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Transport Reviews
ISSN: 0144-1647 (Print) 1464-5327 (Online) Journal homepage: http://www.tandfonline.com/loi/ttrv20
Social force models for pedestrian traffic – state of
the art
Xu Chen, Martin Treiber, Venkatesan Kanagaraj & Haiying Li
To cite this article: Xu Chen, Martin Treiber, Venkatesan Kanagaraj & Haiying Li (2017):
Social force models for pedestrian traffic – state of the art, Transport Reviews, DOI:
10.1080/01441647.2017.1396265
To link to this article: https://doi.org/10.1080/01441647.2017.1396265
Published online: 05 Nov 2017.
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Social force models for pedestrian traffic –state of the art
Xu Chen
a,b
, Martin Treiber
c
, Venkatesan Kanagaraj
c
and Haiying Li
a
a
State Key Lab of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, People’s Republic of
China;
b
School of Traffic and Transportation, Beijing Jiaotong University, Beijing, People’s Republic of China;
c
Institute of Transport and Economics, Technology University of Dresden, Dresden, Germany
ABSTRACT
Pedestrian simulation plays an important role in the fields of
transport station management, building evacuation and safety
management of large public events. Among the continuous
pedestrian flow models, the social force model is one of the most
widespread and supports all of the above use cases. Since its
initial proposal by Helbing and Molnar [(1995). Social force model
for pedestrian dynamics. Physical Review E,51, 4282], many
improvements of the social force model have been put forward
for solving various, but mostly specific, problems. However, an up-
to-date and essentially comprehensive review bringing all the
model variants into a common context is missing. In this paper,
we propose such a framework in terms of assessment criteria for
pedestrian models considering pedestrian attributes, motion base
cases, self-organisation phenomena and some special cases.
Starting with the initial version of Helbing and Molnar [(1995).
Social force model for pedestrian dynamics. Physical Review E, 51,
4282] and the escape panic version of Helbing, Farkas, and Vicsek
[(2000a). Simulating dynamical features of escape panic. Nature,
407, 487–490], we classify the improvements and assess their
degree of the improvements. Further discussion is presented from
the perspectives of description ability, parameter calibration and
flexible application in a complex environment.
ARTICLE HISTORY
Received 26 April 2017
Accepted 15 October 2017
KEYWORDS
Social force model;
pedestrian attributes; motion
base cases; self-organisation
phenomena; special cases
1. Introduction
With the improvement of computer operation efficiency and research of pedestrian traffic
theory, pedestrian simulation has become an important means in operation management
of transport stations (Chen, Li, Miao, Jiang, & Jiang, 2017), building evacuation (Shiwakoti,
Tay, Stasinopoulos, & Woolley, 2017) and safety assurance of large pedestrian flow events
(Wang, Zhang, Cai, Zhang, & Ma, 2013). In the pedestrian simulation, the most critical ques-
tion is the research of pedestrian dynamics. Early studies are mainly based on the empirical
analysis, including the measurement of pedestrian velocity (Tanaboriboon, Hwa, & Chor,
1986), calibration of the fundamental diagram (Fruin, 1971; Garbrecht, 1973; Navin &
Wheeler, 1969; Older, 1968) and division of the service level (Mōri & Tsukaguchi, 1987;
Polus, Schofer, & Ushpiz, 1983). Above-mentioned studies are confined to the static
© 2017 Informa UK Limited, trading as Taylor & Francis Group
CONTACT Xu Chen 14114236@bjtu.edu.cn State Key Lab of Rail Traffic Control and Safety, Beijing Jiaotong
University, Beijing 100044, People’s Republic of China; School of Traffic and Transportation, Beijing Jiaotong University,
Beijing 100044, People’s Republic of China
TRANSPORT REVIEWS, 2017
https://doi.org/10.1080/01441647.2017.1396265
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environment and are difficult to be applied to the pedestrian flow prediction in complex
environments. Therefore, a series of pedestrian dynamics models such as queuing model
(Yuhaski & Smith, 1989), transfer matrix model (Morlok, 1978) and stochastic model
(Ashford, O’Leary, & McGinity, 1976) are proposed. These models take the temporal and
spatial aspects of pedestrians into account, while the self-organisation phenomena in
the crowd are not considered. Henderson (1974) proposed a macroscopic method of mod-
elling the temporal and spatial organisation patterns of the movement, and comparing the
pedestrian population to gas or fluid. This approach is mainly modelled from the perspec-
tive of crowds, which lacks pedestrian heterogeneity and individual behaviours. Hence,
individual-based modelling has gradually become the research hotspot, which is the so-
called pedestrian microscopic simulation model.
From the perspective of space continuity, pedestrian microscopic simulation models can
be divided into continuous models and discrete models. The former include the magnetic
force model (Okazaki & Matsushita, 1993), utility maximisation model (Helbing, 1991)and
social force model (Helbing & Molnar, 1995), the latter include the benefit–cost model
(Gipps & Marksjö, 1985) and cellular automata model (Blue & Adler, 1998). Among these
models, social force model and cellular model are the most popular and have been improved
by lots of researchers to adapt to different environments or to achieve more accurate results.
In addition, many other models emerged in recent years. Hoogendoorn and Bovy (2003)pro-
posed the walking consumption-based model, Robin, Antonini, Bierlaire, and Cruz (2009)and
Liu, Lo, Ma, and Wang (2014) proposed the utility-based discrete choice models, Asano, Iryo,
and Kuwahara (2010) and Guo and Huang (2012) proposed the walking speed-based models,
Zhou, Dong, Wang, Wang, and Yang(2016) and Fu, Song, and Lo(2016) proposed the generic
fuzzy system-based models, Xiao, Gao, Qu, and Li (2016) proposed the local density-based
model, Ma, Lee, and Yuen (2016) proposed the ANN-based model, Fang et al. (2012)and
von Sivers and Köster (2015) proposed the stride length-based models, Lee, Choi, Hong,
and Lee (2007) proposed the data-driven model and so forth. Generally speaking, the research
of pedestrian simulation model receives enough attention. Aiming at these models, research-
ers try to make some reviews so as to provide some insights for future research.
Papadimitriou, Yannis, and Golias (2009) assessed the existing researches on pedestrian
behaviour in urban areas, thereby providing a very comprehensive review of pedestrian
movement models. Schadschneider, Klüpfel, Kretz, Rogsch, and Seyfried (2009) reviewed
numerous models and classified them, which mainly discussed the features of the respect-
ive models with respect to simulating motion in general. Helbing and Johansson (2011)
made a brief review on the social force model, while they paid more attention on the
self-organisation phenomena at the levels of pedestrian, crowd and evacuation rather
than the research history of the social force model. Bellomo, Piccoli, and Tosin (2012)
classified the pedestrian dynamics models from the perspective of model attributes.
Duives, Daamen, and Hoogendoorn (2013) made a relatively comprehensive review of
the crowd motion models, and assessed them from the perspectives of motion-base
cases, self-organisation phenomena and model applicability. Research results provide
some positive insights for future research and the social force model is proved to be
one of the best models. However, they focused on the comparison of different models,
concrete contents and improvements of the models are not presented. Caramuta et al.
(2017) made a review covering the walking pedestrian detection technologies, walking
dynamics models and pedestrian simulation software.
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To sum up, after 20 years of development, social force model has become one of the
most widely used models for simple mathematical formulas and a good ability of describ-
ing movement process. Although lots of improvements have been done, an up-to-date
and essentially comprehensive review is missing for summarising the existing researches
and providing new insights for further research. Therefore, we propose such a framework
in terms of assessment criteria for pedestrian models considering pedestrian attributes,
motion base cases, self-organisation phenomena and some special cases. Beginning
with the initial version of Helbing and Molnar (1995) and escape panic version of
Helbing et al. (2000a), we classify the improvements and assess their degree of the
improvements. Subsequently, we discuss the description ability (i.e. the ability of describ-
ing pedestrian movement process and phenomena), parameter calibration and flexible
application in a complex environment so as to provide some new insights for further
research.
