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Royal Institute of Technology

Bachelor’s Thesis

Evacuating Crowds Using ActiveIntervention: Modelling and Verification

by Simulation

Authors:

Victor Tingström Karl Nyman

[email protected] [email protected]

A thesis submitted in partial fulfilment of the requirements

for the degree of Bachelor of Science in Engineering

May 2013

ROYAL INSTITUTE OF TECHNOLOGY

Abstract

Department of Mathematics

Division of Optimization and Systems Theory

Bachelor of Science in Engineering

Evacuating Crowds: Modelling and Verification by Simulation using Active

Intervention

by Victor Tingström and Karl Nyman

The ability to simulate crowds during emergency evacuations could help designers of

buildings and crowd management systems to decrease the number of casualties in e.g.

fires. With the use of computer simulations of crowd evacuation, costly and possibly

unethical real-world experiments can be avoided.

The proposed crowd evacuation model extends Helbing’s Social Force Model with ob-

stacle and collision avoidance. It also includes elements of cognition such as impatience

and an exit search algorithm. Two active intervention measures are evaluated: The

inclusion of leaders and exit signs.

Simulations verify that the model produces realistic human behaviour. They also imply

that exit signs decrease evacuation time. Leaders decrease evacuation time but above a

threshold ratio of leaders among evacuees, improvement is non-existent. The threshold

ratio is dependent on the number of evacuees in a room and varies between 10% and

30% for 20 to 30 evacuees. It is important to assess in which context the model is used

in order for the assumptions to be valid. Results should be interpreted qualitatively

because quantitative verification has not been made.

http://www.kth.sehttp://math.kth.se/optsyshttp://math.kth.se

Acknowledgements

We would like to thank our supervisor Yuecheng Yang at the Department of Optimiza-

tion and Systems Theory at Kungliga Tekniska Högskolan for his extensive support and

encouragement.

We are also grateful towards Prof. Xiaoming Hu at the Department of Optimization

and Systems Theory at Kungliga Tekniska Högskolan for valuable insight and the op-

portunity of doing this work.

Finally, we would like to thank Mr. Sunil Patel at www.sunilpatel.co.uk for providing

us with the LaTeX template that this thesis paper is based on.

ii

Contents

Abstract i

Acknowledgements ii

List of Figures vi

Abbreviations vii

Symbols viii

1 Introduction 1

1.1 Background and relevance . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Key concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Aims and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Outline of the model CEPABS . . . . . . . . . . . . . . . . . . . . . . . . 2

1.5 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.6 Collaborations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.7 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Literature review 5

2.1 Models of crowd behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Flocking behaviour and SFM:s . . . . . . . . . . . . . . . . . . . . 5

2.1.2 Cellular automata models . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.3 Leader- follower models . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Crowd behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Response to signal systems . . . . . . . . . . . . . . . . . . . . . . 8

2.2.2 Exit choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.3 Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.4 Velocity alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.5 Obstacle avoidance . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Search algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Choice of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 The Model CEPABS 14

3.1 Introduction to the model . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Forces of the SFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

iii

Contents iv

3.2.1 Helbing’s model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.2 Inter-agent forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.3 Physical obstacle forces . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.4 Parameters used in the force model . . . . . . . . . . . . . . . . . . 20

3.3 Forces added to Helbing’s model . . . . . . . . . . . . . . . . . . . . . . . 20

3.3.1 Obstacle avoidance causing retarding . . . . . . . . . . . . . . . . . 20

3.3.2 Wall following . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.3 Speed dampening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.4 Obstacle avoidance . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.5 Desired direction of an agent . . . . . . . . . . . . . . . . . . . . . 23

3.3.6 Moving followers out of the way for leaders . . . . . . . . . . . . . 24

3.3.7 Parameters of the added forces . . . . . . . . . . . . . . . . . . . . 24

3.4 Geometrical representation of a building . . . . . . . . . . . . . . . . . . . 24

3.5 Target points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5.1 Door targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5.2 Viewpoint target points . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5.3 Auxiliary target points . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.5.4 Terminal points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.5.5 Target zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5.6 Exit signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.6 Point Search Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.7 Mental map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.8 Patience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.9 The leader concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.10 Pathfinding Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.11 Pathfollowing Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.12 Exit search strategy for leaders . . . . . . . . . . . . . . . . . . . . . . . . 35

