choice behavior) and his/her interactions with modelling ...based models. macroscopic models...

Modelling of Pedestrian andEvacuation Dynamics

Mohcine Chraibi1, Antoine Tordeux2,Andreas Schadschneider3 and Armin Seyfried1,41Institute for Advanced Simulation,Forschungszentrum Jülich GmbH, Jülich,Germany2School of Mechanical Engineering and SafetyEngineering, University of Wuppertal,Wuppertal, Germany3Institut für Theoretische Physik, Universität zuKöln, Köln, Germany4School of Architecture and Civil Engineering,University of Wuppertal, Wuppertal, Germany

Article Outline

GlossaryDefinition of the Subject and Its ImportanceIntroductionClassification of ModelsAcceleration-Based ModelsVelocity-Based ModelsVelocity Obstacle ModelsOther Velocity-Based ModelsRule-Based ModelsMacroscopic ModelsFuture Directions and Open ProblemsBibliography

Glossary

Pedestrian A pedestrian is a person travelling onfoot. In this article, other characterizations areused, depending on the context, e.g., agent orparticle.

Crowd A large group of pedestrians who havegathered together. Depending on the perspec-tive, more specific definitions exist.

Microscopic models Microscopic models repre-sent each pedestrian separately with individualproperties (e.g., walking velocity or route

choice behavior) and his/her interactions withother individuals. Typical models that belongto this class are cellular automata and the force-based models.

Macroscopic models Macroscopic models donot distinguish individuals. The description isbased on aggregate quantities, e.g., appropriatedensities. Typical models belonging to thisclass are fluid-dynamic approaches.

Acceleration-based models Acceleration-basedmodels are microscopic models defined by asystem of second-order ordinary differentialequations. The resulting acceleration is inte-grated twice to calculate the velocity and posi-tion of modelled pedestrians.

Velocity-based models Velocity-based modelsare microscopic models defined by means ofa system of first-order ordinary differentialequations. The resulting velocity is integratedonce to calculate position of modelledpedestrians.

Rule-based (decision-based) models Rule-based or decision-based models are not basedon differential equations. Instead the dynamicsis based on rules or decisions that the agentsmake in order to determine their new positions,velocities, etc. Typically time is considered tobe discrete in this approach, i.e., it increases incertain discrete time steps Dt at which deci-sions are made.

Cellular automata model Cellular automata aremodels in which space and time are discreteand state variables take on finite set of discretevalues. Usually, a regular square lattice withcells of size 40 cm2 is used. However, hexag-onal lattices have been used too. A discretetime is usually realized through the parallelor synchronous update where all particles orsites are moved at the same time. This intro-duces a timescale.

Fundamental diagram In traffic engineering(and physics): Density dependence of theflow, i.e., a function J(r). Due to the hydrody-namic relation J = rub, equivalent representa-tions used frequently are u = u(r) or u = u(J).

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The fundamental diagram is probably the mostimportant quantitative characterization of traf-fic systems.

Lane formation In bidirectional flows oftendynamically lanes are formed in which allpedestrians move in the same direction.

Bottleneck A bottleneck is in general a limitedresource for pedestrian flows. This can be, forexample, a door, a narrowing in a corridor, orstairs, i.e., locations of reduced capacity. Atbottlenecks jamming occurs if the inflow islarger than the capacity.

Definition of the Subject and ItsImportance

Safety in public spaces, especially at mass eventswhere thousands of people gather in a restrictedarea, is a major challenge for organizers, securitystaff, and other stakeholders. Time and again,overcrowding, heavy congestions, and pushingin crowds lead to injuries or even fatalities. Toname only few recent incidents, we refer to theLoveparade in Duisburg, Germany, the Hajji inMecca, or several large religious gatherings inIndia. Therefore, in recent years we noticed anincreasing activity in research on pedestrian andevacuation dynamics. Besides an improved qual-itative and quantitative understanding of the gen-eral phenomena observed in large human crowds,especially the development of accurate models forpedestrian streams, has become a very active fieldof research. After validation and calibrations,these models can be used for planning of largeevents, the design of transport infrastructures, orthe optimization of evacuation routes. Theseimportant applications have led to the develop-ment of many different models and simulationtools.

In this contribution we will discuss generalaspects of the various modelling approaches thathave been used for the description of pedestrianstreams. We use a classification that allows todistinguish various model classes and discuss themost important representatives of each class inmore detail.

Introduction

Models in pedestrian dynamics describe themovement of single pedestrians, small groups, orthe assembling and dispersal of large gatheringswhich could happen, e.g., outdoor in a city orinside of buildings. They are developed in naturalscience, computer science, engineering, socialsciences, or psychology. The scope and the typeof the models are closely related to the scientificarea or the application and thus are diverse.

In physics, pedestrians are treated as drivenparticles to study the emergence of collective phe-nomena like stop-and-go waves, lane formation,or clogging and intermittent flow at bottlenecks.The objective of minimal modelling approaches isto identify basic microscopic mechanisms andtypes of interaction that are relevant for the emer-gence of the observed collective phenomena. Intraffic engineering, models focus on transportcharacteristics like flow, density, speed, or routechoices. In fire safety science, the process ofbuilding evacuation including behavioral factorslike the reaction time or the influence of smokeand fire is addressed. The models support plannersin the design and dimensioning of pedestrianfacilities in dense cities, high occupancy build-ings, infrastructures for public transport, orlarge-scale events outside. For an overview werefer to the conference series “Traffic and Granu-lar Flow” and “Pedestrian and EvacuationDynamics” (see section “Books and Reviews”).In computer animation, the aim of the models is arealistic appearance of moving agents, while inrobotics the agents should steer in complex envi-ronment safely without collisions with otheragents or moving objects; see, e.g., Ali et al.(2013). In psychology, however, it is of interesthow pedestrians perceive information in complexenvironments and how they gain spatial knowl-edge using it for navigation and wayfinding(Chrastil and Warren 2015; Rio et al. 2014).

The perspective on the real system illustratesthat moving pedestrians constitute a system withmany interesting relations as well as complexphenomena which could depend on a multitudeof environmental, structural, and human factors.Looking on the level of abstraction of actual

2 Modelling of Pedestrian and Evacuation Dynamics

models, one has to note that pedestrians are three-dimensional objects and a complete description oftheir highly developed and complicated motionsequence is rather difficult. Most models in pedes-trian dynamics treat them as points or two-dimensional objects considering the verticalprojection of the body. Besides the complicatedlocomotor systems, pedestrians are equipped withmany senses, cognitive abilities, and memory.Considering the modelling of the space of move-ment, various factors must be taken into account,including the type of building and its facilities,like rooms, corridors, stairs, or ramps with differ-ent spatial structures, lighting, temperatures, andso on. Pedestrians, as humans, are also subject ofpsychology and sociology. Many concepts andterms used by engineers and natural scientists tomodel pedestrian dynamics are borrowed fromthese research fields. Often this is connectedwith an oversimplification; see the discussion inSieben et al. (2017) and Templeton et al. (2015).Psychology considers single pedestrians and theirbehavior and experiences. In pedestrian dynam-ics, the influence of senses like viewing, hearing,or the tactile sense on the movement and cognitiveabilities like wayfinding are of interest. In sociol-ogy, pedestrians are treated as entities, belongingto a group (families or friends) sharing a socialidentity (e.g., protester or police) or followingsocial norms (e.g., waiting in a queue instead ofpushing toward an overloaded exit). Thus manynorms and rules should be considered as well for a

realistic modelling of the movement of crowds,groups, or single pedestrians.