The main contributions of this paper are as follows: (1) proposing an assessment criteria
for pedestrian simulation models, considering the pedestrian attributes, crowd and indi-
vidual motion-base cases, self-organisation phenomena and special cases; (2) making a
review of the social force model, and researchers can learn about the state-of-the-art
easily; (3) providing some insights for the further research of social force model and
other pedestrian dynamics models.
The paper structure is as follows. Section 2 briefly elaborates upon the research meth-
odology and Section 3 describes the assessment criteria. The origin of the social force
model is introduced in Section 4 and the improvements are presented in Section
5. Further discussion is presented in Section 6, and conclusions and future research are
exposed in Section 7.
2. Research methodology
Research methodology of this paper is illustrated in Figure 1. Aiming at the review of social
force model, four questions are raised in sequence (i.e. Q1–Q4 in Figure 1), and corre-
sponding solutions are proposed in Section 3–Section 6. The arrows at the top of the
figure represent the relationships among different sections.
Figure 1. Research methodology.
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With respect to the improvements of the social force model in Section 5, some special
rules are developed:
(1) The information about all of the improvements used in this paper is taken from the
papers describing the respective models. We assume that the descriptions of the
improvements presented in papers agree with the models’implementation and
results;
(2) For a problem, some continuous improvements may be done by the same research
group, and the latest version is adopted;
(3) Different improvements may be made for the same problem, and all of them will be
presented and followed by a discussion;
(4) Parameters with the same meaning may be represented by different letters in differ-
ent papers, and they are unified to avoid ambiguity;
(5) Different papers may use the same letters to represent different parameters, so some
letters are replaced by others to avoid ambiguity;
(6) The value of the parameters will not be listed here and more details can refer to the
corresponding paper;
(7) There are lots of parameters in this paper. For convenience, they are listed in the
Appendix according to the order in which they appear.
3. Assessment criteria
In this section, the full contents of pedestrian attributes, motion base cases, self-organis-
ation phenomena and special environment are introduced in sequence.
3.1. Pedestrian attributes
Pedestrian attributes are inputs of pedestrian dynamics model, which can be divided
into physical attributes and social attributes. Physical attributes include age,
height, shape (i.e. circle, ellipse or others), gender, ability, temper, visual range, mass
and walking speed. Social attributes include profession, aim, familiarity, luggage and
group.
These attributes directly or indirectly affect the pedestrian movement in different ways.
All above-mentioned variables are taken into account for a limited degree of influence. In
social force model, pedestrian attributes including shape, mass, walking speed and group
are considered.
3.2. Motion base cases
Motion base cases are the basic movement forms that the pedestrian shows in the
environment. The motion base cases combined cover the whole range of pedestrian
movement behaviour. Duives et al. (2013) proposed eight kinds of crowd move-
ment base cases. Based on this, the individual motion base cases are supplemented
in this paper, including avoiding, self-stopping, following and overtaking. Figure 2
shows the eight crowd movement base cases and four individual movement base
cases.
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3.3. Self-organisation phenomena
Self-organisation phenomenon is a result of non-linear interactions between many objects
or subjects without the intervention of external influences, and it often causes different
kinds of spatial–temporal patterns of motion (Camazine, 2003). Regarding to the self-
organisation phenomena, different researchers have different classification methods.
Duives et al. (2013) considered six kinds of self-organisation phenomena according to
the place of occurrence (i.e. normal situation, Jamarat bridge and bottleneck). Helbing
and Johansson (2011) divided them into two categories (i.e. crowd and evacuation)
according to the pedestrian density, and 10 kinds of self-organisation phenomena are
considered.
Overall, these researches provide some insights for the classification of the self-organ-
isation phenomena, while there are still some questions: (1) none of them consider the all
known 11 kinds of self-organisation phenomena; (2) in the research of Helbing and
Johansson (2011), the self-organisation phenomena in each category can be further
classified.
Combining with the above researches, a new classification method of 11 self-organis-
ation phenomena is proposed. Specially, the generation conditions and effects of all self-
organisation phenomena are summarised (see Table 1).
3.4. Special cases
Except for the above-mentioned aspects, there are still some special cases should be con-
sidered, such as the pedestrian movement on stairs and emergency evacuation (i.e. con-
sidering the change of desired speed and the effect of evacuation guiders).
Figure 2. Crowd and individual motion base cases.
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4. Starting point of the review
In this section, the initial version of Helbing and Molnar (1995) and the escape panic
version of Helbing et al. (2000a) are presented as the starting point of the review. Social
force models are consistent with the laws of physics (Helbing & Johansson, 2011) and psy-
chology (Wang, 2016; Yerkes & Dodson, 1908).
4.1. Initial version of Helbing and Molnar (1995)
Initial version of Helbing and Molnar (1995) is based on a multi-particle self-driven
system framework, assuming that each pedestrian has the ability to perceive and
respond to the surrounding environment. The interactions between the pedestrian
and environment are described with forces. Forces include the driving force of the
target (Equation (1)), repulsive forces of other pedestrians (Equation (2)), repulsive
Table 1. Self-organisation phenomena.
No. Place
Self-organisation
phenomenon Category Generation condition Effect
1 Corridor Lane formation Crowd Higher relative velocity in
opposite directions
Minimising the frequency and
strength of avoidance
manoeuvres
2 Stripe formation Crowd Crossing flow in only two
directions
Minimising obstructing
interactions and maximising
the average pedestrian speeds
3 Freezing-by-
heating
Evacuation Driving term and dissipative
friction, while the sliding
friction is not required
Decreasing the whole speed
4 Bottleneck Oscillatory flow Crowd Simple pedestrian interactions Reducing frictional effects and
delays
5 Zipper effect Evacuation Provisions have been made for a
decrease in personal space at
the bottleneck, either via a
decrease in the repulsive
forces interpreted by each
individual or a decrease in the
personal space of each
individual at an angle in front
of the individual
–
6 Intermittent flow Evacuation Pedestrian’s impatience and
increase of the driving term or
desired speed
Emerging of pedestrian queue
before the bottleneck
7 Faster=slower Evacuation Pedestrian’s impatience and
increase of the driving term or
desired speed
Slowing down crowd motion or
evacuation
8 Panic Evacuation Pedestrian’s impatience and
increase of the driving term or
desired speed
So high pressures that
pedestrians are crushed or
falling and trampled
9 Open
space
Stop-and-go
waves
Evacuation Relaxation time to adjust the
velocity or acceleration
Emerging of the congestion
10 Turbulence Evacuation Local force-based interaction Random and unintended
displacements into all possible
directions
11 Other Herding Evacuation The influence of the destination
and route choices of others on
the individual is accounted
Pure herding implies that the
crowd is eventually moving
into the same and probably
congested direction, so that
available emergency exits are
not efficiently used
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forces of obstacles and boundaries (Equation (3)), attraction forces of the friends and
exhibitions (Equation (4)). Further considering the direction dependence (Equation (5))
and fluctuation term, the superposition of these forces drives the pedestrian to keep
moving (Equation (6)). In this paper, here afterwards initial version refers to this
model proposed by Helbing and Molnar (1995).
FDi =(vdiei−vi)/
t
,ei=xd−xi
xd−xi, (1)
Fij =−∇
dij Vij[bij (dij )],dij =xi−xj, (2)
Fi
v
=−∇
di
v
Ui
v
(di
v
), di
v
=xi−x
v
, (3)
Fi
b
=−∇
di
b
Wi
b
(di
b
,t), di
b
=xi−x
b
, (4)
m
(ei,F)=1ei·F≥Fcos
w
celse
, (5)
Ftotal =FDi +
Nj
j=1,j=i
Fij
m
(ei,Fij)+
N
v
v
=1
Fi
v
+
N
b
b
=1
Fi
b
m
(ei,Fi
b
)+fluctuations.(6)
Above-mentioned model is idealised and some improvements have been done in the
practical process: (1) the mass of pedestrian is considered; (2) pedestrians are regarded as
circles rather than ellipses; (3) without the considerations of attraction forces of friends and
exhibitions, fluctuation term and direction dependence; (4) the functions of the repulsive
forces are set to exponential. All of these improvements form the initial version (Equations
(7)–(10)).