3.13 Exit search strategy for non-leaders . . . . . . . . . . . . . . . . . . . . . . 35

3.13.1 Prioritising of target points . . . . . . . . . . . . . . . . . . . . . . 36

3.13.2 Leader Following . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.13.3 Single Target Pursuit . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.13.4 Exit Choice Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.14 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Simulation Results 40

4.1 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Verification of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 Arching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.2 Avoidance of obstacles . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2.3 Agent avoidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2.4 Leader shield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3 Results from testing of active intervention measures . . . . . . . . . . . . 45

4.3.1 The effect of leaders . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3.2 The effect of exit signs . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 Discussion 50

Contents v

5.1 Discussion of simulation results . . . . . . . . . . . . . . . . . . . . . . . . 50

5.1.1 Expected agent behaviour . . . . . . . . . . . . . . . . . . . . . . . 50

5.1.2 Effect of active intervention measures . . . . . . . . . . . . . . . . 51

5.2 Comparison with other models . . . . . . . . . . . . . . . . . . . . . . . . 53

5.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.4 Assumptions made in CEPABS . . . . . . . . . . . . . . . . . . . . . . . . 56

5.5 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.5.1 Phenomenological verification . . . . . . . . . . . . . . . . . . . . . 57

5.5.2 Video based verification . . . . . . . . . . . . . . . . . . . . . . . . 57

5.5.3 Evacuation time comparison . . . . . . . . . . . . . . . . . . . . . . 58

5.5.4 Suggestions for improvement . . . . . . . . . . . . . . . . . . . . . 59

5.5.5 Comments on the verification of CEPABS . . . . . . . . . . . . . . 59

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6 Conclusions 61

List of Figures

3.1 Wall influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Agent avoidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Obstacle avoidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.1 Building geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Exit sign placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 Arching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4 Obstacle avoidance 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43



4.7 Agent avoidance 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.8 Agent avoidance 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.9 Leader shield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.10 Evacuating 20 people with 1 leader . . . . . . . . . . . . . . . . . . . . . . 46

4.11 Evacuating 20 people with 2 leaders . . . . . . . . . . . . . . . . . . . . . 46





4.16 Leader to follower 20 people . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.17 Leader to follower 30 people . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.18 Evacuating 20 poeple with no signs . . . . . . . . . . . . . . . . . . . . . . 48

4.19 Evacuating 20 poeple with signs . . . . . . . . . . . . . . . . . . . . . . . 48





vi

Abbreviations

SFM Social Force Model

CAM Cellular Automata Model

CEPABS Crowd Evacuation of Pedestrians using Agent Based Simulation

vii

Symbols

ri position of agent i m

Ri radius of agent i m

vi velocity of agent i ms−1

v0i desired speed of agent i

rtargeti target position of agent i

mi mass of agent i kg

τ acceleration time s

pind individualism parameter

ppatiencei patience level of agent i

Rsight sight radius of agent i m

Si mental map of agent i

e0i desired direction of agent i

Rij combined radii of agent i and j m

dij distance between agent i and j m

diW distance between agent i and obstacle W m

κ constant of friction kgm−1s−1

A constant of inter agent forces N

B constant of inter agent forces m

K spring constant kgs−2

rtarget target point of an agent

Aagent.ret constant of agent retarding force N

Aobst.ret constant of obstacle retarding force N

Aobst.ret,close constant of close obstacle retarding force N

h constant of close obstacle force m

Awall.foll constant of wall following force N

viii

Symbols ix

c constant for wall following force

Aobst.av constant of obstacle avoiding force N

Aagen.av constant of agent avoiding force N

δ constant for g′

and g′iW m

Aspeed.dampening constant of speed dampening force kg/m

Ashield constant of leader shield force N

di,virt.obst distance between agent i and a virtual obstacle m

ri,virt.obst combined radii of agent i and a virtual obstacle m

rij vector between agent i and j

nij normal vector between agent i and j

tij tangential vector between agent i and j

riW vector between agent i and an obstacle W

niW normal vector between agent i and an obstacle W

tiW tangential vector between agent i and an obstacle W

f ij forces between agent i and j N

f iW forces between agent i and obstacle W N

∆vnji the scalar product of the relative speed between agent i and j and nij

∆vtji the scalar product of the relative speed between agent i and j and tij

∆vnWi the scalar product of the relative speed between agent i and obstacle W niW

∆vtWi the scalar product of the relative speed between agent i and obstacle W and tiW