The high complexity of the system in combi-nation with the very different scopes of modellingillustrates why in pedestrian dynamics many dif-ferent approaches for models are developed andwhy a huge number of model variants exist. Oftena model or a variant of a model is introduced todescribe a special phenomenon or to consider anapplication-driven influence on the movement. Italso explains the lacking generality of actualapproaches.

Classification of Models

An often used categorization in pedestriandynamics takes three different levels of behaviorinto account (Fig. 1). At the strategic level, pedes-trians decide which activities they like to performand the order of these activities. With the choicesmade at the strategic level, the tactical level con-cerns the short-term decisions made by the pedes-trians, e.g., choosing the route taking into accountobstacles, density of pedestrians, etc. Finally, theoperational level describes the actual walkingbehavior of pedestrians, e.g., their immediatedecisions necessary to avoid collisions, etc.

The processes at the strategic and tacticallevel are usually considered to be exogenous tothe pedestrian simulation. Here informationfrom other disciplines (sociology, psychology,

Choice of activities

Schedule of activities

Choice of activity area

Route choice

Walking

Interaction with environment

Performing activity

Network topologytimetable

Geometry

ObstaclesNeighbours

Characteristics ofthe pedestrians

Strategic level

Tactical level

Operational level

Modelling of Pedestrianand EvacuationDynamics, Fig. 1 Thedifferent levels ofmodelling of pedestrianbehavior. (According toDaamen (2004) andHoogendoorn et al. (2002))

Modelling of Pedestrian and Evacuation Dynamics 3

etc.) is required. In the following we will mostlybe concerned with the operational level,although some of the models that we are goingto describe allow to take into account certainelements of the behavior at the tactical levelas well.

Modelling on the operational level is usuallybased on variations of models from physics.Indeed the motion of pedestrian crowds has cer-tain similarities with fluids or the flow of granularmaterials. The goal is to find models which are assimple as possible but at the same time can repro-duce “realistic” behavior in the sense that theempirical observations are reproduced. Therefore,based on the experience from physics, pedestriansare often modelled as simple “particles” that inter-act with each other. Extensions to models of“intelligent” particles with numerous parametersand mechanisms for the interaction with theneighbors and the environment are usually calledmulti-agent systems see, e.g., Kielar et al. (2016a)and Ondrej and Pettré (2010)). Agents interactwith each other and with the environment. Theycan have an internal state reflecting, e.g., theirgoals and general behavior.

There are several characteristics which can beused to classify the modelling approaches:

• Microscopic vs. macroscopic: In microscopicmodels each individual is represented sepa-rately which allows to introduce differenttypes of pedestrians with individual propertiesin a natural way. In macroscopic models dif-ferent individuals cannot be distinguished.Instead the state of the system is described bytime and space averages of density, velocity,and flow.

• Discrete vs. continuous: Each of the threebasic variables for a description of a systemof pedestrians, namely, space, time, and statevariable (e.g., velocities), can be either discrete(i.e., an integer number) or continuous (i.e., areal number). Here all combinations are possi-ble. In a cellular automaton approach, all vari-ables are by definition discrete, whereas inhydrodynamic models all are continuous.These are the most common choices, butother combinations are used as well.

A discrete time is usually realized through theparallel or synchronous update where all par-ticles are moved at the same time. This intro-duces a timescale.

• Deterministic vs. stochastic: The dynamics ofpedestrians can either be modelled determinis-tically or stochastically. In the first case, themovement at a certain time is completely deter-mined by the present state. In stochasticmodels, the movement is determined by prob-abilities such that the agents can react differ-ently in the same situation. Such stochasticbehavioral rules often generate a rather realisticrepresentation of complex systems like pedes-trian crowds. On the other hand, thestochasticity in the models reflects the lack ofknowledge of the underlying physical pro-cesses that, e.g., determine the decision-making of the pedestrians.

This “intrinsic” stochasticity should be dis-tinguished from “noise.” Sometimes externalnoise terms are added to the macroscopicobservables, like the position or velocity.Often the main effect of these terms is toavoid certain special configurations which areconsidered to be unrealistic, like completelyblocked states. Otherwise the behavior is verysimilar to the deterministic case. For truestochasticity, on the other hand, the determin-istic limit usually has very different propertiesfrom the generic case.

• Rule-based vs. acceleration-based vs. velocity-based: The motion of the pedestrians to a newposition can be determined in different ways. In alarge class of models, first the new velocity oracceleration is determined which then allows(by integration) to calculate the new position ofa pedestrian. Usually this requires the solution ofa first-order and second-order differential equa-tion. Such models can be classified as velocity-based and acceleration-based models, respec-tively. In most models that do not belong tothese two classes, the determination of the newposition does not require the solution of a differ-ential equation. Instead the new position is spec-ified by certain rules which, e.g., take intoaccount the desired direction of motion and theposition of other pedestrians and obstacles. This


can be seen most clearly in cellular automatamodels. Often in these models, time is taken asa discrete variable. Models in this third class canbe called rule-based models or decision-basedmodels. They focus on the intrinsic propertiesof the agents, and thus the rules are often justifiedfrom psychology. It should be mentioned thatsometimes a clear distinction between velocity-and acceleration-based models on one hand andrule-based models on the other cannot always bemade and several models combine aspects ofboth approaches.

In acceleration-based models, the basicinteractions are typically described by forces:agents “feel” a force exerted by others and theinfrastructure. It is a physical approach basedon the observation that the presence of othersleads to deviations from a straight motion. Inanalogy to Newtonian mechanics, this changeof speed, i.e., the acceleration, is explained by aforce. The equations of motion allow to deter-mine the velocity and new position of eachagent by integration or solving second-orderdifferential equations. In velocity-basedmodels, the speed and the direction of anagent depend on the distance and position ofthe surrounding neighbors and obstacles. Theequations of motion determine directly thevelocity without referring to an acceleration.Therefore they are typically given by first-order differential equations. They are generallybased on collision avoidance and speed

optimization principles. In the simplest models,the speed is directly proportional to the distanceto the next agent or obstacle in front, while thedirection is the one maximizing the speed to thedesired destination. In contrast to rule-basedmodels, acceleration- and velocity-basedmodels emphasize the extrinsic properties andtheir relevance for the motion of the agents.