FDi =mi(vdiei−vi)/
t
, (7)
Fij =Aiexp ( −1ij /Bi)en
ij ,1ij =dij −(ri+rj), (8)
Fi
v
=A
v
exp ( −1i
v
/B
v
)en
i
v
,1i
v
=di
v
−ri, (9)
Ftotal =FDi +
Nj
j=1,j=i
Fij +
N
v
v
=1
Fi
v
.(10)
4.2. Escape panic version of Helbing et al. (2000a)
In the initial version, pedestrians are under the condition of low density. With the
increase of density, pedestrians start to contact with others, which promotes the for-
mation of the escape panic version. Here, the normal contact force and tangential
sliding friction force are taken into account (Helbing et al., 2000a). Equations (11) and
(12) are descriptions from the perspective of normal force and tangential force, and
Equations (13) and (14) are descriptions from the perspective of contact and non-
contact. The function g(x) only plays roles when the pedestrian contacts with others
(Equation (15)).
Fij =[Aiexp ( −1ij/Bi)+(−1ij kn)g(1ij )]en
ij +(−1ijkt)g(1ij )vt
jiet
ij,1ij =dij −(ri+rj), (11)
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Fi
v
=[A
v
exp ( −1i
v
/B
v
)+(−1i
v
kn)g(1i
v
)]en
i
v
+(−1i
v
kt)g(1i
v
)vt
v
iet
i
v
,1i
v
=di
v
−ri, (12)
Fij =Aiexp ( −1ij /Bi)en
ij +[(−1ijkn)en
ij +(−1ijkt)vt
jiet
ij]g(1ij ), 1ij =dij −(ri+rj), (13)
Fi
v
=A
v
exp ( −1i
v
/B
v
)en
i
v
+[(−1i
v
kn)en
i
v
+(−1i
v
kt)vt
v
iet
i
v
]g(1i
v
), 1i
v
=di
v
−ri, (14)
g(x)=0, x≥0
1, x,0
.(15)
5. Improvements of the social force model
On the basis of above-mentioned two versions, the improvements are presented from four
aspects.
5.1. Pedestrian attributes
In social force model, pedestrian attributes including shape, mass, walking speed and
group are considered. Among these attributes, mass and walking speed are physical attri-
butes obtaining from the empirical studies. Therefore, no improvement focused on them
and more attention is paid on the pedestrian shape and group.
5.1.1. Pedestrian shape
In reality, pedestrians are ellipses with different widths on the two-dimensional plane. In
the model, some modifications are made to improve computation efficiency. In the
escape panic version, pedestrians are represented by circles. In the later research,
researchers try to use ellipse (Johansson, Helbing, & Shukla, 2007; Shukla, 2005) and
three circles (Langston, Masling, & Asmar, 2006; Smith et al., 2009) to represent the ped-
estrians (see Figure 3).
5.1.1.1. Ellipse representation. According to the calculation formula of the semi-minor
axis bij (Equation (16)), the interaction force can be calculated by Equation (17)
andyij ;dij −vjDt. For more details, one can refer to Johansson et al. (2007) and
Shukla (2005).
2bij =
(dij +dij −vjDt)2−vjDt2
, (16)
Figure 3. Pedestrian representation methods in the simulation.
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Fij =Aie−bij/Bidij +dij −yij
4bij
×dij
dij
+dij −yij
dij −yij
.(17)
5.1.1.2. Three-circle representation. There are two kinds of forces in the representation of
three circles being a motive force (Equation (18)) and interaction force between ped-
estrians (Equation (19)). The latter one can be further divided into the normal force
(Equation (20)), sliding friction force (Equation (21)) and physical damping force (Equation
(22)). The superposition of these forces drives the pedestrian to keep moving. Further-
more, pedestrians representing by three circles have the characteristic of rotation. The
moment of interaction force is calculated by Equation (23) and that of motive force is cal-
culated by Equation (24).
FDi =mi(vdiei−vi)/
t
, (18)
FC=FN+FT+FD, (19)
FN=kN(rikjk −dikjk)en
ikjk , (20)
FT=kT(rikjk −dikjk)vRTikjk et
ikjk , (21)
FD=cDPvRNikjk en
ikjk, (22)
MC=Ric ×FC, (23)
MM=kM(
u
i−
u
D).(24)
5.1.1.3. Discussion. There are three kinds of pedestrian representations being circle, ellipse
and three circles. Circular representation is most common and efficient although there are
some calculation errors. Ellipse representation has the opposite performance. Three-circle
representation is a kind of compromise of other two representation methods. On the one
hand, it can describe the shape of pedestrians more accurately. On the other hand, it can
ensure the simulation efficiency. In recent years, the three-circle representation has gradu-
ally become a popular method. All in all, the pedestrian representation is closely related to
simulation accuracy and efficiency. For a large-scale crowd simulation in stations, stadiums,
theatres and so forth, it is better to use the circular representation for saving computation
time. For pedestrian simulation in a local environment, both of the elliptical and three-circle
representations are suitable (Zanlungo, Ikeda, & Kanda, 2011).
5.1.2. Group
In reality, lots of pedestrians in a crowd are moving in groups, such as friends, couples and
families. In the initial version, they put forward the concept of group, but did not model it.
On the basis of empirical research, Moussaïd, Perozo, Garnier, Helbing, and Theraulaz
(2010) and Xu and Duh (2010) proposed different modelling methods.
5.1.2.1. Moussaïd et al. They formulated a new interaction term Fgroup describing the
response of pedestrian ito other group members (Equation (25)). There are three
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parts in the interaction term: (1) group members want to turn their gazing direction to
see their partners. The greater
a
i, the less comfortable is the turning for walking. They
assumed that pedestrian iadjusts his position to reduce the head rotation
a
i. This is
modelled by the acceleration term (Equation (26)); (2) pedestrian ikeeps a certain dis-
tance to the group’s centre of mass, and the average distance to the centre of mass
increases with group size. This is modelled by the acceleration term (Equation (27));
(3) a repulsive effect so that group members do not overlap each other (Equation (28)).
Fgroup =Fvis
i+Fatt
i+Frep
i, (25)
Fvis
i=−
b
1
a
ivi, (26)
Fatt
i=qA
b
2ui, (27)
Frep
i=
k
qR
b
3wij.(28)
5.1.2.2. Xu and Duh. They proposed the concept of bonding force. Bonding means that
two or more members in a group tend to be together and maintain a certain distance.
Notably, they only focused on two-pedestrian bounded groups. The calculation of
bonding force is shown in Equation (29). Furthermore, the repulsive force between the
group members is also considered, while the magnitude of that is set to the half of the
normal situation considering the intimate relationship (Equation (30)). Subsequently, the
interaction force between the group members is the superposition of above-mentioned
two forces (Equation (31)).
kbond
ij =−Ciexp (1ij /Di)en
ij ,1ij =dij −(ri+rj), (29)
fbond
ij =Fij/2, (30)
Fbond
ij =kbond
ij +fbond
ij .(31)
5.1.2.3. Discussion. There are two methods for simulating the group behaviour. Regard-
ing the applicable scope, the former can be applied to dyad, triad and four, while the
latter can only be applied to dyad. With respect to the modelling method, both of them
consider the attraction force for maintaining a certain distance between the group
members and the repulsive force for avoiding overlapping. However, the expressions
of the above-mentioned two forces in different researches are different. Furthermore,
the former research considers the desire of that pedestrians want to turn their gazing
direction to see their partners. In the control experiments of Moussaïd et al., the
spatial pattern of the group is mainly influenced by parameter
b
1, which means the
desire of seeing partners is important in modelling the group behaviour. Therefore,
the research of Moussaïd et al. is better.
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5.2. Motion base cases
Eight crowd movement base cases and four individual movement base cases are intro-
duced in Section 3.2. The initial version can simulate most of the crowd motion base
cases and Duives et al. (2013) also illustrate this point, and later improvements focused
on the bidirectional flow and bottleneck flow. Regarding the individual motion base
cases, the initial version can simulate the avoiding behaviour and later improvements
focused on the self-stopping and overtaking behaviour.