B the building set

O the outside set

WB the set of all visual obstacles in the building

Wi the set of all visual obstacles in room i

PD,i the set of points of all door targets in room i

PV,i the set of viewpoint target points in room i

daux auxiliary point offset m

PA,i the set of all auxiliary points in room i

dterminal offset distance of a terminal point m

PF the set of all terminal points

wzone width of a target zone m

hzone height of a target zone m

dzone offset distance of a door target zone m

Symbols x

Ppot the set of all potential targets in a room

L symbol for a line or line segment

Rmax,vis the maximum distance at which two points can be considered visible m

Pintersect the set of intersection points between two lines

Arect symbol for a rectangle

Pvisible set of all currently visible targets of an agent

Pknown the set of all known target points of an agent

Pvisited the set of visited target points of an agent

Pinvalid the set of invalidated target points of an agent

ppatience,min threshold patience level

〈v〉 mean speed of an agent over some past time steps ms−1

GD the door target graph

Rfollow,max minimum radius for leader influence m

rleader position of a leader

Chapter 1

Introduction

1.1 Background and relevance

Reliable and accurate tools for simulating crowd evacuation are essential for aiding the

design of buildings in purpose of minimising casualties and evacuation time. Notable

disasters in the past, like The Station Night Club Fire, Rhode Island, USA with 100

diseased and the World Trade Center bombings could have had more positive outcomes

if further pre-emptive actions had been taken[1]. Software for simulating pedestrian

behaviour under influence from active intervention during emergencies could play a

major role in this by enabling repeated testing of proposed active intervention measures

for minimising evacuation time and casualties. This is especially important considering

the limited abilities of conducting accurate real-world experiments due to ethical aspects

of exposing test subjects to hazard and the difficulty of producing realistic behaviour of

subjects in arranged situations.

Previously, hand book-calculation has been the primary tool when considering the lay-

out of buildings from an evacuation safety perspective [2–5]. A major drawback of these

are that they provide no insight of the behaviour of evacuees during evacuation. Com-

putational models for simulating crowd evacuation have been around since at least the

70s, with Takahashi and Helbing as notable predecessors [5]. Many of these models have

however met criticism for making unreasonable assumptions producing results inconsis-

tent with realistic human behaviour. Haraala and Hagelin note that slight variations in

parts of the model Steps results in very different outcomes [3]. Therefore, it is important

1

Chapter 1. Introduction 2

to develop models that more accurately reproduce the behaviour of evacuees, something

that is requested by several developers of computational models [6, 7]. That being said,

computational models have been used when designing major event buildings such as

Swedbank Arena (nowadays called Friends Arena) and Wembley National Stadium [8],

and the authors have found new models as recently as 2012.

1.2 Key concepts

Before moving on, two key terms in the report are clarified:

Agent: The entity in this crowd evacuation model that represents an evacuating human

being. Different definitions of an agent are used in the literature. The specific use in

this thesis is specified in chapter 3.

Active intervention: Any measures taken by a designer of crowd evacuation systems in

order to reduce casualties and evacuation time. Common examples are exit signs, signal

systems and rescue personnel.

1.3 Aims and objectives

The aim of this Bachelor’s Thesis is to extend the Social Force Model (abbreviated SFM)

proposed by Helbing with means of active intervention, in purpose of analysing the effect

of the latter. The specific active intervention measures are the addition of exit signs and

trained leaders. Furthermore, agents are assumed to have no knowledge of the floor plan

prior to evacuation and the model should be adapted to a floor plan that requires the

evacuees to move through a minimum of three doors before being evacuated.

1.4 Outline of the model CEPABS

The model proposed in this thesis is called CEPABS. A more thorough description of

CEPABS and a review of previous work will be treated in subsequent chapters, but the

main features of CEPABS are briefly outlined below. CEPABS:


• Uses a Newtonian force model to describe close personal interactions, making it

suitable for analysis of close scale behaviour of crowds.

• Uses a leader-follower model for rescue personnel and directed exit signs as means

of active intervention.

• Does not require evacuees to have any prior knowledge of the building floor plan

and manages to evacuate evacuees in absence of trained leaders or directions.

• Takes into account variations of characteristics of evacuees such as body size,

desired movement speed and sight range.

• Exhibits experimentally verified [5, 6] features of evacuating crowds, such as arch-

ing at narrow passages and obstacle avoidance.