• Heuristic vs. first-principles: Heuristicmodels typically include several interactionterms considered to be relevant. These interac-tions are defined by parameters of the modelthat are used to fit empirical data. First-principle models are derived from certain pos-tulates considered to be fundamental. Often aclear distinction between the two approaches isnot possible.

• High vs. low fidelity: Fidelity here refers to theapparent realism of the modelling approach.High-fidelity models try to capture the com-plexity of decision-making, actions, etc. thatconstitute pedestrian motion in a realisticway. In contrast, in the simplest models, pedes-trians are represented by particles without anystimulus-response mechanism like viewing orhearing and cognitive capabilities like spatialknowledge or decision-making. The introduc-tion of “internal states” of an agent whichinfluence their behavior can be interpreted assome kind of “intelligence” that leads to morecomplex approaches, like multi-agent models.

Pedestrian models

Microscopic

Macroscopic

Discrete

(Mixed)

Continuous

Stochastic

Deterministic

Decision-based

Velocity-based

Acceleration-based

Modelling of Pedestrian and Evacuation Dynamics,Fig. 2 Different possible combinations of characteristicsof the microscopic pedestrian models. Based on the pro-posed model classification, a large number of differentapproaches can be distinguished. Here we show only

(part of) the classification scheme for microscopic models,but macroscopic models present similar classification. Themost common approaches are microscopic discrete, sto-chastic, rule-based models and continuous, deterministicacceleration-based models


Fig. 2 shows the variety of model classesobtained from this classification. There are exam-ples for almost every class shown here. It shouldbe mentioned that a clear classification accordingto the characteristics outlined here is not alwayspossible. In the following we will describe somemodel classes in more detail.

Acceleration-Based Models

A popular class of microscopic models for pedes-trian dynamics is defined by the so-calledacceleration-based, also known as force-based,models, where the movement of pedestrians isdefined by a superposition of exterior forces.Given initial values of the state variables, theresulting second-order ordinary differential equa-tions can be solved numerically to obtain thetrajectories of pedestrians.

In general, the acceleration of pedestrian i at agiven time step is directly proportional to a driv-ing force, pointing to a specific target. In mostknown acceleration-based models, this forcemodel is an exponential acceleration from zeroto a certain desired speed. However, in the pres-ence of other pedestrians or static obstacles, thetrajectory of pedestrian i deviates from a straightline, due to different repulsive forces acting oni from its environment.

For the solution of the equations of motion ofNewtonian many-particle systems, the well-founded molecular dynamics technique exists.However, in most studies so far, the distinctionsbetween pedestrian and Newtonian dynamicswith respect to the principles actio equals reactio,superposition of forces, or the meaning of themass in the second law are not discussed in detail.

The mathematical expression of the repulsiveforce differs from one model to another. In gen-eral, its magnitude increases with decreasinginter-distance of two pedestrians. This assures acertain volume exclusion of pedestrians, althoughin contrast to cellular automata models 6.1, pedes-trians may still collide or even overlap with eachother. In section “Limitations of Acceleration-Based Models” numerical difficulties and

properties of acceleration-based models will bediscussed in more details.

In the following we give a brief overview ofsome important milestones in the development ofacceleration-based models.

Herai-Tarui Model (HTM)In 1975 Hirai and Tarui proposed the first knownmicroscopic force-based model to investigate themovement of pedestrians in a 2D space (Hirai andTarui 1975). In their seminal work, the authorsinvestigated several aspects of human’s behav-ioral motion. On one side, the proposed modelconsidered the movement of pedestrians on the“operative level,” where pedestrians move indirection of an assigned goal while avoiding col-lisions with other pedestrians or obstacles. On theother side, the model incorporates differentaspects of the “tactical level” of human behavioras well, e.g., group behavior and the influence ofguiding signs on agent’s way-finding – aspectswhich recently caught the attention of the com-munity of pedestrian dynamics with severalemerging studies.

The equation of motion of the HTM is

mid2xidt2

þ vidxidt

¼ f1i þ f2i þ j, (1)

where mi, vi and xi are the mass, coefficient ofviscosity, and the position of the individual i,respectively. f1i and f2i are external forces actingupon the individual i, implementing differentmodelling concepts. f1i is a force required by theindividual i to form a group together with otherindividuals and to move forward, while f2i is aforce exerted by the environment around theindividual i.

The mean forces f1i and f2i are defined as asuperposition of other forces, expressing differentideas. The “group” force, for instance, consists ofthree components:

f1i ¼ fai þ fbi þ f ci, (2)

where fai is a driving force causing i to moveforward with constant speed. fbi is an interactionforce acting between the individual i and other


individuals. Depending on the distance betweentwo pedestrians, it can be attractive or repulsive.Finally, fci is the group force, which acts only ifsome individuals move together in a restricteddomain.

Both forces fai and fci are distance and angledependent. This expresses the observation that theinfluence of other pedestrians on i is greater in itsdirection of motion than on the side. No forces acton individuals from behind.

In analogy to the “group force,” the “environ-mental” force f2i is defined as the sum of otherforces, each with a specific effect, e.g., repulsionfrom walls and attraction toward signs. For moredetails the reader is referred to the original paper.

The last main term in the model j is a randomforce acting upon i to consider uncertainties in themodelled behavior. Its direction varies stochasti-cally, and its magnitude is a function of the dis-tance between i and the wall.

Hirai and Tarui showed that their modelexhibits, to some extent, realistic evacuationbehavior in a simplified train station. The authorsvalidated their model with respect to experimentsperformed on rats, making the assumption thathuman’s behavior in “panic”-like situations maybe comparable to animal’s dynamics.

Although the model was not elaborated furtherwhether in the original paper or in the literature, itshows some original concepts that were investi-gated lately in different works, e.g., group behav-ior, route choice modelling, attraction to a site,memory effects of objects, etc.

Social Force Model and Its DerivativesThe social force model (Helbing and Molnár1995) (SFM) is the most investigatedacceleration-based model for pedestrian dynam-ics, probably due to the simplicity of its definition,compared to the HTM, since it only implementsthe movement of pedestrians on the operativelevel. Besides, unlike the HTM, an exponentialacceleration of i from zero to a predefined maxi-mal speed is chosen (Pipes 1953). The main equa-tion of motion is given by

d2xidt2

¼ f vi,Dvij,Dxij� �þ v 0ð Þ

i � vit

, (3)

where f is the sum of the interactions of i with itsenvironment, t is a time constant, and vi = dxi/dt.In the original paper, these forces are described bythe negative gradient of a potential field withelliptical equipotential lines.

