Modeling Pedestrian Dynamics on Triangular Grids

Available online at www.sciencedirect.com

ScienceDirect Transportation Research Procedia 2 (2014) 327 – 335

The Pedestrian and and Evacuation Evacuation Dynamics Dynamics 2014 The Conference Conference on in Pedestrian 2014 (PED2014) (PED2014)

Modeling pedestrian dynamics on triangular grids Minjie Chen a,∗, G¨unter B¨arwolff b , Hartmut Schwandt a a MA b MA

6-4, Institut f¨ur Mathematik, Technische Universit¨at Berlin,Straße des 17. Juni 136, 10623 Berlin, Germany 4-5, Institut f¨ur Mathematik, Technische Universit¨at Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany

Abstract We study the modeling of pedestrian dynamics on triangular grids. We consider the cases both with and without orientation. The geometry of a path with orientation may be represented by a special triangular grid with a continuously reducing number of grid cells in the inner side. In addition to the path orientation, a fictional path force will be applied to described the position deviation in this system. In the second case, we consider geometry with no clear orientation. We present here a computation model on the general triangular grid for the step configuration on the local operational level. © 2014 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). c 2014 The Authors. Published by Elsevier B.V.  Peer-review of Department of Transport & Planning Faculty of Civil Engineering and Geosciences Peer-reviewunder underresponsibility responsibility of PED2014. Delft University of Technology

Keywords: pedestrian dynamics; modeling; path-oriented coordinate system; triangular grid; step calculation

1. Introduction Pedestrian dynamics are often simulated on geometrical settings in two dimensions. The modeling technique of pedestrian dynamics can be either macroscopic or microscopic, discrete or continuous (see Daamen et al. (2002) for an overview). In microscopic modeling, pedestrians will be simulated separately and are thus often called “agent”s, with each of them having independent characteristics, sometimes subject to additional behavioral rules Moussa¨ıd et al. (2011). In discrete modeling, the geometry is usually discretized into a two-dimensional grid. A special category of this is the cellular automaton models in which homogene grids are applied and pedestrians as simulation agents are to obey a set of universal rules. Cellular automaton models have also been defined on rectangular grids (for details see Burstedde et al. (2001) and Schadschneider (2002) and Kirchner et al. (2004)) as a special case of homogene grids. Generally speaking, the state transition (or evolution) of the grid positions can be used to describe the dynamics of the system. The grid is not limited to be rectangular. ∗

Corresponding author. Tel.: +49-30-314-28565. E-mail address: [email protected]

2352-1465 © 2014 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of Department of Transport & Planning Faculty of Civil Engineering and Geosciences Delft University of Technology doi:10.1016/j.trpro.2014.09.024

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2. Model 2.1. Path-oriented system Under certain circumstances, rectangular (or homogene) grids prove to be less favourable in the modelling. This is especially the case when pedestrians have only one main moving direction, but this direction cannot be represented or approximated directly by any straight line. This can be often observed when pedestrians walk along a path or corridor. In the current text, this main direction will be referred to as the orientation of the path. Gloor et al. (2004) argued that for the pedestrians walking on a path there is a fictional “path force”, under whose influence the pedestrians will be drawn toward the centre line of the path. Hence, in addition to the fictional axis represented by the orientation, a second axis can be defined by this path force. Consequently, given the orientation of the path and the deviation, a path-oriented coordinate system can be constructed. The position in a path-oriented coordinate system can be decided by the position on the centre line of the path and the deviation from it. Assuming this centre line to be a plane curve S , we can consider the whole area of the path as a tubular neighbourhood of this curve (see Chen et al. (2009) for technical details). In addition, we request that this curve be regular and have unchanged sign of curvature. If the curvature changes the sign, the centre line can be first decomposed into curve segments which meet this requirement, and after discretization, the components of the original curve can be connected to each other to restore the original geometry. 2.1.1. Discretization scheme y (m − 1, m + n − 1) (m + n) · d

x

(m − 1, n − 1) n·d

d

(0, 0)

0

S0 S1

S

Sm

S0 S1

S

Sm

Fig. 1. Discretization in the path-oriented coordinate system and its equivalent representation in the Euclidean coordinate system. The original centre line of the path S may or may not overlap with an equidistant curve.