Matlab TMis used as the programming language of implementation, but the program

can easily be realised in other languages.

1.5 Limitations

In this thesis, and consequently in CEPABS, only single floor buildings are considered.

Furthermore, communication between evacuees is limited to wanting to maintain per-

sonal space. Physically, evacuees are modelled as circular 2D objects, while other authors

use more authentic body shapes [3, 9]. They are also modelled to have 360◦ angle of

vision and no moment of inertia.

1.6 Collaborations

The model consists of two weakly separated parts: the main model and active inter-

vention. The main model is made jointly with Leijonmarck and Olerg̊ard, while active

intervention is carried out by the authors. Besides the work with the main model part,

there has been additional cooperation with Leijonmarck and Olerg̊ard. (Date of publish-

ing of Leijonmarck and Olerg̊ard is currently unknown but is estimated to be available

in KTH database DIVA by the latest 2014.)


1.7 Outline of the thesis

In Chapter 2 a review of previous models and theories of human behaviour during

emergency evacuations is presented. Chapter 3 contains a description of the model

CEPABS and in Chapter 4 simulation results are presented. In Chapter 5 the model

CEPABS and the results from the simulations are discussed and analysed and in Chapter

6 the thesis is summarised, with some suggestions for extensions.

Chapter 2

Literature review

Preceding this work, there has been research governing the behaviour of humans in

various settings as well as several approaches to modelling crowd evacuation. In this

section approaches used in this thesis and alternative approches are outlined. Findings

regarding typical behaviour of individuals and crowds are also mentioned, in purpose of

discussing assumptions made in the model.

2.1 Models of crowd behaviour

2.1.1 Flocking behaviour and SFM:s

Flocking behaviour, as described by Reynolds in his seminal article ”Flocks, Herds, and

Schools: A Distributed Behavioral Model” captures some key characteristics of a collec-

tive flock based on a few basic principles [10]. Collision avoidance is the notion of flock

members trying to avoid colliding with other flock members. Velocity matching means

that members try to align velocity (direction and speed) with nearby members. Flock

centering is the notion of flock members attempting to stay close to others.

It is not obvious to what extent these principles can be applied to human crowd be-

haviour, but an adaptation is made by Helbing in his SFM [6]. The SFM models

evacuees as particle-like objects subject to ”forces” constructed to reflect desired prop-

erties of human interactions and behaviour. The agents are assumed to possess mass, a

body radius, a (variable) desired velocity, a finite reaction time and an ”individualism”

5

Chapter 2. Literature review 6

parameter. Some SFM:s add or omit parameters. A few key forces in the model are

worth mentioning, considering that they are used in the model CEPABS of this thesis.

The ”personal distance” force is meant to reflect the notion of humans wanting to main-

tain a certain personal distance to each other and which is usually modeled as directed

along the distance vector between two agents and decaying with distance. Osiragi and

Olfati Saber discuss forces with exponential and inversely proportional decay as well

as linear combinations of the two [9, 11]. Osiragi chose to adapt an exponentially de-

creasing force arguing that this way unbounded forces are avoided. This decay is used

in Helbings model and CEPABS. Two forces that only apply when two agents are in

contact with each other are a Hooke’s Law-like ”compression force” and a tangential

interagent damping like force with the purpose of slowing down agents ”rubbing their

bodies against each other” (authors’ interpretation). The finite reaction time of humans

and delay in motor skills hypothetically give rise to a force that works to change the

current velocity to the desired velocity, with a certain delay. What determines the de-

sired velocity varies between models and is usually affected by several parameters.

Since SFM:s take into account many physical interactions at a high resolution in time

and space (only limited by accuracy and step size of numerical solvers), they can poten-

tially, using well designed forces and tuned parameters describe realistic kinematics of

humans in a crowd. This virtue is also a shortcoming when it comes to computational

cost; solving differential equations numerically and calculating forces between many ad-

jacent agents is inevitably consuming and limits the scalability of a model. This is likely

one reason why other approaches exist along side the SFM.