Formally, the social force model is the sum ofrepulsion with the neighbours and obstaclesweighted by the field of vision and a drivingforce for the desired direction (see Fig. 3). Themathematical expression of the driving force inEq. (3) is systematically used in all known force-based models, as it corresponds well to human’sacceleration in free movement (Moussaïd et al.2009).

The SFM in its original form was shown toreproduce two qualitative phenomena, namely,oscillations at bottlenecks and lane formations.

In high-density situations, the abovementioned“social” forces, initially defined to model naviga-tion of pedestrians around obstacles (and otherpedestrians), show that they may not be sufficientto guarantee a certain volume exclusion of pedes-trians. Therefore, additional physical (contact)forces, e.g., friction and compression, were intro-duced later on in Helbing et al. (2002). Besidesthis addition, the qualitative form of the repulsiveforces in the generalized SFM was assumed to beexponential. In Chen et al. (2017) a comprehen-sive review of the evolution of the SFM over theyears is given. Therein, the authors show thedifferent modifications and enhancements of thegeneric model.

Limitations of Acceleration-Based ModelsThe appeal of the force-based models is givenmainly by the analogy to Newtonian dynamics.In particular scenarios, where a pedestrian in acrowd has limited freedom, this analogy can bejustified. However, unlike particles or solidobjects, pedestrians can show a complex andnon-smooth dynamics while moving in space,especially in situations where prompt decisionsare made, like, for example, stopping.


Acceleration-based models describe particle withinertia which do not stop instantly.

Describing particles with inertia often leads tooverlapping and oscillations of the modelledpedestrians. Depending on the strength of therepulsive forces, the modelled body of pedestrianscan be excessively overlapped and hence violatethe principle of volume exclusion. In Kretz(2015), it was shown that in an 1D-scenario,oscillations in the movement of pedestrians areinherent. Pedestrians do not stop and keepmovingindependently of the actual situation. In somesituations pedestrians perform repetitive back-ward and forward movement due to, e.g., highrepulsive forces. In Chraibi et al. (2010) it wasfound that parameter of the repulsive forces can beadequately adjusted to mitigate overlapping andoscillations, without annihilating them totally.Furthermore, velocity-based models can bedescribed by construction volume exclusion (seethe next section).

In Köster et al. (2013), Köster et al. analyzedthe properties of the SFM from a numerical per-spective. They showed that in order to overcome

the numerical difficulties that result in a straight-forward implementation of the SFM, severalchanges in the model are necessary. Hence, theso-called mollified SFM, which removes discon-tinuities from the original equation of motion,shows better numerical behavior. Furthermore, itallows the usage of high-order fast-convergingnumerical solvers (usually an Euler scheme witha very small time step is used to numerically solvethe equation of motion, which results in very highcomputational overhead), which leads to adecrease in simulation time and increase in theaccuracy of the obtained results. With the pre-sented model, collisions and oscillations couldbe mitigated as well, especially in critical situa-tions, e.g., near intermediate goals.

Another issue of acceleration-based models,which is the superposition principle, wasdiscussed in Seitz et al. (2016). The authors sug-gest a change in the modelling perspectivetowards other paradigms, which may inspire newmodels.

Modelling of Pedestrianand EvacuationDynamics,Fig. 3 Illustration for thesocial force model. Theacceleration rate of apedestrian is the sum ofexponential repulsionweighted by the field ofvision and a driving force


Velocity-Based Models

In contrast to acceleration-based models, thevelocity in velocity-based (VB) models is instan-taneously adjusted to the neighborhood and theenvironment with no inertia or an implicit imple-mentation of a reaction time. Such a modellingapproach is largely inspired by motion planning inrobotics, since robots move with almost no inertiaand react instantaneously to a command. Con-straints on the velocity in case of contact allowto model hard-core body exclusion (volumeexclusion) and to describe collision-free pedes-trian dynamics with no overlapping (see Mauryet al. (2008) for a general framework).

Generally speaking VB approaches are basedon visual considerations (local interaction) andoptimization techniques under collision avoid-ance constraints. Mathematically, VB models aredifferential equations of the first order. A typicalexample is

dxidt

¼ V xj � xi, vj� �

, j�Ni

� �: (4)

Here the velocity is a function V depending onthe relative positions xj � xi and velocities vj ofthe neighbors j ϵ Ni of the considered pedestrian i.

Velocity Obstacle Models

Pioneer VB models inspired from the field ofrobotics are velocity obstacle models (Fiorini andShiller 1998). The velocity in this model class isdetermined by minimizing the deviation to thedesired speed while avoiding collision. The colli-sion possibilities are evaluated over horizon timesby assuming constant velocities of the neighborsand carrying out the velocity obstacle cones VO(i)(see Fig. 4). No collision occurs when the velocityis set outside the cones. The velocity is then thesolution of the optimization problem

Vi ¼ argmin k v� v0 kv=2 vo ið Þf g

, (5)

which minimizes the deviation to the desiredspeed v0. Similar optimization problems coupledto exclusion constraints in case of contact aresolved in Maury and Venel (2011) for evacuationdynamics.

Velocity obstacle models can show so-calledping-pong undesired oscillatory effects (i.e., twopedestrians switching from one direction toanother). This phenomenon results from theimplicit definition of velocities in the model (thespeed of a pedestrian depends on the speed of theneighbor and vice versa). The reciprocal velocityobstacle model (RVO) allows to avoid thisundesired behaviour by taking into account thefact that the neighbors make similar collisionavoidance reasoning (van den Berg et al. 2008).However, this formulation only guarantees hard-core body exclusion under specific conditions.The optimal reciprocal collision avoidance(ORCA) overcomes this limitation and providessufficient and efficient conditions for movingagents to avoid collisions among one another(van den Berg et al. 2011). ORCA modellingapproach and its variants are frequently used todescribe pedestrian dynamics (see, e.g., Guy et al.(2009, 2010a, b), Kim et al. (2014), Mordvintsevet al. (2014), and Paris et al. (2007)).

xi

xj

xk

vi

vj

vk

vj

vk

VOj

VOk

Modelling of Pedestrian and Evacuation Dynamics,Fig. 4 Velocity obstacle cones VO(i) for the pedestrian i.No collision occurs when the velocity is set outside thecones.


Other Velocity-Based Models

Other VB models consider the velocity as theproduct of a speed V (scalar) by a directione (such that ||e|| = 1):

dxidt

¼ V xj � xi, vj� �

, j�Ni

� �� e xj � xi, vj

� �, j�Ni

� �: (6)

Both speed V and direction e depend on therelative positions of the neighbors and their veloc-ities. For the synthetic-vision-based steeringmodel (Ondrej and Pettré 2010), the speed V is afunction of the time-to-interaction (TTI, a quantityclose to the time-to-collision) with the neighbors,while the direction e is a function of the TTI, thebearing angle and its derivation

V i ¼ u0 1� exp �minj

TTIi,j

� �� and ei

¼ F TTIi,j, ai,j, _ai,j� �

: (7)

The bearing angle a is the angle between thecurrent direction and the direction to the neigh-bour (see Fig. 5). It remains constant in case ofcollision, while it varies sharply for safe crossings.