The path spanned by maximum deviation can now be divided into m stripes, which are separated by m + 1 equidistant curves. For simplicity’s sake, these equidistant curves will be named as S 0 , S 1 , . . . , S m with the index 0, . . . , m starting from the inner boundary of the original path. Obviously, in nontrivial cases (with the curvature of the centre line being non-zero), the equidistant curves in the inner side have shorter lengths. We can intuitively divide the equidistant curve S i (i = 0, . . . , m) by arc length further into n + i parts. This is to say, the innermost curve S 0 of the path will be divided into n parts, and for the outermost curve S m , there will be m + n parts. We approximate the curves S 0 , . . . , S m by straight line segments. By this means, the original path can be represented as a set of triangles. This topology can be reorganized in the Euclidean coordinate system, in which the two triangles bounded by two neighbouring equidistant curves sharing a common edge now form a quadrilateral. We will label these quadrilaterals

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by their x- and y-positions, for x = 0, . . . , m − 1 and y = 0, . . . , n + x. See Fig. 1 for an illustration. In each of these quadrilaterals, if we associate the triangle in the negative direction of x with an additional bit 0, and in the positive direction, an additional bit 1 respectively, the triangles in the path-oriented coordinate system can be easily labelled by 3-tuples. Technically speaking, we may choose m and n independent of each other. However, if we divide the innermost curve S 0 into n segments of equal arc length (written as d), we shall have the choice of a second parameter m that the length m · d can be roughly equal to the width of the original path. In this case, the triangle indexed (x, y, 0) and its neighbour with the index (x, y, 1) or (x, y + 1, 1) can form a quadrilateral which has approximately the shape of a square. The physical length may be further rescaled by the edge length d in the respective axis. In the current text, discrete lengths without unit will be applied. 2.1.2. Position transition in the coordinate system We now consider two types of position transition in the path-oriented coordinate system. These happen in the direction of the orientation of the centre line and in the direction of its deviation. (m + n) · d

(m + n) · d

n·d

(m + n) · d

n·d

(m + n) · d

n·d

n·d

d

d

d

d

0

0

0

0

S0 S1

S

Sm

S0 S1

S

Sm

S0 S1

S

Sm

S0 S1

S

Sm

Fig. 2. Position transition (Fwd) in the primary direction of the path-oriented coordinate system and position transition (Dev+ and Dev− ) in the secondary direction measured as deviation from the centre line of the path. Fwd is expressed in the first two subfigures (counting from the left), the start position has additional bit 1 and 0 respectively; the rest two subfigures refer to Dev+ and Dev− . We notice that Dev− is for (1, 4, 1) and (3, 5, 1) undefined. Concerning (0, 3, 0), (2, 4, 0) and (4, 5, 0) there exist two possibilities for the execution of Dev+ .

Main direction. In this case, the position transition takes place in the same stripe bounded by two neighbouring equidistant curves S i and S i+1 , i = 0, . . . , m − 1. Then obviously we can define the position transition ⎧ ⎪ ⎪ 0), if b = 1, ⎨(x, y, Fwd(x, y, b) = ⎪ (1) ⎪ ⎩(x, y + 1, 1), if b = 0. Alternatively, (1) can be rewritten as ¯ b) ¯ Fwd(x, y, b) = (x, y + b, if we write 1¯ = 0 and 0¯ = 1. See the first two subfigures (from the left) of Fig. 2. The backward position transition can be formulated in an analogous way. Secondary direction. Now we consider the situation that the pedestrian moves exclusively under the influence of the so-called path force, in other words, with deviation only. The third subfigure of Fig. 2 explains the positive deviation from a position with additional bit 1 or the negative deviation from a position with additional bit 0. This can be

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formulated as Dev+ (x, y, 1) = (x + 1, y, 0), Dev− (x, y, 0) = (x − 1, y, 1).

(2)