5.2.1. Bidirectional flow
Bidirectional flow is widespread in reality and the initial version can simulate this move-
ment. However, the pedestrian will not try to avoid the oncoming pedestrians, which pro-
duces an unrealistic conflict. Therefore, researchers made improvements from four
aspects: (1) adding a component to the desired velocity (Smith et al., 2009); (2) adding
new forces to change the pedestrian’s trajectory (Lee, Kim, Chung, & Kim, 2016); (3)
counter-flow-based active decision (Heliövaara, Korhonen, Hostikka, & Ehtamo, 2012); (4)
incorporating a dynamic navigation filed (Jiang, Chen, Wang, Wong, & Cao, 2017). The illus-
tration of these schemes is shown in Figure 4.
5.2.1.1. Smith et al. Firstly, they calculated the closest point on the vector vj−vifrom
pedestrian i. This point defines the closest point of approach, i.e. the smallest possible dis-
tance between pedestrian iand jgiven their current velocities. In effect this takes into
account the projected positions of pedestrians. Then, a unit vector uwas calculated in
the direction of pedestrian ifrom the closest point of approach. A vector was then
added as an extra component to the desired velocity of the pedestrian iand jin the direc-
tion of uand−u, respectively, ensuring that the avoidance action is away from the col-
lision. The magnitude of the correction vector was taken as 0.5 of the initial desired
velocity magnitude of the pedestrian.
5.2.1.2. Lee et al. They added following effect and evasive effect to the initial version to
explain the bidirectional flow. Both of the effects consider the influences of other ped-
estrians from the perspectives of relative displacement and moving direction. These influ-
ences are described by corresponding forces (Equation (32)–(35)). Then these forces are
Figure 4. Four schemes for simulating bidirectional flow.
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added to the initial version.
Fs
follow =Eiexp[(−1ij)/Fi]followi,1ij =dij −(ri+rj), (32)
Fv
follow =Giexp[(−1ij)/Hi]attj,1ij =dij −(ri+rj), (33)
Fs
evade =Eiexp[(−1ij)/Fi]evadei,1ij =dij −(ri+rj), (34)
Fv
evade =Giexp[(−1ij)/Hi]refj,1ij =dij −(ri+rj).(35)
5.2.1.3. Heliövaara et al. The area in front of the pedestrian iis divided into three over-
lapping sectors and each covers a 2
u
wide sector. By calculating the utility values of all
sectors, the pedestrian makes a decision: move straight on, dodge to the right or dodge
to the left. Notably, the calculation process of this model is complicated.
5.2.1.4. Jiang et al. They incorporated a dynamic navigation field to describe the ped-
estrian movement direction resulting from the decision-making processes.
5.2.1.5. Discussion. There are four methods for improving the simulation of bidirectional
flow, and all of them have pros and cons. The former two methods have consistency in
the modelling idea. The advantages are that calculation process is simple and par-
ameters are easy to calibrate. The disadvantage is that the pedestrian can only avoid col-
lision with one pedestrian at a time, which, especially in the case of dense crowds, may
not be enough to get realistic results (Heliövaara et al., 2012; Lee et al., 2016 ). Compared
with the former two methods, the third one can avoid multiple pedestrians at the same
time. The drawbacks are that lots of parameters are difficult to calibrate and it only
applies to the short distance interaction. Different from the former three methods, the
last method is creative for incorporating a dynamic navigation filed, while the navigation
field should be defined before the simulation and the computational workload during
the simulation process increases.
5.2.2. Bottleneck flow
Bottleneck flow does not belong to the set of crowd motion base cases. More precisely, it is
a combination of several crowd motion base cases and can be seen in reality frequently.
The most common example is the process of boarding and alighting in subway stations.
During this process, lane formation is observed. According to the pedestrian organisation
pattern, the lane formation can be further divided into the unidirectional lane, mixed lane
and separate lane (Guo, 2014). The initial version can reproduce the mixed lane while does
not do well in other two kinds of lanes. Then some improvements should be done for
simulating these kinds of lanes.
Guo (2014) maintained that the main improvements should focus on the interactions
between pedestrian with other pedestrians and boundary/obstacle, and proposed two
new interaction terms (Equation (36) and (37)). The first term of Equation (36) represents
a repulsive action force and describes the psychological tendency of pedestrians iand jto
stay away from each other. The second term of Equation (36) takes effect only when ped-
estrians iand jare not in the same direction. That is to say, if pedestrian iand jare in the
same direction, then the second term is equal to zero. The second term represents a
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sliding action force which guarantees that pedestrian itends to move towards a lateral
direction as he or she faces pedestrian jin the other direction and his or her movement
is blocked by the pedestrian j. Moreover, the sliding action force becomes stronger
when the distance between them decreases. In this way, he or she can go around the ped-
estrian in the other direction.
Fij =Aiexp ( −1ij /Bi)en
ij +Jiexp ( −1ij /Ki)et
ij,1ij =dij −(ri+rj), (36)
Fi
v
=A
v
exp ( −1ij /B
v
)en
ij +J
v
exp ( −1ij /K
v
)et
ij,1i
v
=di
v
−ri.(37)
5.2.3. Overtaking behaviour
Pedestrians with high walking speed are used to overtake the pedestrians with lower
walking speed. One example can be observed at subway stations during the peak
hours, in which pedestrians rush along the platform and try to catch the train. Aiming
at this behaviour, Yuen and Lee (2012) made some improvements on the basis of the
initial version.
Modelling of overtaking behaviour can be divided into two parts: (1) the direction of
overtaking and (2) the degree of overtaking. Here, we can assume that a pedestrian is
walking from left to the right. Regarding the direction of overtaking, it can be divided
into upward and downward. Since the overtaking process is influenced by the surrounding
pedestrians and boundaries, the direction can be calculated according to the force that
the pedestrian suffered. Although Yuen and Lee considered four kinds of forces: (1)
upward social repulsive force, (2) downward social force, (3) upward boundary repulsive
force and (4) and downward boundary repulsive force, its essence is the resultant force
that the pedestrian suffered in the vertical direction in the social force model (Equation
(38)). If the direction of the resultant force is upward, the pedestrian will walk upward,
and vice versa. Regarding the degree of overtaking, it is determined by a two-dimensional
Gaussian function (Equation (39)), the format of which is the product of the horizontal
dimension Gaussian function and the vertical dimension Gaussian function shown in
Equations (40) and (41). According to the direction and degree of overtaking, the overtak-
ing force can be calculated and added to the initial version (Equation (42)).
FMi =(
Nj
j=1,j=i
Fij +
N
v
v
=1
Fi
v
)·up, (38)
FOi =
Nj
j=1,j=i
a
r
·
u
(x)·
u
(y), (39)
u
(x)=1
2
ps
2
x
exp −(x−
m
x)2
2
s
2
x
, (40)
u
(y)=1
2
ps
2
y
exp −(y−
m
y)2
2
s
2
y
, (41)
F=FDi +Fij +Fi
v
+FMi ·FOi.(42)
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5.2.4. Self-stopping behaviour
In reality, the pedestrian will choose to stop sometimes when facing pedestrians from
other directions or approaching pedestrians in the same direction. With respect to this
behaviour, Seyfried, Steffen, and Lippert (2006) and Parisi, Gilman, and Moldovan (2009)
proposed different improvements.
5.2.4.1. Seyfried et al. When the slowing-down condition is reached (i.e. each pedestrian
has a certain personal space, if someone invades the pedestrian’s personal space), the vel-
ocity of the pedestrian is set to zero.
5.2.4.2. Parisi et al. They incorporated the concepts of respect factor RFand respect dis-
tance Dri (Equation (43)). If any other pedestrians/boundaries touch the respect area, the
magnitude of the desired velocity is set to zero until the respect area is free again.
Dri =RFri.(43)
5.2.4.3. Discussion. From the perspective of modelling idea, both of them consider that
pedestrians require space to move. The required space refers to the personal space in
the former model, and refers to the respect area in the latter model. However, the
improvements made by Parisi et al. is better than that made by Seyfried et al. for produ-
cing a continuous slowing down instead of a sudden stop.