In other VB approaches, the speed depends, inanalogy to optimal velocity (OV) models in traf-fic, on the minimal distance in front (Tordeux et al.2016), while the direction is, as in acceleration-based models, a sum of exponential repulsionwith the neighbors (Dietrich and Köster 2014). Ifei is the direction of the pedestrian i, ei,j the direc-tion from j to i, and si,j = ||xj � xi|| the spacingdistance between pedestrians i and j, and if thepedestrians are considered as discs of diameter ‘,then the pedestrians in front of the pedestrian i arethe set J= {j, ei � ei,j � 0 and e⊥i � ei,j �‘/si, j}. Theminimum spacing in front is si = minjϵ J� si,j (seeFig. 6). The optimal velocity VB model withexponential repulsion is then.

V i ¼ f sið Þ and ei

¼ 1

N

v0k v0 k þ A

Xj

exp �si,j=B� � !

: (8)

Here N is a normalization constant and f is theoptimal velocity function. Any positive OV func-tion f depending on the minimal spacing in front siensures collision-free dynamics as soon as f(s)= 0for all s� ‘, independently to the direction model(Tordeux et al. 2016). Note that in contrast to VB

α

α̇ > 0

α̇ = 0

α̇ < 0

Modelling of Pedestrian and Evacuation Dynamics,Fig. 5 Bearing angle a of two pedestrians. The pedes-trians collide when the bearing angle is constant (i.e., whenits derivative vanishes)

�ei,j

ei

xi

xk

xj

si

Modelling of Pedestrian and Evacuation Dynamics,Fig. 6 Minimal spacing in front. The pedestrians in frontof the pedestrian i (i.e., into the gray area) are the set J= {j,ei � ei,j� 0 and e⊥i � ei,j� ‘/si,j}, while the minimum spacingin front is si = minj� J si,j


models described above, the velocity in thismodel solely depends on the relative positions ofthe neighbors, and not on their velocities, makingthe system of velocities explicitly defined, in con-trast to the velocity obstacle or the synthetic-vision-based steering models.

Models with NoiseVB models do not incorporate relaxation or delaymechanisms and consequently do not describestop-and-go wave phenomena. However, relaxa-tion and diffusion processes can be introduced byusing specific noises in the dynamics. Forinstance, in Tordeux and Schadschneider (2016),the speed is proportional to the distance spacingand has an additive noise

dxidt

¼ 1

Txiþ1 � xi � ‘ð Þ þ ei,

deidt

¼ � 1

bei þ a

dWi

dt

8><>: (9)

i + 1 being the pedestrian directly in front of thepedestrian i. The parameter T is the time gap and ‘is the pedestrian size, a and b are parametersrelated to the noise, and Wj is a Wiener process.

Classically the noise is white in acceleration-based models (see, e.g., (Helbing and Molnár(1995)). Here the noise is a Brownian one relaxedat the second order to describe the autocorrelationof the speed residuals. Simulation results show thatthe relaxed noises allow to describe oscillatingtraffic waves (stop-and-go dynamics) (Tordeuxand Schadschneider 2016). Thus this modellingAnsatz of stop-and-go waves for pedestriandynamics is noise-induced. It contrasts with themodelling of stop-and-go for traffic flow, generallydone by means of instability and phase transitions.

Rule-Based Models

In contrast to the models discussed in sections“Acceleration-Based Models and Velocity-BasedModels,” another class of models is not based ondifferential equations. Instead the dynamics isbased on rules or decisions that the agents makein order to determine their new positions, velocity,etc. Therefore these models can be called

decision-based models or rule-based models.Typically time is considered to be discrete in thisapproach, i.e., it increases in certain time steps Dtsuch that decisions are only made at discretetimes. Since this approach does not lead to adifferential equation-based description of thedynamics, these models could be interpreted aszeroth-order models. The time step corresponds toa natural timescale Dt which could, e.g., be iden-tified with some reaction time. This can be usedfor the calibration of the model which is essentialfor making quantitative predictions.

Cellular AutomataCellular automata (CA) are rule-based dynamicmodels that are discrete in space, time, and statevariable which in the case of traffic usually corre-sponds to the velocity. In computer simulationsthe discrete time is usually realized through aparallel or synchronous update where all pedes-trians move at the same time. A natural spacediscretization into cells can be derived from themaximal densities observed in dense crowdswhich gives the minimal space requirement ofone person. Usually each cell in the CA can onlybe occupied by one particle (exclusion principle)so that this space requirement can be identifiedwith the cell size. In this way, a maximal densityof 6.25 P/m2 (Weidmann 1993) leads to a cell sizeof 40 � 40 cm2. The exclusion principle and themodelling of humans as non-compressible parti-cles mimics short-range repulsive interactions,i.e., the “private-sphere.”

The dynamics is usually defined by rules whichspecify the transition probabilities for the motion toone of the neighboring cells (Fig. 7). The modelsdiffer in the specification of these probabilities aswell in that of the “neighborhood.” For determin-istic models all except of one probability are zero.

The first cellular automata (CA) models (Blueand Adler 2000; Fukui and Ishibashi 1999b;Klüpfel et al. 2000; Muramatsu et al. 1999) forpedestrian dynamics can be considered two-dimensional variants of the asymmetric simpleexclusion process (ASEP) (for reviews, see Blytheand Evans (2007), Derrida (1998), and Schütz(2001)) or models for city or highway traffic(Biham et al. 1992; Chowdhury et al. 2000;Nagel and Schreckenberg 1992) based on it. Most


of these models represent pedestrians by particleswithout any internal degrees of freedom. They canmove to one of the neighboring cells based oncertain transition probabilities which are deter-mined by three factors: (1) the desired direction ofmotion, e.g., to find the shortest connection,(2) interactionswith other pedestrians, and (3) inter-actions with the infrastructure (walls, doors, etc.).

Blue-Adler ModelThemodel of Blue and Adler (2000, 2002) is basedon a variant of the Nagel-Schreckenberg model(Nagel and Schreckenberg 1992) of highwaytraffic. Pedestrian motion is considered in analogyto a multilane highway. The structure of the rules issimilar to the basic two-lane rules suggested inRickert et al. (1996). The update is performed infour steps which are applied to all pedestrians inparallel. In the first step, each pedestrian chooses apreferred lane. In the second step, the lane changesare performed. In the third step, the velocities aredetermined based on the available gap in the newlanes. Finally, in the fourth step, the pedestriansmove forward according to the velocities deter-mined in the previous step.