This case involves the position transition into a neighbouring stripe. Similarly, the last subfigure of Fig. 2 describes the positive deviation from a position with additional bit 0 or the negative deviation from a position with additional bit 1. The position transition takes place in the same stripe, however, we need to pay additional attention to the centre − n+x−1 of the stripe. If n + x is an even number, we leave Dev− (x, n+x 2 , 1) undefined. If n + x is odd, Dev (x, 2 , 0) may n+x−1 n+x+1 take either (x, 2 , 1) or (x, 2 , 1). Otherwise, we define ⎧ ⎪ ⎪ 1), if y < n+x−1 ⎨(x, y, + 2 , Dev (x, y, 0) = ⎪ ⎪ ⎩(x, y + 1, 1), if y ≥ n+x , 2 ⎧ (3) n+x−1 ⎪ ⎪ (x, y, 0), if y ≤ , ⎨ − 2 Dev (x, y, 1) = ⎪ ⎪ ⎩(x, y − 1, 0), if y ≥ n+x+1 . 2 From a technical perspective, a successful position transition according to (1), (2) and (3) shall request that the new position (x, y) and the additional bit be well-defined in the coordinate system, owing to this consideration, some positions will not be involved in the operations of Fwd, Dev+ and Dev− , cf. Fig. 2. The operations of Fwd and Dev will be later called substeps as well. 2.1.3. Effect of the path force Generally speaking, a navigation module of the system should take care of the step configuration (and possibly routing as well) of the pedestrians in the simulation. In our simplified model, we assume that each pedestrian has an ideal, pre-configured walking speed v obeying a Gaussian distribution. Adequate measurements of empirical data (see for example Plaue et al. (2012)) can be carried out to locate system parameters like v. The operation Fwd will be responsible for the projection of v on the grid. On the other hand, to demonstrate the impact of the path force, a secondary component v x of the pedestrian’s temporary velocity can be defined to be proportional to the pedestrian’s deviation from the centre line of the path (with x¯ addressing the x-position of the centre line) v x = ( x¯ − x) · a

(4)

with a constant parameter a. The Dev-operations are associated with v x , whereas the Fwd-operation leads partially to deviation as well, which influences the next Dev-operation through the update of the x-position. 2.2. Triangular grid Sometimes no moving direction of the pedestrians seems to be dominant or can be identified clearly in a scenario. In this case, triangular grids can be given consideration for the modelling. With (irregular) triangular grids, it will also later become possible for hybrid methods in combination with the continuous model Huth et al. (2014) based on finite volume method. 2.2.1. Substep calculation The geometry settings of the simulation or, as in any real-world scenario, the architectural layout of the environment can be Delaunay-triangulated into a triangular grid. This task can be easily performed by the grid generating tool Gmsh Geuzaine and Remacle (2009). The step choice of the simulation agent (pedestrian) can be understood as a sequence of position transitions from neighbouring grid cells sharing a common edge. This kind of elementary position transition will be called substep as well. The simplest strategy for substep is to choose the neighbouring grid cell in the direction of local moving direction. However, this does not take into consideration of the interaction of the agents in the simulation, so unless the density is at a very low level that the pedestrians may walk in straight lines without disturbance, this strategy will not be much useful. In simulation experiments the agents often block the passage among one another and this can result in unnecessary system-wide deadlocks. For a reasonable substep

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Minjie Chen et al. / Transportation Research Procedia 2 (2014) 327 – 335 C

X

D

C

OCBD

E

x O1,x

O2,n

OACE

O1

OABC

O2

B

O O1,n

A

B

O2,x

OBAF

O3,x

O3,n

A O3

n

F

Fig. 3. (a) Four cell positions in a triangular grid, The centroids of the neighbouring triangles (cell positions) are connected. (b) A close-up of the cell position ABC at which a temporary direction x is indicated.

calculation, we shall consider all possible neighbours. As shown in Fig. 3, starting from position ABC, there are three possible direct neighboring positions CBD, ACE and BAF. Let the temporary destination be written as X (and after normalization, x). For the sake of notational simplicity, the neighbouring grid cells CBD, ACE and BAF will be addressed by the subscript 1 , 2 and 3 respectively. In addition, we assume that X is not inside or on any edge of ABC (otherwise the temporary destination would be identical—in discrete sense—with the current grid cell position). The three possible substep choices can be considered as position difference between the centroid of the current grid cell O and the centroids O1 , O2 and O3 of the neighbouring positions respectively, see Fig. 3. These can be expressed as vectorial components in the temporary direction x and its normal direction n: −−−→ −−−−→ −−−−→ OO1 = OO1,x + OO1,n ,

−−−→ −−−−→ −−−−→ OO2 = OO2,x + OO2,n ,

−−−→ −−−−→ −−−−→ OO3 = OO3,x + OO3,n .

(5)

O2

O

O1

x

Δy

X

O3 Δx

Fig. 4. Possible substeps on a square grid. The temporary destination X requests a position transition (Δx, Δy).

For the corresponding position transitions, we now consider the probabilities of the substep choices p1 , p2 , p3 ∈ [0, 1],

(6)

p1 + p2 + p3 = 1.

(7)

Naturally, the deviation from the proposed temporary direction should be avoided or at least reduced whenever possible, so we expect in the normal direction: −−−−→ −−−−→ −−−−→ p1 · OO1,n + p2 · OO2,n + p3 · OO3,n = 0.