2.1.2 Cellular automata models

Cellular automata models (abbreviated CAM) divide a floor plan into a (usually uniform)

grid in which agents move and naturally have a computational advantage over SFM:s. In

CAM:s, agents and objects occupy grid cells and one cell can never be occupied by more

than one agent or object. CAM:s determine the move of an agent in each time step by

some algorithm which usually evaluates a cost function and/or follow a logical decision

chain (of an ”if-then” type), sometimes incorporating stochastic elements. Haraala and

Hagelin point out that the (cellular automata) model Steps is highly sensitive to changes

in the grid size, and that another CAM Simulex over-predicted evacuation time in the

soccer stadium scenario investigated by them [3]. It should be noted, however, that


Haraala and Hagelin report the Steps model to give accurate results with the right

tuning. Furthermore, the authors of this thesis encountered far more CAM:s than SFM:s

during the research of crowd evacuation models, implying that CAM:s do have some

virtues.

2.1.3 Leader- follower models

Some crowd evacuation models include variations of the leader-follower mechanisms

[4, 6, 7, 12, 13]. In a leader-follower model at least one agent, the follower follows at

least one leader to some extent. At one extreme, a majority of agents are assumed

to unconditionally follow a small number of leaders, who themselves consider no other

agents in the choice of route. At another extreme, there are no distinct leaders and all

agents choose their targets mainly independent of each others. Some models allow for

variable leader-follower characteristics [6], some use leaders for specific grouping phases

of an evacuation and some have no pronounced leaders, but make their agents obey

social hierarchies [4, 7].

The leader-follower model by Pelechano and Badler deserves a more thorough explana-

tion [12]. In Pelechano and Badler’s model, there are three kinds of agents: Followers,

untrained leaders and trained leaders. Followers are evacuees without a leader disposi-

tion that are considered to freeze in panic (do nothing) unless there is a nearby leader,

whom they follow. Untrained leaders, people who possess leader characteristics but do

not know the building before hand employ a depth-first search to make their way out

of it (it should be noted that building in the model had a restricted maze-like geometry

composed of square rooms, which allowed for this simplified approach). Trained leaders,

like fire fighters, know the building floor plan and will take the optimum route for getting

out. Pelechano and Badler compared different leader/non-leader ratios and found that

a ratio up until 10 % gave a vastly decreased evacuation time, while a higher ratio gave

a marginal decrease. As a reference, Can and Qingge explored the same ratio in their

model and found that an exaggerated number of leaders gave increased evacuation time,

reasoning that it would make followers indecisive in choosing leaders causing them to

switch which leader to follow frequently [7]. This comparison highlights the significance

of the implementation of a leader-follower Model.


2.2 Crowd behaviour

2.2.1 Response to signal systems

When modelling crowd evacuation with active intervention one needs to know what effect

various forms of signal systems such as exit signs and sound signals has on evacuees.

Sound signals

In a study of the World Trade Centre Bombings, it was reported that no more than

14% of evacuees had recalled hearing a sound alarm [2]. It is mentioned that the low

percentage might be due to damages to the sound alarm system. However, only 1-5 %

(depending on tower) of respondents stated that the primary reason for evacuating was

hearing a sound alarm, giving reasons such as ”being told to evacuate” or ”feeling in

danger” as the most important. One should however be careful interpreting the survey

as representative of other evacuations since the WTC was a multi-floor building, the

building was damaged in an unusual way and other warnings might have reached the

evacuees before the sound alarm. Proulx and Laroche further state that one of the

most common emergency alarm sounds was not recognised as such by a majority of

test subjects in a survey [14]. They did however mention that an alternative sound was

perceived by more test subjects as an alarm signal, and that when a signal was perceived

by an evacuee, he or she would have an increased feeling of urgency to leave. This is also

supported by a testimonial of an evacuee in the WTC bombings in a report by Averill

et. al [2]. From this it tempting to conclude one of the following: Commonly used

sound alarms are generally not perceived by evacuees and as such is not very effective,

or evacuees have not received sufficient training to correctly respond to signals. However,

considering the small amount of information and inconclusive indications, one should

further research the effect of sound signals before relying on them in a model.

Effect of exit signs

In the WTC bombings, between 33 and 17 % of survivors reported having been helped by

photoluminescent markings (exit signs), where the higher percentage was found among

evacuees in a building in which lights had been lost [2]. Kluepfel, further cites Abe in his


Doctoral Thesis, reporting that 53% of evacuees followed either exit signs or instructions

from staff in a simulated evacuating setting [15]. The survey does not discern what part

of the percentage was due to exit signs, but the two sources together indicate that people

evacuating buildings during emergencies do make use of exit signs. Unfortunately, no

copy in English of the original source (Abe) of Kluepfel could be found.