In counterflow situations, head-on conflictsoccur. These are resolved stochastically, and withsome probability opposing pedestrians are allowedto exchange positions within one time step. Notethat the motion of a single pedestrian (notinteracting with others) is deterministic otherwise.

In the Blue-Adler model, motion is notrestricted to nearest neighbor sites. Instead pedes-trians can have different velocities umax whichcorrespond to the maximal number of cells theyare allowed to move forward. In contrast to

vehicular traffic, acceleration to umax can beassumed to be instantaneous in pedestrian motion.

Floor Field CAIn the floor field CA (Burstedde et al. 2001, 2002;Kirchner and Schadschneider 2002; Schadschneider2002) the transition probabilities to neighboringcells are no longer constant but vary dynamically.This is motivated by the process of chemotaxis (see(Ben-Jacob (1997) for a review) used by someinsects (e.g., ants) for communication. They createa chemical trace to guide other individuals to foodsources. In the approach of Burstedde et al. (2001),the pedestrians also create a trace which, however, isonly virtual although one could assume that it cor-responds to some abstract representation of the pathin the mind of the pedestrians. This reduces interac-tions to local ones that allow efficient simulations inarbitrary geometries since, e.g., it does require tocheck whether the interaction between particles arescreened by walls, etc. The number of interactionterms always grows linearly with the number ofparticles.

The translation into local interactions isachieved by the introduction of so-called floorfields. The transition probabilities for all pedes-trians depend on the strength of the floor fields intheir neighborhood in such a way that transitionsin the direction of larger fields are preferred. Thedynamic floor field Dij corresponds to a virtualtrace which is created by the motion of the pedes-trians and in turn influences the motion of otherindividuals. Furthermore it has its own dynamics.Diffusion and decay lead to a dilution and finallythe vanishing of the trace after some time. Thestatic floor field Sij does not change with timesince it only takes into account the effects of the

pij

pi+1j

pi−1j

pij−1 pij+1

0

0

0

0

Modelling of Pedestrianand EvacuationDynamics, Fig. 7 Aparticle, its possibledirections of motion, andthe corresponding transitionprobabilities pij for the caseof a von Neumannneighborhood


surroundings. Therefore it exists even without anypedestrians present. It allows to model, e.g., pre-ferred areas, walls, and other obstacles. Figure 8shows the static floor field used for the simulationof evacuations from a room with a single door. Itsstrength decreases with increasing distance fromthe door. Since the pedestrian prefers motion intothe direction of larger fields, this is already suffi-cient to find the door.

Coupling constants control the relative influ-ence of both fields. For a strong coupling to thestatic field, pedestrians will choose the shortestpath to the exit. This corresponds to a “normal”situation. A strong coupling to the dynamic fieldimplies a strong herding behavior where pedes-trians try to follow the lead of others. This oftenhappens in emergency situations.

The model uses a fully parallel update. There-fore conflicts can occur where different particleschoose the same destination cell. Conflicts havebeen considered a technical problem for a longtime, and usually the dynamics has been modifiedin order to avoid them. The simplest method is toupdate pedestrians sequentially instead of using afully parallel dynamics. However, this leads toother problems, e.g., the identification of the rel-evant timescale. Therefore it has been suggestedin Kirchner et al. (2003a) to take these conflictsseriously as an important part of the dynamics.

For the floor field model, it has been shown inKirchner et al. (2003b) that the behavior becomesmore realistic if not all conflicts are resolved in thesense that one of the pedestrians is allowed tomove, whereas the others stay at their positions.Instead with probability m, which is called frictionparameter, the movement of all involved pedes-trians is denied (Kirchner et al. 2003b) (seeFig. 9). This allows to describe clogging effectsbetween the pedestrians in a much more detailedway. Although a local effect, it can have enormousinfluence on macroscopic quantities like flow andevacuation time.

The original floor field model has beenextended in various ways. An intrinsic problemof CA models is the limited spatial resolutionwhich is a consequence of the discreteness inspace. This makes it, e.g., impossible to representa corridor of width 1 m by 40� 40 (cm)2 cells. Toavoid this problem, models with smaller cells(e.g., 40 � 40 (cm)2) have been proposed(Kirchner et al. 2004). In this case pedestrianscorrespond to extended particles that occupymore than one cell (e.g., four cells).

Modelling of Pedestrian and Evacuation Dynamics,Fig. 8 Left: Static floor field for the simulation of anevacuation from a large room with a single door. Thedoor is located in the middle of the upper boundary, andthe field strength is increasing with increasing intensity.Right: Snapshot of the dynamical floor field created bypeople leaving the room

t t + 1

µ

Modelling of Pedestrianand EvacuationDynamics,Fig. 9 Refused movementdue to friction parameter m


The lattice structure introduces a spatial anisot-ropy. As a consequence straight motion in direc-tions not parallel to the lattice axis is difficult toachieve. This can to some degree be cured by aproper choice of the transition probabilities(Burstedde et al. 2001; Schultz and Fricke2010). For similar reasons in some models, agentsare allowed to move further than just to nearestneighbor cells similar to the Blue-Adler model(Kirchner et al. 2004; Kretz 2007; Kretz andSchreckenberg 2006; Luo et al. 2018).

Most variants, however, have introduced othertypes of interactions. For counterflow situationsthe inclusion of anticipation effects (Nowak andSchadschneider 2012; Suma et al. 2011) leads to amore realistic behavior. Another interesting exten-sion is the inclusion of game theory, e.g., in theresolution of conflicts (Bouzat and Kuperman2014) or the choice of exits in an evacuation(Lo et al. 2006).

Other CA ModelsThere are many other cellular automata modelsthat have been proposed in the last 20 years. Mostof them can be considered generalizations of theASEP. One of the earliest is the Fukui-Ishibashimodel (Fukui and Ishibashi 1999a, b) which isbased on a two-dimensional variant of the ASEP.Originally it was used to study bidirectionalmotion in a long corridor. The motion is determin-istic; only sidestepping to avoid oppositely mov-ing pedestrian is stochastic. Various extensionsand variations of the model have be proposed,e.g., an asymmetric variant (Muramatsu et al.1999) where walkers prefer lane changes to theright, different update types (Fang et al. 2003),simultaneous (exchange) motion of pedestriansstanding “face to face” (Jian et al. 2005), or thepossibility of backstepping (Maniccam 2005).The influence of the shape of the particles hasbeen investigated in Nagai and Nagatani (2006).Also other geometries (Muramatsu and Nagatani2000b; Tajima and Nagatani 2002) and extensionsto full two-dimensional motion have been studiedin various modifications (Maniccam 2003, 2005;Muramatsu and Nagatani 2000a).