(8)

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Next, we study the main direction toward the destination. In nontrivial cases, among the three possible choices, one of −−−→ them has the least angular difference with vector x. Without loss of generality, let this be OO1 , cf. the right subfigure of Fig. 3. Obviously, this position transition should be preferred. To establish a third relationship for p1 , p2 and p3 , we first consider the situation on a square grid (see Chen et al. (2010) for technical details). There we suggested a −−−→ −−−→ −−−→ balancing mechanism among the three substep choices OO1 , OO2 , OO3 with the corresponding probabilities p1 , p2 and p3 :  p1 1 +



 (Δx −

1)2

+ (Δy −

1)2

 + p2 1 +



 (Δx)2

+ (Δy −

1)2

 + p3 1 +



 (Δx −

1)2

+

(Δy)2

=



(Δx)2 + (Δy)2 , (9)

−−−→

cf. Fig. 4. We notice that the rest distance O1 X = (Δx − 1)2 + (Δy − 1)2 is the shortest among the three possible substeps, under certain circumstances—although not always (the reason for this is to form an overall natural trail toward the destination, see the explanation below)—p1 receives more weight. This explains that very often a diagonal −−−→ −−−→ substep choice (in nontrivial cases Δx, Δy  0) should be preferred. Substeps OO2 and OO3 are axis-parallel, by choosing these the agents moves in a more or less “zig-zag” manner on the rectangular grid. The balancing mechanism in Chen et al. (2010) is designed to reduce this unnatural phenomenon, and overall, the individual agent is expected to exhibit a natural trail of all possible position transitions toward a given destination. To achieve this, we need to differ the discrete length of a step choice (that is, the number of the substeps contained in this step in a simulation cycle) from its physical length on the grid. In (9), the term 1 (with three occurrences on the left side of the equation) denotes that all these substeps have a discrete length of 1 (defined in number√of substeps in a simulation cycle). However, we recall that the diagonal substep has an effective physical length of 2 (after being rescaled by the unit length of the grid cells). Hence, the left side of (9) can be rewritten as          −−−→ −−−→ −−−→ p1 β OO1 + (Δx − 1)2 + (Δy − 1)2 + p2 OO2 + (Δx)2 + (Δy − 1)2 + p3 OO3 + (Δx − 1)2 + (Δy)2 , √

the multiplicator β = 22 is introduced to compensate the effective physical length of a diagonal position transition. In analogy, we propose now on the triangular grid    −−−→ −−−−→ −−→ −−−→ −−−−→ −−−→ −−−−→ p1 · β O1 X + OO1,x + p2 · O2 X + OO2,x + p3 · O3 X + OO3,x = OX

(10)

with β being a compensation multiplicator for choosing the neighbouring position with the least angular difference toward destination, defined as

1 3

hA , (|BC| + |CA| + |AB|)

where hA denotes the altitude in ABC through A in length. β can be even further simplified to be assumption that all grid cells in the triangular grid are of similar shapes and sizes.

√2 , 3

under the

Given (7), (8) and (10), p1 , p2 and p3 can be decided. Obviously in case (6) is not satisfied, the solution should be adjusted. For i = 1, 2, 3, if pi < 0, we set pi := 0 and if pi > 1, p := 1. In extreme cases, there could be two p’s no less than 1, these should be manually corrected again. In addition, we recall that the case under study refers to three accessible neighbouring grid cell positions. If there is only one grid cell position accessible, the substep solution is obvious. If there are two, (10) may not need our attention. If no neighbouring grid cell position can be accessed, a substep execution is not possible at the moment. Whether a substep operation can be carried out successfully plays an important role in the system update of the simulation, this will be explained in the next section. The same applies to the Fwd- and Dev-operations in Section 2.1.