2.2.2 Exit choice

In section 2.2.1 the effect of exit signs was discussed. It is important to know on what

grounds evacuees prioritise exits. The survey of Abe as cited in Kluepfel show that after

Exits signs and instructions from staff, distance to an exit was the second most common

reason for choosing an exit [15]. Following others (6,7%) and picking an exit which was

not crowded (0.7%) was given as other reasons. These reasons are stated as perceived

primary reason by the evacuees, but this shouldn’t exclude that several reasons are

considered simultaneously. Following others is in compliance with the velocity matching

and leader-follower concepts, while ”exit was not crowded” is investigated by Aik [16].

Aik showed both through simulation (a CAM) and experiment that some evacuees (in

an arranged experiment should be noted) chose a more remote exit if the closest exit was

the most crowded [16]. Although not conclusive from these studies, until more research

has been done, it would be worthwhile to investigate a model using exit choice based

on: visibility of signs, proximity of exits and crowdedness. This is done in CEPABS.

2.2.3 Communication

Most models include communication in one way or another. It could either be indirect,

in form of wishing to maintain personal space (SFM) or direct by sharing information

of a floor plan as is done in [12]. A report from the WTC bombings implies that

communication among evacuees is substantial, as do Pan citing Sime [4]. It is however

reasonable to believe that the nature of the evacuation matters here. It is perhaps less

likely for direct communication to occur in a high-density sports stadium stampede than

in an office building where evacuation starts with a rumour of a fire in a remote part

of the building. As interesting and important as may be in many evacuation settings,

communication is mostly neglected in the version of CEPABS used in this thesis.


2.2.4 Velocity alignment

Velocity alignment in humans could be seen as the tendency to follow others. In Helbing’s

model velocity alignment is dependent on an individualism parameter, which weighs the

desired direction of an agent between that of his neighbours and that of his own (in case

he would be by himself) [6].

Olfati Saber has extensively treated consensus algorithms for multi-agent networks, in-

cluding under what circumstances convergence is achieved [11]. A consensus problem

arises in velocity matching when agents are arranged according to their neighbours in

a network (preferably viewed as a graph) and set their desired velocity to the mean

of its neighbours (which can be defined in a multitude of ways). Olfati-Saber showed

that convergence is not guaranteed for any network, but for a fully connected graph the

velocity converges to one common velocity. In case of one leader and a graph where

every node (agent) can be connected to the leader with a directed path, convergence is

also reached, but the convergence rate is not mentioned.

In CEPABS none of the sufficient conditions for convergence are always satisfied (glob-

ally) and so the results of Olfati Saber are of limited interest, but they could well be

used the analysis of crowd evacuation models, especially those facing consensus problems

other than for velocity alignment [4].

2.2.5 Obstacle avoidance

Obstacle avoidance is the notion of crowd members wanting to avoid colliding with

obstacles and other crowd members. The term can also include the way an agent chooses

its path in order to circumvent an obstacle. Obstacle avoidance might not be a feature

unique to crowds, but nevertheless is of importance in crowd simulations. A majority of

crowd evacuation models investigated in the literature study do not explicitly mention

obstacle avoidance, but Olfati Saber and Reynolds treated it in the broader context of

flocking behaviour[11, 17].

Reynolds presented an obstacle avoidance in which obstacles are represented as com-

positions of spheres. A simplified reformulation of the principle is described as follows.

An agent would have as a goal to keep an imaginary cylinder of free space in front of


it. First, an agent finds the vector between its position and the centre of each obsta-

cle. It then calculates the length of the orthogonal complement of the projection onto

the centre line of its free space cylinder. If the length of the orthogonal complement is

greater than the sum of the radius of the agent’s free space cylinder and the radius of

the obstacle, the obstacle is not considered. Otherwise the agent steers sideways of the

obstacle to whichever side is nearer the target.

Olfati Saber introduced the concept of β-agents, in contrast to α-agents who are ordinary

flock members [18]. β-agents are virtual agents that are created slightly within the

boundary of an obstacle at the point closest to an α-agent. β and α agents in Olfati

Saber’s model follow a flocking protocol, similar to that in [10] but including a consensus

algorithm as described in [11]. Altogether, the interactions between β and α agents,

which includes close distance repulsion and a form of velocity alignment, causes α agents

to move around obstacles.

In CEPABS, a simplified and modified version of Reynold’s model is used, adapted to

the employed SFM. Olfati Saber’s obstacle avoidance as presented in [18] is not used in

CEPABS, but many aspects are in practice similar in both models.