In the Gipps-Marksjö model (Gipps andMarksjö 1985), interactions between pedestrians

are assumed to be repulsive, anticipating the ideaof social forces (see section “Acceleration-BasedModels”). To each cell a score is assigned basedon its proximity to other pedestrians. This scorerepresents the repulsive interactions, and theactual motion is then determined by the competi-tion between these repulsion and the gain ofapproaching the destination. The pedestrian thenselects the cell of its nine neighbors (Moore neigh-borhood) which leads to the maximum benefitwhich is given by the difference between thegain of moving closer to the destination and thecost of moving closer to other pedestrians. Thisleads to a deterministic model dynamics.Updating is done sequentially to avoid conflicts.

Related Approaches

Lattice-Gas ModelsConceptually, lattice-gas models are very similarto cellular automata models. For the modelling ofpedestrian dynamics, the main difference is usu-ally the position of the particles. In lattice-gasmodels the pedestrians are located on the verticesof the lattice, whereas they are located on the facesof the lattice in CA. Apart from this, the definitionof the dynamics is analogous.

However, there are models of pedestriandynamics which take inspiration from lattice-gasmodels that have been introduced for the descrip-tion of classical hydrodynamics (Frisch et al.1986), e.g., the model introduced by Marconiand Chopard (2002) which is based on a sequenceof collision and propagation steps.

Semi-Discrete ModelsIn Yamamoto et al. (2007) an approach based onreal-coded cellular automata (RCA) has been pro-posed. In contrast to a genuine CA, the velocitiesof the agents in a RCA are not discrete, and agentscan move in any direction. Therefore the velocityis not given by the number of lattice sites the agentmoves. The new position is determined from thenon-integer velocity by using a rounding proce-dure. RCA models have the advantage of makingthe motion isotropic irrespective of the latticestructure.


In the Optimal Steps Model (Seitz and Köster2012), the pedestrian movement takes place incontinuous space and is discretized in a way thatreflects the movement in steps. The model is builtin order to achieve natural grid-free trajectorieswithout introducing social force-type differentialequations or complex steering behaviors.

Macroscopic Models

Fluid-Dynamic ModelsPedestrian dynamics has some obvious similari-ties with fluids. For example the motion aroundobstacles appears to follow “streamlines.” There-fore it is not surprising that, very much like forvehicular dynamics, the earliest models of pedes-trian dynamics took inspiration from hydrody-namics or gas-kinetic theory (Bellomo et al.2012; Helbing 1992; Henderson 1974). Typicallythese macroscopic models are deterministic,force-based, and of low fidelity.

Motivated by the observation that the velocitydistribution functions of crowds at low densitieshave a good agreement with the Maxwell-Boltzmann distribution (Henderson 1971), Hen-derson has developed a fluid-dynamic theory ofpedestrian flow (Henderson 1974). Describing theinteractions between the pedestrians as a collisionprocess where the particles exchange momentaand energy, a homogeneous crowd can bedescribed by the well-known kinetic theory ofgases. However, the interpretation of the quanti-ties is not entirely clear, e.g., what the analoguesof pressure and temperature are in the context ofpedestrian motion. Temperature could be identi-fied with the velocity variance, which is related tothe distribution of desired velocities, whereas thepressure expresses the desire to move against aforce in a certain direction.

Hughes (2000, 2002, 2003) has derived a fluid-dynamical model which he calls “thinking fluid”based on a function measuring the discomfort of apedestrian at a given density and the tendency tominimize the estimated travel time. It leads to anonlinear system of partial differential equationsinvolving an eikonal equation.

The applicability of classical hydrodynamicmodels is based on several conservation laws.The conservation of mass, corresponding to con-servation of the total number of pedestrians, isexpressed through a continuity equation of theform

@r r,tð Þ@t

þ▽:J r,tð Þ ¼ 0, (10)

which connects the local density r(r,t) with thecurrent J(r,t). This equation can be generalized toinclude source and sink terms. However, theassumption of conservation of energy andmomentum is not true for interactions betweenpedestrians which in general do not even satisfyNewton’s third law (“actio= reactio”). In Helbing(1992), several other differences to normal fluidswere pointed out, e.g., the anisotropy of interac-tions or the fact that pedestrians usually have anindividual preferred direction of motion.

Gaskinetic ModelsIn Helbing (1992) a fluid-dynamical descriptionwas derived on the basis of a gaskinetic modelwhich describes the system in terms of a densityfunction f (r, v, t). The dynamics of this function isdetermined by Boltzmann’s transport equationthat describes its change for a given state as dif-ference of inflow and outflow due to binarycollisions.

An important new aspect in pedestrian dynam-ics is the existence of desired directions of motionwhich allows to distinguish different groups m ofparticles. The corresponding densities fm changein time due to four different effects:

1. A relaxation term with characteristic time tdescribes tendency of pedestrians to approachtheir intended velocities.

2. The interaction between pedestrians ismodelled by a Stosszahlansatz as in theBoltzmann equation. Here pair interactionsbetween types m and u occur with a total ratethat is proportional to the densities fm and fu.

3. Pedestrians are allowed to change from type mto u which, e.g., accounts for turning left orright at a crossing.


4. Additional gain and loss terms allow to modelentrances and exits where pedestrians can enteror leave the system.

The resulting fluid-dynamic equations are sim-ilar to that of ordinary fluids although, due to thedifferent types of pedestrians, one actually obtainsa set of coupled equations describing severalinteracting fluids. Equilibrium is approachedthrough the tendency to walk with the intendedvelocity, not through interactions as in ordinaryfluids. Momentum and energy are not conservedin pedestrian motion, but the relaxation toward theintended velocity describes a tendency to restorethese quantities. Unsurprisingly for a macroscopicapproach, the gas-kinetic models have problemsat low densities. For further discussion, see, e.g.,Helbing (1992).

Network and Queuing ModelsAnother class of models is based on ideas fromqueuing theory. Typically rooms are representedas nodes in the queuing network, and links corre-spond to doors. In microscopic approaches eachagent chooses a new node, e.g., according to someprobability (Lovas 1994). In Treuille et al. (2006),a fluid-dynamic model is transformed into a par-ticle system in a network. Some explicit particle-to-particle interactions are considered, forinstance, to enforce a minimum distance betweenpedestrians.

Queuing models are generally simple to imple-ment and fast to simulate. They are particularlysuitable to large-scale simulation of pedestrianmovement. For instance, the queue simulationmodel introduced by Lämmel et al. is used forthe evacuation of the city of Padang (Indonesian)in case of a tsunami warning (Lämmel et al. 2009).The affected population comprised approximately330,000 individuals, and the overall network con-sists of 6289 nodes, 16,978 links, and more than900,000 cells in an area of roughly 7 � 4 km2.Few minutes (approximately 400 s) were neces-sary in 2011 to simulate the full evacuation on astandard computer.