Minjie Chen et al. / Transportation Research Procedia 2 (2014) 327 – 335

2.3. Update scheme In our model, the step execution in the simulation cycle is composed of a series of substeps. Substeps are constructed to reflect the original intended position change. With multiple substeps as operations for the agents (pedestrians) in the simulation cycle, it is possible for us to simulate pedestrians with heterogeneous characteristics (for example, walking speed). In other words, pedestrians with different speeds cover different distances in the same time interval, it is obvious that pedestrians will undertake different numbers of substeps as elementary operations. A valid update scheme should take care of the execution of all the elementary operations of all the agents in the simulation smoothly. Unlike on the triangular grid in Section 2.2, in the path-oriented system in Section 2.1 two kinds of operations are involved. The agent has, in addition to the usual forward position change (in form of the Fwd-operations) defined by its speed v, also a deviation movement (4) (in form of the Dev-operations). The magnitude of the latter is dependent of the width of the path and the path force constant a. The agent has v + |v x | (rounded as an integer number) substeps to carry out in a simulation cycle. Let vmax denote the maximum speed of all the agents and v xmax the maximum possible deviation in a simulation cycle. We divide the simulation cycle further into u equal subintervals with u ∈ N, u ≥ vmax + v xmax . The agent will now execute its configured v + |v x | operations in the u subintervals. The next code fragment describes how the substep operations of the pedestrians can be executed within a simulation cycle. procedure operate: parameter: a collection of agents mark all agents as “unprocessed”; i ⇐ u; while i ≥ 1 do process all agents marked “unprocessed” by calling procedure operate agent sequentially with parameter i; i ⇐ i − 1; enddo return

The next update scheme processes the agents separately: procedure operate agent: parameter: agent, i if destination reached mark as “processed”; else count (in the current simulation cycle) the successful operations in the main direction s and in the secondary direction s x ; if v−s+vi x −sx ≤ 1; execute an operation in the main direction with probability v−s ; i execute an operation in the secondary direction with probability fi fi

v x −s x ; i

return

The parameter i serves as a time stamp. In the above procedure operate agent, unsuccessful substeps in the current simulation cycle contribute to a higher probability that a substep should be attempted again (since v, v x , s and s x remain unchanged but i is decremented). This reduces the occurrence of conflict notably. In the procedure operate agent, if there is a reason for a pedestrian to apply an operation in a subinterval in the simulation cycle, that is,

333

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Minjie Chen et al. / Transportation Research Procedia 2 (2014) 327 – 335 v−s+v x −s x i

≤ 1, we should still decide whether this is to take place in the main or the secondary direction. Therefore, v−s = v−s the operation in the main direction will be carried out with a probability for v−s+vi x −sx · v−s+v i , and naturally x −s x v x −s x for the operation in the secondary direction, i . Since the pedestrians in the simulation are not expected to have a homogeneous velocity, the substep execution in the procedure operate agent varies among the agents substantially. This should be considered as an advantage in the modelling, since the likelihood of a local conflict among pedestrians in their substeps will be clearly reduced. In this sense, our model is completely different from the usual cellular automaton models. The main concern in our model is the execution of the substeps as elementary operations on the lowest simulation level, whereas the step conflict solution must be provided on higher levels of the simulation (in that in every simulation step a well-calculated temporary destination must be provided for every agent). The actual conflict in substep execution helps us to decide how these substeps are to be performed in order that the pre-configured step choices can be most faithfully carried out. 3. Preliminary results

Fig. 5. Simulation examples of path-oriented coordinate system.

In Fig. 5 we present some test results respecting Section 2.1. Pedestrians were simulated to start from the lower border of the grid and moved in the direction of the orientation of the centre line of the path, and finally they left the scenario at the upper-left corner. Counting from the left, the first three subfigures show the cases of low, middle and high entrance rates of the pedestrians, with no path force present. In every case, the pedestrians tend to group into stripes after somehow a stabilization of the system. The other three subfigures combine the entrance rate with path force: 4) low entrance rate, path force in the middle range, 5) middle entrance rate, path force in the low range, 6) high entrance rate, maximum path force (that is, the agent always strove to get to the centre of the path; at the same time, when this was impossible—owing to the high density in the system—agents still moved forward at a slightly reduced average speed). Currently we are still missing real-world empirical data for the calibration of the system parameters. However, in the simulation examples, patterns resembling real-world observation can be roughly identified. Fig. 6 refers to the simulation (respecting Section 2.2) of an experiment carried out during the Lange Nacht der Wissenschaften (German: Long Night of the Sciences) 2010 at Technische Universit¨at Berlin. In the experiment, volunteer visitors of the event were given instructions to demonstrate the intersection of two groups of pedestrians. The left subfigure shows the triangular grid of the location according to architectural plan. The simulated trajectories of the pedestrians in two groups are presented in the right subfigure. Qualitative study of this experiment via for example, the density estimator introduced in Plaue et al. (2011), is currently being prepared. Acknowledgement The authors gratefully acknowledge the support of Federal Ministry for Economic Affairs and Energy of Germany for the project VP2653402RR1 and federal state of Berlin/Investitionsbank Berlin for the project 10153525.

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Fig. 6. A simulation example of two intersecting pedestrian groups.

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