential must both be at the same new potential as each other at some later time (noting time is a measure of potential by Hypothesis 3), the distance between potentials must be proportional to pedestrian speed irrespective of the path followed by a pedestrian. Thus we write p 1 p = u2 + v 2 (∂φ/∂x)2 + (∂φ/∂y)2
(3)
where φ has been scaled appropriately. Eqs. (1)–(3) and the usual equation of continuity combine to form the governing equations for pedestrian flow ∂φ ∂φ ∂ ∂ ∂ρ 2 2 + ρg(ρ)f (ρ) + ρg(ρ)f (ρ) =0 (4) − ∂t ∂x ∂x ∂y ∂y and 1 g(ρ)f (ρ) = p , (∂φ/∂x)2 + (∂φ/∂y)2
(5)
R.L. Hughes / Mathematics and Computers in Simulation 53 (2000) 367–370
369
where the factor g(ρ) accounts for the tempering behaviour at high densities as referred to in Hypothesis 3. This formulation can be easily extended to crowds involving multiple pedestrian types. Observations of pedestrian motion suggest that f (ρ) and g(ρ) can be approximated by A, ρ ≤ ρtrans r A ρtrans , ρtrans < ρ ≤ ρcrit (6) f (ρ) = ρ r ρtrans ρcrit (ρmax − ρ) , ρcrit < ρ ≤ ρmax A ρ 2 (ρmax − ρcrit ) and
ρ ≤ ρcrit 1, g(ρ) = ρ(ρmax − ρcrit ) , ρcrit < ρ ≤ ρmax ρcrit (ρmax − ρ)
(7)
where A = 1.4 ms−1 , ρtrans = 0.8 ms−2 , ρcrit = 3.0 ms−2 , and ρmax = 5.0 ms−2 typically. The detailed forms of Eqs. (6) and (7) depend on the properties of the pedestrians being studied. However, the forms given are close to the behaviour observed for most crowds and are mathematically convenient. At low densities, less than ρtrans , the speed of pedestrians is constant. For these densities, the speed is limited by the ability of pedestrians to move there limbs quickly. For densities greater than ρtrans , the speed of pedestrians is limited by fear of collision, and interference between pedestrians occurs. The speed of pedestrians decreases with in increased density until pedlock occurs at a density of ρmax . Flows of pedestrians at conditions near pedlock are often frightening to those involved and hence evasive action is taken by many pedestrians, thereby increasing the function g(ρ) above unity. Consideration of the form of Eq. (6) shows that while f (ρ) is continuous, f 0 (ρ) is not continuous at ρ = ρtrans , or ρ = ρcrit . Avoiding issues associated with the behaviour at ρ = ρcrit , it is clear that (ρf (ρ))0 is positive for ρ < ρcrit and negative for ρ > ρcrit . Thus following [1], disturbances are swept downstream for ρ < ρcrit but propogate upstream for ρ > ρcrit . Critical conditions therefore occur when ρ = ρcrit .
3. A solution method An interesting method of solving Eqs. (4) and (5) is conformal mapping. Despite the time dependent nature of the formulation of Eq. (4) and its non-linearity, Eqs. (4) and (5) are conformally mappable. To understand the application of conformal mapping it is necessary to consider the behaviour for ρ ≤ ρcrit , ρtrans < ρ ≤ ρcrit and ρcrit < ρ ≤ ρmax separately. For ρ ≤ ρcrit , pedestrians walk with a constant speed irrespective of density. They walk in straight lines between their origin and destination, only changing direction when forced to do so by boundary geometry. The path taken is the shortest path. For ρtrans < ρ ≤ ρcrit , Eqs. (4) and (5) are conformally mappable as formulated in terms of φ and ρ. Under conformal mapping φ remains unchanged but ρ scales as the square of the Jacobian of the map. By Eq. (5), the speed of pedestrians, f , also scales but as the inverse of the Jacobian.
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R.L. Hughes / Mathematics and Computers in Simulation 53 (2000) 367–370
For ρcrit < ρ ≤ ρmax , Eqs. (4) and (5) are again conformally mappable. Under conformal mapping φ remains unchanged but (ρmax − ρ) scales as the Jacobian of the map. For these densities, the motion of ‘holes’ in the crowd are of more physical significance than the motion of pedestrians in the crowd and so the scaling of (ρmax − ρ) rather than ρ is not surprising. The application of this solution technique is straightforward. The only difficulty involves the choice of boundary conditions. For example, in studying the motion of pilgrims over the Jamarat bridge near Mecca, the attitude of pilgrims to their objective is critical. Pilgrims are required to stone three pillars in turn. If pilgrims are assumed to set as their objective, the stoning of the next pillar then a crowd of pilgrims is predicted to develop in front of each pillar, with low density on the sides of each pillar. However, if pilgrims were to see the leaving of the whole site as their objective, there would be a low density in front of each pillar and high density at the sides as pilgrims, positioning themselves to move to the next pillar, throw their stones as they walk past and route to the next pillar. In practice the former behaviour is observed. 4. Extensions There are two important extensions to this study that need to be noted. Firstly, it has been assumed that the speed of pedestrians is only a function of the density of pedestrians. Many situations exist where the speed, f , also depends on position because of non-uniformity of the surface on which the pedestrians walk. In such cases Eqs. (4) and (5) are still correct, but they are no longer conformally mappable. Secondly, in many cases more than one type of pedestrian are involved. In such cases Eqs. (4) and (5) hold for each type of pedestrian with two equations for each pedestrian type. Surprisingly, it is well established from observations [2] that the only change in the speed f (ρ) is that the ρ now refers to the total density of pedestrians, not the density of each type of pedestrian. This unexpected behaviour results from the way pedestrian crowds walk through each other. 5. Conclusions Despite popular belief the behaviour of pedestrians is rational and easily formulated mathematically. The resulting equations have been shown to be highly non-linear. However, despite this nonlinearity and possible time dependence, the equations have the remarkable property of being conformally mappable. The greatest difficulty in applying this formulation involves the appropriate choice of boundary conditions to match the psychological state of the pedestrians. As shown earlier, the psychological state of pedestrians can completely change the flow pattern, as illustrated by a case study of flow over the Jamarat bridge. There is great scope for improving the safety of pedestrians at many major events. As the location of pedestrian accidents can often be accurately predicted, those changes required for safety can be implemented inexpensively. Fascinating problems involving the prevention of major accidents await anyone who chooses to persue a study of these crowds. References [1] M.J. Lighthill, G.B. Whitham, On kinematic waves: I flood movement in long rivers; II theory of traffic flow on long crowded roads, Proc. R. Soc. A 229 (1955) 281–345. [2] T. Ando, H. Ota, T. Oki, Forecasting the flow of people, Railway Res. Rev. 45 (1988) 8–14 (in Japanese).