5.3. Self-organisation phenomena
Eleven kinds of self-organisation phenomena are introduced in Section 3.3. According to
Helbing and Molnar (1995), the initial version can reproduce the lane formation and oscil-
latory flow. In the following subsections, the reproductions of other self-organisation
phenomena are introduced. Specially, all of them are occurred in evacuation scenarios
(see Table 1), so these improvements are based on the escape panic version.
5.3.1. Freezing-by-heating
Lane formation is observed in the corridor frequently. With the increase in density, lanes
are destroyed by increasing fluctuation force and a solid state is formed. In order to repro-
duce this phenomenon, Helbing, Farkas, and Vicsek (2000b) define the fluctuation
strength as follows (Equation (44)), where niwith 0 ≤ni≤1 measures the nervousness
of the pedestrian. The parameter
h
0means the normal and
h
max denotes the maximum
fluctuation strength.
h
i=(1 −ni)
h
0+ni
h
max.(44)
5.3.2. Intermittent flow, faster=slower, panic
All of these three self-organisation phenomena emerge at the bottleneck. If the overall
flow towards a bottleneck is higher than the overall outflow from it, a pedestrian queue
emerges. With the pedestrian density continues to increase, pedestrians will compete
for the same few gaps. This typically causes body interactions and frictional effects,
which can slow down crowd motion or evacuation (i.e. faster=slower). A possible
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consequence of these coordination problems is intermittent flows (Helbing, Buzna,
Johansson, & Werner, 2005). Furthermore, due to the decline in speed, stopped ped-
estrians cannot see the reason for the temporary slowdown, and get impatient and
pushy, which may trigger the panic in the worst case. These phenomena can be repro-
duced by adjusting the desired speed of pedestrians (Equation (45) and (46)).
v0
i(t)=(1 −ni(t))v0
i(0) +ni(t)vmax
i, (45)
ni(t)=1−vid(t)/v0
i(0).(46)
5.3.2.1. Discussion. Among these three phenomena, faster=slower is the most wide-
spread and can be used as an indicator to quantitatively evaluate the pedestrian dynamics
model. In Helbing et al. (2000a), this phenomenon is observed in the case of desired speed
equals to 1.5 m/s. In Parisi and Dorso (2005), the desired speed for that is a little higher but
basically consistent. In Lakoba, Kaup, and Finkelstein (2005), the desired speed for that is
3 m/s, which means their models need improvements.
5.3.3. Stop-and-go waves, turbulence
Pedestrians are involuntarily moved when they are densely packed, and as a consequence,
interactions increase in areas of extreme densities, which lead to an instability of ped-
estrian flows. When the average density is increasing, sudden transitions from laminar
to stop-and-go waves and turbulence are observed. In order to reproduce this phenom-
enon, Yu and Johansson (2007) improved the interaction between pedestrians. Without
the consideration of sliding friction force, the interaction between pedestrian is shown
in Equation (47). In the extremely crowded areas, the speeds of pedestrians are very
low, so the small change of dij may not lead to sufficient forces for the occurrence of tur-
bulence for the repulsive force is increased linearly. In the revised model (Equation (48)),
the small changes of dij will change the repulsive forces greatly and lead to sudden invo-
luntary displacements. With the influence of such strong reactions, the motion of ped-
estrians near iwill be affected and will further spread the irregular displacements to a
larger area. Thus, the turbulent motion of pedestrians will be triggered. Furthermore,
the angular dependence Q(
w
ij)is also considered here.
Fij =[Aiexp ( −1ij/Bi)+(−1ij kn)g(1ij )]en
ij ,1ij =dij −(ri+rj), (47)
Fij =FQ(
w
ij) exp[−dij/D0+(D1/dij)k]en
ij .(48)
5.3.4. Herding
If the pedestrian is not sure where to go, there is a tendency to show a herding behaviour,
i.e. to imitate the behaviour of others. For example, pedestrians try to leave a smoky room
and find one of the invisible exits. Each pedestrian may either select an individual direction
or follow the average direction of his neighbours in a creation radius, or try a mixture of
both. In order to reproduce this phenomenon, Helbing et al. (2000a) improved the direc-
tion of desired speed (Equation (49)), where pireflects the degree of panic of pedestrian i.
e0
i(t)=Norm[(1 −pi)vi+pikv0
j(t)li].(49)
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5.4. Special cases
Here, some improvements for special cases including pedestrian movement on stairs and
emergency evacuation are presented in sequence.
5.4.1. Pedestrian movement on stairs
Pedestrian movement is not limited to the 2D movement within rooms or corridors,
and there are lots of special movement spaces such as the stairs. Aiming at the ped-
estrian movement on stairs, Qu, Gao, Xiao, and Li (2014)madesomeimprovements.In
order to better describe the pedestrian movement, each pedestrian is represented by
three circles. Pedestrian movement on stairs can be divided into three parts: (1) select-
ing optimal direction, (2) calculating the self-driven force and (3) calculating the
contact force. The walking direction is calculated by Equations (50) and (51). According
to the principle of no collision with each other, put the first two items of Equation (50)
into the third item, we can calculate Dt, and calculate the optimal direction
ei=(cos
a
∗,sin
a
∗) consequently (Equation (51)). The calculation of driving force is
shown in Equations (52) and (56). Equation (52) defines the pedestrian’s horizontal
footstep, which is restricted by the tread depth Dand riser heights Hof the step.
Here,
b
is the included angle between the optimal direction
a
∗and the x-coordinate.
If the pedestrian is obstructed by other pedestrians or obstacles, he/she will slow
down and avoid collision, and the footstep length does not exceed the maximal
distancef(
a
∗). Equation (53) defines the desire speed of the pedestrian. On the one
hand, the pedestrian does not want to collide with others during the relaxation
time dh/
t
. On the other hand, pedestrian’s speed is assumed to not exceed a
maximum velocity vmax
iand the horizontal maximum velocity is vmax
icos
u
. Here, the
relaxation time
t
is influenced by movement direction, stair slopetan
u
, pedestrian
mass miand cumulative moving height H(Equation (54)). The cumulative moving
height is calculated by Equation (55). Then the driving force is calculated by Equation
(56). In the case of crowded, the contact force is calculated by Equation (57). Further-
more, considering some contact forces do not point to the pedestrian’s centre (i.e.
each pedestrian is an ellipse), we need to calculate the rotation of the pedestrian
according to Equation (58).
lixm(t+Dt)=lixm(t)+vixDt,liym(t+Dt)=liym+viyDt(m,n[{1, 2, 3})
ljxn(t+Dt)=ljxn(t)+vjxDt,ljyn(t+Dt)=ljyn+vjyDt
(lixm(t+Dt)−ljxn(t+Dt))2+(liym(t+Dt)−ljyn(t+Dt))2=(rim +rjn)2,
(50)
f(
a
imjn)=min {viDt,dmax}, f(
a
)=min {f(
a
imjn)},
a
∗=arg min {d(
a
)}
d(
a
)=d2
max +f(
a
)−2dmaxf(
a
)cos(
a
0−
a
),
(51)
dh=min {nD/cos
b
,f(
a
∗)}, (52)
vdes
i=min {dh/
t
,vmax
icos
u
},
u
=H/D,(53)
t
=
t
0on the ground
(1 +
a
1tan
u
+
b
1mi+
g
1H(t))
t
0upstairs
(1 +
a
2tan
u
+
b
2mi+
g
2H(t))
t
0downstairs
⎧
⎨
⎩
,(54)
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H(t)=h0+tih(ti)ifv(ti).vmin
h0if v(ti)≤vmin
,(55)
FDi =(FDicos
b
cos
u
,FDisin
b
cos
u
)=mi(vdiei−vi)/
t
,(56)
FC
ij =
m,n
Fimjn=
m,n
kng(lim−ljn)eimjn,(57)
Mi=
m,n,j
Fimjndimjn(m=2).(58)
5.4.2. Emergency evacuation
Pedestrian simulation is not limited to the normal situation, but also emergency evacua-
tion. Regarding to the emergency evacuation, current improvements focus on three
aspects: (1) herding (2) desired speed and (3) guiders. The improvements for herding
have been introduced in Section 5.3.4. Therefore, only the improvements for desired
speed and guiders are introduced in this part.