2.3 Search algorithms

In models where movement is determined by potential fields, such as Simulex [5] or just

by force interactions with other agents there is no need for strategies for finding a way

out of a building. In other models agents need to follow some algorithm. Pelechano

and Badler mention a depth first algorithm, usable if there is a well defined portal graph

(a graph of ”rooms” and ”portals” between rooms) [12]. How realistic this algorithm

is for a human beings is however not discussed. Qingge uses the A* algorithm for the

trained leaders, but in order to be useful for way finding an agent needs to know the

building before hand [7]. In contrast to the depth first algorithm A* does not require

the building to be described by a network, but rather needs the floor plan to be divided

into grid cells.

Pan and Pelechano and Badler primarily make agents follow others in lack of options

and Pan employs ”random search” if an agent is completely ”individualist” (as may

occur in his model) [4, 12], although it is not closely described how this works. Pan


does however mention two algorithms for updating perception of an agent which would

trigger a desired direction of movement: A target search algorithm and a ray tracing

algorithm. In the target search algorithm an agent looks for nearby potential targets

(such as doors, meeting spots or other evacuees in Pan) and checks if the line of sight

to those targets are obstructed or not. The Ray Tracing algorithm checks in which

direction an agent can move by shooting rays of vision at different angles in front of the

agent to see if there is an obstacle obstructing its potential path.

The authors of this thesis see the need of continuing the development of such algorithms

in order to avoid unrealistic behaviour in some geometries. Therefore an extension to a

combination of some previously existing algorithms is proposed in chapter 3.

2.4 Choice of parameters

Most crowd evacuation models will contain parameters that determine how the model

works. Sometimes heuristics can be used for certain parameters (i.e. reaction time, sight

range) and sometimes iterative tuning can be done until the model works as ”desired”.

Another approach, proposed by Osiragi uses regression to fit parameters of a model to

experimental data [9]. In Osiragi’s work, this method is used for describing pedestrian

movements in a non-emergency situation. However, since parameters can be expected to

be interdependent, introduction of new parameters or changing the experimental setting

could alter the configuration of parameters. In that sense, the method is not considered

more adaptable than other options. In this thesis, most parameters are based on values

from other models, or tuned by trial an error. CEPABS is not expected to be used

for real-world building planning until tested and refined (which would include a strict

revision of parameters).

2.5 Summary

In this chapter, two major approaches to modelling crowds were presented: SFM and

CAM, of which the SFM is used in this thesis. Furthermore, leader-follower models

have been discussed. A search algorithm with resemblance to that of [12] is used in

CEPABS was mentioned and is further described in chapter 3. Some aspects of crowd

behaviour were discussed, of which exit choice selection and velocity alignment play


important roles in CEPABS. Finally, methods for finding parameters were discussed,

where iterative refinement is mostly used in CEPABS.

Chapter 3

The Model CEPABS

3.1 Introduction to the model

CEPABS is an SFM based on Helbing’s model [6], extended with a mental map, patience

and a set of exit search strategies. The forces of the model are described in sections

3.2.1 and 3.3, the mental map concept is described in section 3.7, patience is described

in section 3.8 and exit search strategies are described in sections 3.12 and 3.13.

Since agents are subject to forces in an SFM, the movement of agents is calculated by

solving a system of ordinary differential equations (also called an ODE) arising from

Newtons second law of motion. For the remainder of this thesis it is assumed that the

ODE solver uses a constant step size. This step size is from now on referred to as the

time step and is also taken as the least increment that any other part of the model, i.e.

the mental map, can be updated. There is nothing saying a solver with variable time

step cannot be used, but then the time step concept might need a revision.

Certain mechanisms like patience and the mental map are updated at update times

which are taken to be evenly spaced at intervals greater or equal to the time step.

Before explaining CEPABS in detail, some key concepts of the model are defined. Unless

otherwise stated, all quantities are elements in R2. Distance will always mean the

Euclidean distance, ∗ will always mean the complement of a set and bold font will

always mean a vector quantity.