Lastly, multi-scale models on network may bementioned. Multi-scale (or hybrid) models com-bine models formulated at different levels of

aggregation yielding in a compromise betweencomputational complexity and accuracy(Crociani et al. 2016; Lämmel et al. 2014). Ensur-ing consistency across levels, and especially tran-sitions from a level to another, both in terms ofparameter calibration and in terms of state vari-ables, makes the use of such models howeverrelatively difficult.

Future Directions and Open Problems

Data-Driven ModellingThe increase of real and experimental databaseand computational capacity of the computersmakes nowadays possible the development ofmachine learning approaches for prediction ofpedestrian movement. Oppositely to physicalmodels, data-driven approaches are deliberatelycomplex and have a high number of parameterswith no possible physical interpretations (seeFig. 10). Their calibration (or training) generallyrequires a very large amount of data. Accuratelytrained, the very high plasticity of the predictionmodels allows in theory to describe any type ofpatterns.

The methodological approach with data-drivenapproaches consists in partitioning the dataset intraining, testing, and even predicting subsets.Bootstrap techniques are used to evaluate thequality (precision) of the prediction (Kohavi1995). Data-driven modelling approach is usedfor predictions of pedestrian movements, notablyin complex geometries (Das et al. 2015), for themotion planning of robots moving in a crowd(Chen et al. 2018), or for modelling of walkerpose (Fragkiadaki et al. 2015). Simplestapproaches are based on artificial neural networks(see, e.g., Das et al. (2015) and Ma et al. (2016)),while the most sophisticated prediction models liein Long-Short-Term Memory networks (Alahiet al. 2016) or deep reinforcement learning tech-niques (Chen et al. 2017).

The predictions with machine learningapproaches are generally more precise than thoseof physical models; however their robustness fordifferent types of facilities and walking situationsremains to be demonstrated. However, the


development of data-driven approaches isexpected to grow during the next years. This canbe for prediction of pedestrian movement in com-plex facilities, and potentially also of evacuations,as well as for motion planning of robots moving ina crowded environment. Yet the calibration(training) and validation of machine learningmodels are difficult tasks. One of the reasons liesin the fact that the amount data necessary forcalibration depends on the complexity of themodelling approach. A too complex approach,for instance, leads to overfitting of the data usedfor the calibration and bad prediction of new data.

Validation and CalibrationAll models should be validated before they areused in practical applications. If quantitative pre-dictions are required, e.g., estimates of evacuationtimes, the models also should be calibrated prop-erly. So far there seems to be no real consensushow this can be achieved.

As discussed in the companion article (Bolteset al. 2018), empirically several interesting collec-tive phenomena are observed in crowd motion,e.g., clogging at bottlenecks, stop-and-go motion,dynamical lane formation in counterflow,etc. Currently the qualitative reproduction ofthese effects is often used as criterion for thevalidity of a model.

The main problem with calibration is the lackof reliable empirical data (Seyfried andSchadschneider 2009). As described in Bolteset al. (2018) even for fundamental diagram, i.e.,the density dependence of the flow, currently noconsensus exists. Published data for the funda-mental diagram – and other quantities as well –differ substantially. The origin of these discrepan-cies is – up to now – not clear and one of the mostimportant areas of research. The situation is thusvery different from that in vehicular traffic. Here abasic consensus about the quantitative dependen-cies exists, although there is still some debateabout the interpretation of the data.

The lack of generally accepted quantitativeempirical results causes problems for the calibra-tion of models. In highway traffic the empiricalfundamental diagram is the main source for cali-bration. However, the published fundamental dia-grams of pedestrian streams differ quantitatively somuch that they cannot be considered to yield reli-ably calibrated models. Therefore the developmentof a proper procedure formodel calibration remainsone of the most important issues of pedestriandynamics (Seyfried and Schadschneider 2009).

As outlined in the introduction, differentresearch areas pursuing different objectives con-tribute to the modelling of pedestrian dynamics.This diversity makes it difficult to establish a

INPUTSystem state at t

Positions/velocitiesof surroundding neigh-bors and obstacles

OUTPUTSystem state t + 1

Position, velocity oracceleration rate ofthe pedestrians

Classical models

Acc = f(x, ...) or Speed = g(x, ...)with parameters v0, d0, τ , ...

Explicit nonlinear function

Data-driven approaches

Non-explicit nonlinear function

Modelling of Pedestrian and Evacuation Dynamics,Fig. 10 Minimalistic illustrative scheme for the distinc-tion between classical models calibrated by few

meaningful parameters, and data-driven modelling, forwhich the number of parameters is deliberately large


consensus with respect to minimal standards or ageneral accepted suite of benchmarks for models.

Route Choice and NavigationWhile most models focus on the operational level,only few works exist to model route choices andnavigation of agents on the tactical or strategiclevel; see section “Classification of Models.”

For application of evacuation, the shortest pathis a standard approach (Dijkstra 1958). A moreadvanced method is the representation of routesaccording to graphs (see Höcker (2010) andHöcker et al. (2010). To optimize the travel timeby considering waiting times, e.g., for the selec-tion of doors or alternative routes, see KemlohWagoum et al. (2012), Kretz (2009) and Kretzet al. 2015).

Several approaches exist taking humanwayfinding abilities into account. The impact ofdiscrete choices at the knots of the network wasmodelled by Løvs (1998). Herding as a strategy ofunfamiliar agents was used in Ehtamo et al.(2010) by introducing game theoreticalapproaches to model imitation of decisions ofother agents. In Kneidl (2013)) spatial inaccura-cies concerning metrics and direction were usedfor wayfinding. This approach was extended byKielar et al. (2016b) to agents with differentknowledge. To enable agents finding an exit with-out any knowledge, search strategies like randomsearch and nearest room/door heuristic were usedin von Sivers et al. (2016). Models for the percep-tion of signage including the dependency on thesigns’ properties, the distances, the observationangles to signs, or lighting conditions could befound in Filippidis (2006), Nassar (2011), and Xieet al. (2007).

An approach to model partial knowledge inte-grating visibility graphs with a cognitive mapapproach including landmarks, generalizedknowledge like the functionality of corridors,and stairs and signage can be found in Andresenet al. (2018).

Simulation FrameworksFor application, several models are implementedin different software. Besides commercial soft-ware several open-source projects like Menge

(Curtis et al. 2016), FDS + Evac (Korhonenet al. 2010), VADERE (Köster et al. 2017), orJuPedSim (Kemloh Wagoum et al. 2015) areavailable. These frameworks enable the user todefine the number and initial positions of pedes-trians, the parameter of the models, and the spatiallayout of the buildings with the definition ofusable exits. Kielar et al. (2016a) give an exten-sive overview of the existing pedestrian simula-tion frameworks. Meanwhile they introduceMomenTUMv2, a generic, modular, and flexibleframework for agent-based simulation thatenables users to couple different models in aneasy way.

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