5.4.2.1. Desired speed. In the case of emergency evacuation, pedestrian’s desire speed will
change. In Section 5.3.2, Helbing et al. (2005) incorporated the concept of nervousness for
changing the pedestrian’s desired speed (Equations (46) and (47)). Yang, Dong, Wang,
Chen, and Hu (2014) maintained that the nervousness/excitement is not only impacted
by the realisation degree of desired velocity itself, but also influenced by the surrounding
pedestrians and environment. Then a modified time-dependent parameter to reflect the
motivation/excitement is proposed (Equation (59)). Here, three coefficients are positive
constants between 0 and 1, and
x
1+
x
2+
x
3=1. h(x) is a piecewise function defined
by (Equation (60)). The first term in the right-hand side of Equation (59) reflects the
effect of realisation degree of desired velocity on the pedestrian’s motivation. v0
j(t)is
the average velocity of pedestrians who are in the vision field of pedestrian i. If the ped-
estrians within the valid influence domain of pedestrian ihave a larger average speed than
that of i, the current pedestrian iwho becomes nervous or excited will accordingly
increase his/her desired velocity. The third term is the effect of the surrounding environ-
ment such as an annunciators or displayers on the motivation. It is assumed as distributes
between 0 and 1. Finally, we can update the desired speed according to Equation (61).
ni(t)=
x
1·hv0
i(t)−vid(t)
v0
i(t)
+
x
2·hv0
j(t)−vi(t)
v0
j(t)
+
x
3·
j
(t), (59)
h(x)=0, x,0
x,x≥0
, (60)
v0
i(t)=(1 −ni(t))vmin
i+ni(t)vmax
i.(61)
5.4.2.2. Guider. In the case of emergency evacuation, guiders are set up to speed up the
evacuation process. Regarding the influences of guiders, there are two kinds of improve-
ments. Hou, Liu, Pan, and Wang (2014) maintained that the presence of guiders changes
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the desired movement direction, and there are two movement trends of the pedestrian:
(1) moving toward the position of the guider; (2) keeping the same movement direction
with the guider. Then the desired movement direction of a pedestrian is calculated by
Equation (62), where
r
is a distance-dependent parameter. Furthermore, if there are
several guiders, the pedestrian need to select one from them according to Equation (63).
ei=qer+(1 −q)en
ir, (62)
pr=e−dir /
Nr
r=1
e−dir .(63)
Different from the research of Hou et al., Yang et al. (2014) used the forces to represent
the influences of guiders, and there are two kinds of forces: (1) interaction force Fir
between the pedestrian iand guider r; (2) the navigation force Fr
iof the guider r(Equation
(64)), which is influenced by the position and velocity of them. Then these two forces and
the driving force calculated by Equation (71) are superimposed to calculate the resultant
force (Equation (65)).
Fr
i=
a
i·mi·[−b1(xi−xr)−b2(vi−vr)], (64)
Ftotal =
b
i·FDi +
Nj
j=1,j=i
Fij +
N
v
v
=1
Fi
v
+Fir +Fr
i.(65)
6. Discussion
In this section, the description ability, parameter calibration and flexible application in a
complex environment are discussed.
6.1. Description ability
Description ability is the ability of describing pedestrian movement process and
phenomenon. Table 2 makes a summary of the improvements in Section 5. The
second column is the category of the improvements and the third column is the specific
field of improvements (number in the brackets indicated the number of improved
methods for each problem). Furthermore, the improvement methods or conclusions
are presented in the fourth column, which can provide some insights for further research
and application.
The initial version and escape panic version can describe some motion base cases and
reproduce some self-organisation phenomena. Further considering these improvements,
the social force model has the strong description ability currently. However, there are still
some problems that need to be further addressed: (1) existing researches focus on several
motion base cases, whether the social force model can describe other motion base cases
such as rounding a corner, entering, exiting and random flow accurately should be further
validated; (2) some improvements should be done to reproduce the self-organisation
phenomenon of zipper effect; (3) for each item, there are several improved methods.
Although a brief discussion has been presented in each subsection, a comprehensive
and quantitative comparison considering simulation efficiency and accuracy should be
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done. The necessary steps are bringing these models in the same one example, calibrating
the parameters of each model in turn, developing a general simulation platform and
running these models in the same one computer.
6.2. Parameter calibration
In this subsection, 10 parameters in the escape panic version are presented, followed by
the introduction of parameter calibration methods and sensitivity analysis methods.
6.2.1. Parameter list
Parameters in the escape panic version are shown in Table 3, which can be divided into
five categories.
(1) Pedestrian mass miand radius ri
In the initial version, the pedestrian mass is simplified as unit mass. In the later research,
the pedestrian mass is set up according to the reality. Normally, the pedestrian mass is
subject to the normal distribution. So does the radius.
(2) Strength Ai/A
v
and characteristic distance Bi/B
v
of the forces
Parameters Ai/A
v
and Bi/B
v
are the strength and characteristic distance of the inter-
action forces. In the escape panic version, they are set to 2000 and 0.08, respectively. In
the pedestrian simulation software SimWalk (Kuligowski, Peacock, & Hoskins, 2005), they
are set to a series of discrete values. Normally, the value of parameter Ai/A
v
is between
0.3 and 2.1, and that of Bi/B
v
is between 0.18 and 0.3. Furthermore, a list for the different
environment is presented (Daamen, 2004).
Table 2. Summary of the improvements of social force model.
NO. Category Specific filed Improvement method/conclusion
1 Pedestrian
attributes
Pedestrian shape (3)
Pedestrian group (3)
Circle (efficient) ellipse (accurate) three circles (balance of
efficiency and accuracy)
Adding new forces, and the model of Moussaïd et al. is
better
2 Crowd motion base
cases
Bi-directional flow (4) Four different methods
Bottleneck (1) Adding new forces
3 Individual motion
base cases
Overtaking behaviour (1) Determining the direction and degree of overtaking in turn
Self-stopping behaviour (2) Setting the speed or desired speed to zero, and the latter
one is better
4 Self-organisation
phenomena
Freezing-by-heating (1) Incorporating the fluctuation strength
Intermittent flow, faster =
slower, panic (1)
Adjusting the desired speed by incorporating the time-
dependent parameter reflecting the nervousness
Stop-and-go waves,
turbulence (1)
Changing the form of the interaction forces
Herding (1) Adjusting the desired speed by incorporating the parameter
reflecting panic
5 Special cases Pedestrian movement
on stairs (1)
Selecting optimal direction, calculating the self-driven force
and the contact force in turn
Emergency: desired speed (2),
guider (2)
Adjusting desired speed according to the pedestrian itself
and surrounding environment; the role of guider is
described by forces
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(3) Relaxation time
t
Relaxation time is related to the pedestrian response time, inertia and environment.
Through repeated experiments under the normal environment, Li (1982) found that the
relaxation time is subjected to the lognormal distribution rather than the normal
distribution, and the average relaxation time is 0.4 s. Of course, pedestrian relaxation
time changes with environments. Furthermore, Johansson, Duives, Daamen, and
Hoogendoorn (2014) explored the many roles of the relaxation time in force-based
models.
(4) Desired speed vdi
Pedestrian’s desired speed is influenced by many factors: (1) pedestrian character-
istics: lots of pedestrian characteristics, such as gender, age, luggage and action
ability, affects the pedestrian’s desired speed (Franěk, 2013). These differences will be
described by a distribution of the desired speed; (2) surrounding density: as the sur-
rounding density increases, pedestrian’s desired speed will decline; (3) specific environ-
ment: pedestrian’s desired speed is different in the normal and emergency evacuation.
In the case of emergency evacuation (Yang et al., 2014), pedestrian’s desired speed is
the combination of his own speed and the average speed of the surrounding
pedestrians.
(5) Compression coefficient knand friction coefficient kt
These two parameters are seldom been researched separately. Normally, they are cali-
brated by the parameter calibration methods.
6.2.2. Parameter calibration methods
Until now, several calibration methods have been proposed, and they can be
divided into three categories: pedestrian trajectory-based parameter calibration,
density distribution-based parameter calibration and parameter calibration in special
cases (see Table 4). Furthermore, some discussions of observed/experimental data,
evaluation index, updating algorithm and simulation process control are presented as
follows:
Table 3. Parameters in the escape panic version.
No. Parameter Parameter description Value
1miPedestrian mass 80(±?) kg
2riPedestrian radius 0.3 (±0.05) m
3AiStrength of social repulsive force between pedestrian iand j2000 N
4BiCharacteristic distance between pedestrian iand j0.08 m
5A
v
Strength of social repulsive force between pedestrian iand boundary/
obstacle
v
2000 N
6B
v
Characteristic distance between pedestrian iand boundary/obstacle
v
0.08 m
7
t
Pedestrian relaxation time 0.5 s
8vdi Magnitude of desired speed of pedestrian i0.8 m s
−1
9knBody compression coefficient 1.2 × 10
5
kg s
−2
10 ktSliding friction coefficient 2.4 × 10
5
kg m
−1
s
−1
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(1) Observed/Experimental data
Pedestrian trajectory and density distribution are the main collected data for parameter
calibration. Some other data such as the number of the evacuated pedestrian are also
useful in some special cases.
(2) Evaluation index (fitness function)
Evaluation indexes are used to judge the simulation accuracy, which are closely related to
the type of parameter calibration method. With respect to the pedestrian trajectory-based
methods, evaluation indexes focus on the variables related to acceleration, velocity, dis-
tance and position. With respect to the density distribution-based methods, evaluation
indexes include the difference of scaled density distribution and difference of local density.
(3) Updating algorithm
Updating algorithms are used to search the optimal parameters, and they can be divided
into three categories: maximum likelihood estimation, general heuristic algorithm (simple
evolutionary algorithm, greedy approach, simulated annealing, genetic algorithm, covariance
matrix adaptation, et.) and new heuristic algorithm (differential evolution algorithm). Com-
pared with the general heuristic algorithm, the new heuristic algorithm performs better in sol-
ution quality and convergence rate (Li, Zhao, He, Chen, & Xu, 2015; Zhong & Cai, 2015).
Furthermore, Zhong, Hu, Cai, Lees, and Luo (2015) proposed a new algorithm (DESAP) combin-
ing differential evolution algorithm, sensitivity analysis and Powell’s method. The differential
evolution algorithm is used to update the population, the sensitivity analysis is used to dyna-
micallyanalyse the current population and identify the important parameters, and thePowell’s
method is used to fine-tune the important parameters of the best individual. This algorithm
can speed up the convergence rate and provide more information about parameters.
(4) Simulation process control
Simulation process control is not presented in Table 4, while it is important for par-
ameter calibration. There are two kinds of control methods: controlling one pedestrian
at a time (i.e. each pedestrian is simulated separately while the other pedestrians are
kept on the actual trajectory), controlling all pedestrians at once (i.e. all pedestrians are
simulated simultaneously). According to Rudloff, Matyus, Seer, and Bauer (2011), the
latter method is better.
6.2.3. Sensitivity analysis methods
Sensitivity analysis is crucial to identify the important parameters. Currently, two typical
methods have been proposed.
(1) Experimental contrasting method (Rudloff et al., 2011)
Setting a set of values for each parameter (around the optimal value, ±30%, ±20%,
±10%) and other parameters are fixed, then calculating the difference between the
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Table 4. Review on the parameter calibration method.
Year Researcher
Observed/
Experimental data
Evaluation index
(fitness function) Updating algorithm Calibrated parameter
Pedestrian trajectory-
based methods
2006 Hoogendoorn and
Daamen
Pedestrian
trajectory
Difference of
acceleration
Maximum likelihood estimation Desired speed vdt, Interaction strength Ai,
Interaction distance Bi, Relaxation time
t
,
Acceleration time Ti, Anisotropy factor
h
i
2007 Johansson, Helbing,
and Shukla
Pedestrian
trajectory
Difference of distance Simple evolutionary algorithm Interaction strength Ai, Interaction distance Bi,
Anisotropy parameter
l
2012 Daamen and
Hoogendoorn
Pedestrian
trajectory
Difference of
acceleration
Maximum likelihood estimation Desired speed vdt, Interaction strength Ai,
Interaction distance Bi, Acceleration time Ti
2011 Rudloff, Matyus, Seer,
and Bauer
Pedestrian
trajectory
Average square
distance
Mean percentage of
time error
MATLAB function fminsearch
(number optimisation with a
Nelder-Mead algorithm)
Seven parameters in a variant of the social force
model
2014 Wolinski, Guy, Olivier,
Lin, and Manocha
Pedestrian
trajectory
Absolute difference
metric
Path length metric
Inter-pedestrian
distance metric
Progressive distance
metric
Vorticity metric
Fundamental diagram
metric
Greedy approach
Simulated annealing
Genetic algorithm
Covariance matrix adaptation
Five models, and each model has several
parameters
Density distribution
based methods
2015 Zhong and Cai Density distribution Difference of scaled
density distribution
Differential evolution algorithm Interaction strengthAi, Interaction distance Bi,
Body force coefficient kn, Sliding friction force
coefficient kt
2015 Zhong, Hu, Cai, Lees,
and Luo
Density distribution Difference of local
density
DESAP (Differential evolution and
Sensitivity Analysis and Powell’s
method)
Interaction strength Ai, Interaction distance Bi,
Body force coefficient kn, Sliding friction force
coefficient kt
Parameter calibration
in special situations
2015 Li, Zhao, He, Chen, and
Xu
Number of
evacuated
pedestrian
Difference of
evacuated
pedestrian
Differential evolution algorithm Desired speed vdt , Interaction strength Ai,
Interaction distance Bi, Body force coefficient
kn, Sliding friction force coefficient kt
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simulation result and actual data, and drawing the curve for each parameter. In this way,
the important parameters can be identified.
(2) Entropy-based method (Zhong et al., 2015)
The main idea is to measure the importance of a parameter by its randomness (or
diversity) among the target vectors in the current population. If a parameter has
small randomness (i.e. distributed within a narrow interval), it is likely to be an important
parameter. Otherwise, if a parameter is distributed randomly, it may have negligible
influence on the output. Here, the entropy is used to measure the randomness of
each parameter.
6.3. Flexible application in complex environment
There are lots of complex environment in reality, such as the subway stations, buildings,
airports and Mecca pilgrimage. Taking the subway stations for example: (1) there are
different types of pedestrian flows: inbound, outbound and transfer. Furthermore, ped-
estrians show different characteristics; (2) there are different kinds of crowd motions:
straight flow, rounding a corner, entering, exiting, bi-directional flow in the corridor, cross-
ing flow and random flow on the platform. Specially, the bottleneck flow is formed during
the process of boarding and alighting; (3) there are different kinds of individual motions:
avoiding other pedestrians, self-stopping before the boarding points, following other ped-
estrians in the corridor, overtaking for catching the train; (4) on the basis of pedestrian
movement, almost all pedestrian self-organisation phenomena can be observed in
subway stations; (5) there are lots of stairs connecting different levels in subway stations.
All in all, complex environments such as subway stations almost integrate all possible
situations.
Facing these complex environments, one or a few improvements are obviously not
enough. Then, we have to consider the flexible application of the social force model
and its improvements. The process can be divided into three steps: (1) analysing the poss-
ible pedestrian movement situation in the environment; (2) for each situation, adding the
corresponding improvements to the initial social force model in turn. Of course, there are
several versions of social force models for different scenarios in the same one environ-
ment; (3) calibrating the parameters in each scenario.
7. Conclusions and future research
For providing an up-to-date and essentially comprehensive review of the social force
model, a framework in terms of the assessment criteria for pedestrian models considering
pedestrian attributes, motion base cases, self-organisation phenomena and special cases is
proposed. Starting with the initial version of Helbing and Molnar (1995) and escape panic
version of Helbing et al. (2000a), the improvements are classified and evaluated. The
description ability, parameter calibration and flexible application in a complex environ-
ment are further discussed.
Regarding the improvements for each problem, they have been discussed at the end of
each subsection. On the whole, the improved social force model has a strong ability in
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describing pedestrian movement and reproducing self-organisation phenomena, while
further research should be done on some problems, such as the validation of application
of the social force model in other motion base cases, reproduction of the zipper effect and
a quantitative comparison of different improved methods for each item. With respect to
the parameter calibration in different environment, 10 parameters in the escape panic
version are analysed, parameter calibration methods are summarised and sensitivity analy-
sis methods are presented. Specially, there are many complex environments in reality and
we have to take a flexible application of the social force model and its improvements.
Current work has laid a good foundation for the further research of social force model,
while there are still some questions that need to be considered: (1) some other improve-
ments of the social force model are not included due to the reasons of research topic and
limited space; (2) current assessment criteria and review are confined to the qualitative
level and they should gradually shift to the quantitative level in a near future.
Disclosure statement
No potential conflict of interest was reported by the authors.
Funding
This work was supported by State Key Lab of Rail Traffic Control and Safety [RCS2017ZT003].
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Appendix
Meaning of the letters
FDi Driving force of the target point
Fij Interaction force between pedestrian iand pedestrian j
Fi
v
Interaction force between pedestrian iand boundary/obstacle
v
Fi
b
Attraction force between pedestrian iand friend/exhibition
b
Ftotal Resultant force suffered by pedestrian i
t
Relaxation time of pedestrian i
vdi Magnitude of the desired speed of pedestrian i
eiDirection of the desired speed of pedestrian i
viCurrent speed of pedestrian i
xdPosition of the target point
xiPosition of pedestrian i
Vij A monotonic decreasing function for the interaction between pedestrian iand j
Ui
v
A monotonic decreasing function for interaction between pedestrian iand boundary/
obstacle
v
Wi
b
A monotonic increasing function for interaction between pedestrian iand friend/
exhibition
b
bij Semi-minor axis of the elliptical equipotential lines
dij Distance vector pointing from pedestrian jto i
di
v
Distance vector pointing from boundary/obstacle
v
to pedestrian i
di
b
Distance vector pointing from friend/exhibition
b
to pedestrian i
m
(ei,F) Direction dependent weight
miMass of pedestrian i
dij Distance between pedestrian iand j
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di
v
Distance between pedestrian iand boundary/ obstacle
v
riRadius of pedestrian i
en
ij Normalised vector on the normal direction between pedestrian iand j
en
i
v
Normalised vector on the normal direction between pedestrian iand boundary/obstacle
v
AiStrength of social repulsive force between pedestrian iand j
A
v
Strength of social repulsive force between pedestrian iand boundary/obstacle
v
BiCharacteristic distance between pedestrian iand j
B
v
Characteristic distance between pedestrian iand boundary/obstacle
v
knBody compression coefficient
ktSliding friction coefficient
et
ij Normalised vector on the tangential direction between pedestrian iand j
et
i
v
Normalised vector on the tangential direction between iand boundary/obstacle
v
vt
ji Tangential velocity difference between pedestrian iand j
vt
v
iTangential velocity difference between iand boundary/obstacle
v
DtTime step
vjCurrent speed of pedestrian j
FCInteraction force between pedestrians only in the three-circle model
FNNormal force between pedestrians only in the three-circle model
FTSliding friction force between pedestrians only in the three-circle model
FDPhysical damping force between pedestrians only in the three-circle model
MCMoment of the interaction force
MMMoment of the motive force
kNNormal spring constant
kTTangential dynamic spring constant
rikjk Sum of the radius of one circle kin pedestrian iand another circle kin pedestrian j
dikjk Distance between one circle kin pedestrian iand another circle kin pedestrian j
en
ikjk Normalised vector pointing from circle kin pedestrian ito another circle kin pedestrian j
et
ikjk Normalised vector on the tangential direction between one circle kin pedestrian iand
another circle kin pedestrian j
vRTikjk Magnitude of tangential relative speed between one circle kin pedestrian iand another
circle kin pedestrian j
vRNikjk Magnitude of normal relative speed between one circle kin pedestrian iand another
circle kin pedestrian j
cDP Magnitude of the damping parameter
Ric Radial vector from the pedestrian centre to the point of contact
kMEmpirical spring constant for motive moment
u
iCurrent directional angle of pedestrian i
u
DDesired directional angle of pedestrian i
Fgroup Interaction force between pedestrian iand other group members
Fvis
iForce describing the pedestrian desire of seeing his partners
Fatt
iAttraction force of the group’s centre of mass
Frep
iRepulsive force between the group members
a
iAngle of the head rotation in the social groups
b
1Strength of the social interactions between group members
b
2Strength of the attraction effect
b
3Strength of the repulsive effect
qA0-1 variables, whether the distance between pedestrian iand group’s centre of
mass exceeds a threshold value
qR0-1 variables, whether the pedestrian ioverlap with group member j
uiUnit vector pointing from pedestrian ito the centre of mass
wij Unit vector pointing from pedestrian ito the group member j
kbond
ij Bonding force between pedestrian iand group member j
fbond
ij Repulsive force between pedestrian iand group member j
Fbond
ij Interaction force between pedestrian iand group member j
CiStrength of bonding force between pedestrian iand group member j
DiCharacteristic distance between pedestrian iand group member j
Fs
follow Following force considering the relative displacement
Fv
follow Following force considering the moving direction
Fs
evade Evading force considering the relative displacement
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Fv
evade Evading force considering the moving direction
EiStrength of following/evading force considering the relative displacement
FiCharacteristic distance considering the relative displacement
GiStrength of following/evading force considering the moving direction
HiCharacteristic distance considering the moving direction
JiStrength of sliding action force between pedestrian iand jin the bottleneck flow
KiCharacteristic distance between pedestrian iand jin the bottleneck flow
J
v
Strength of sliding action force between pedestrian iand boundary/obstacle
v
in the
bottleneck flow
K
v
Characteristic distance between pedestrian iand boundary/obstacle
v
in the bottleneck
flow
FMi Resultant force that pedestrian isuffered in the vertical direction
FOi Magnitude of the overtaking force
u
(x) Horizontal dimension Gaussian function, xis the horizontal distance between pedestrian semi-
colon iand j
u
(y) Vertical dimension Gaussian function, yis the vertical distance between pedestrian iand j
a
Constant
r
Density of the specific area
ux,uySet to zero as pedestrian iis located at the origin
s
x,
s
yParameters respectively related to a product of parameters and the desired speed of
pedestrian i
Dri Respect distance
RFRespect factor
h
iFluctuation strength
niNervousness of pedestrian i
h
0,
h
maxNormal and maximum value of the fluctuation strength
v0
i(t) Magnitude of adjusted desired speed of pedestrian i
v0
i(0) Magnitude of initial desired speed of pedestrian i
vmax
iMagnitude of the maximum desired speed of pedestrian i
ni(t) Time-dependent parameter reflecting the nervousness of pedestrian i
vid(t) Projection value of the velocity at the desired direction
FMagnitude of the maximum repulsive force
D0,D1Constants
Q(
w
ij) Angular dependence function
e0
i(t) Direction of desired speed of pedestrian i
piDegree of panic of pedestrian i
v0
j(t) Average speed of surrounding pedestrians in the vision range
dhDistance between pedestrian iand the first obstacle in the desired direction
DTread depth
HRaiser height
FC
ij Total contact force between pedestrian iand j
FimjnContact force between circle mand n
MiMoment of the contact force of pedestrian i
vi(t) Magnitude of current speed of pedestrian i
j
(t) Effect of the surrounding environment
vmin
iMagnitude of the minimal desired speed of pedestrian i
erDirection of desired speed of guider r
en
ir Normalised vector on the normal direction between pedestrian iand guider r
qDistance dependent parameter
prProbability of selecting guider r
dir Distance between pedestrian iand guider r
Fir Interaction force between pedestrian iand guider r
Fr
iNavigation force of guider r
a
i0-1 variables, whether the pedestrian ican get the information of guider r
b1,b2Positive constants reflecting the weights of navigational feedback
b
iDegree of pedestrian’s willingness to follow the guider r
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