14

Chapter 3. The Model CEPABS 15

Definition 3.1. An agent is defined as a set containing a position, an effective body

radius, a velocity, a desired speed, a target point, mass, a reaction time, a patience level,

an individualism factor, a sight range, a mental map, a body and a desired direction. The

position is a measure of the agent’s location. The effective body radius is an estimate of

the body of a human being approximated as a circle (in the plane of the floor). The body

of an agent is a circle with the agent’s position as the centre point and the effective body

radius as the radius. The velocity is the time derivative of the position. The desired

speed is the speed at which an agent prefers to move. The target point is a point to

which an agent strives to move. The reaction time is an estimate of how quickly an agent

can adapt its velocity to its desired speed in its desired direction. The patience level is

a measure of how slowly an agent can move towards its target point before becoming

impatient. The individualism factor is a measure of how keen an agent is to follow other

agents. The mental map is a collection of information about the environment the agent

is situated in. The desired direction is the direction between from the agent’s position

to its target point.

The vector position is denoted r, the scalar effective body radius is denoted R, the vector

velocity is denoted v, the scalar desired speed is denoted v0, the vector target point is

denoted rtarget, the scalar mass is denoted m, the scalar reaction time is denoted τ , the

scalar patience level is denoted ppatience, the scalar individualism factor is denoted pind,

the sight range is denoted Rsight, the set mental map is denoted S and the vector desired

direction is denoted e0.

In CEPABS, from tweaking and recommendations in [6], the masses of agents were taken

to be uniformly distributed among agents between 60 and 80 kg, the body radii were

taken to be uniformly distributed between 0.25 and 0.35 m and the desired speeds were

taken to be uniformly distributed between 3 and 6 ms−1.

When there is more than one agent, each agent gets an index subscript i to denote the

i:th agent in the sequence of agents.

Obstacles are walls and other static objects such as pillars and are an integral part of

the model. Obstacles are represented in two ways, depending on use. Both definitions

are given below.


Definition 3.2. A fundamental physical obstacle is defined as an agent with v and R

identically zero at all times.

Although a fundamental physical obstacle does not move or have a body radius, it affects

agents with forces just as any regular agent would.

Definition 3.3. A physical obstacle is the union of one or more fundamental physical

obstacles.

When fundamental physical obstacles are arranged together, they can form represen-

tations of obstacles, such as e.g. walls if arranged along a straight line. The use of

fundamental physical obstacles is explained in the sections 3.3.2, 3.3.1 and 3.3.4. This

representation of obstacles resembles the β-agent concept described by Olfati Saber in

chapter 2, but here agents are not created in response to a proximate α-agent (using

the terminology of Olfati Saber). Instead, they are always existent but only influence

an agent if the agent gets sufficiently close to the wall 3.1. In addition, this means

that there is not only one β-agent per α-agent and that the β-agent does not follow an

α-agent as it moves along the wall.

Figure 3.1: The influence of physical targets on an agent. The circle encloses thearea of wall-agent influence. Red dots are inside and black dots are outside the radius

of influence.


The second kind of obstacle interferes with the vision of agents. The building blocks

of these obstacles, in comparison with fundamental physical obstacles, are straight line

segments and are called fundamental visual obstacles.

Definition 3.4. A visual obstacle is the union of one or more connected fundamental

visual obstacles.

Visual obstacles can be used to approximate many kinds of geometrical shapes and they

can be either closed or open. The use of visual obstacles is described in section 3.6.

3.2 Forces of the SFM

The equations of motion used to model the behaviour of the agents are inspired by

the SFM of Helbing, but with a few additions in purpose of making the agents behave

more naturally than they in the original model. Despite the fact that the equations of

motion are modelled as forces, it is important to note that not all of these forces are

actual forces, but rather forces acting upon the agents in a way that motivates them to

perform an action like slowing down or avoiding an obstacle.

3.2.1 Helbing’s model

The SFM originally used by Helbing is

midvidt

= miv0i e

0i − vi(t)τ

+∑j 6=i

fij +∑W

fiW (3.1)

v0i is the desired speed of agent i, e0i is the desired direction of agent i and τ is en

estimate of the time in seconds it takes for an agent to accelerate to it’s desired velocity

v0i e0. The first term on the right hand side of the equation is a velocity correcting term

which works to align an agent’s velocity with its desired velocity. The second last term

is the sum of forces acting on agent i from all its neighbouring agents j and the last

term is the sum of forces acting on agent i from all neighbouring walls W. In the original

formulation in [6], what constitutes a neighbour is not described. In CEPABS there are

two definitions of two points being neighbours, depending on if the points consist of two


agents or one agent and a physical obstacle. The definitions are general, but are here

used in the context of agents and walls.

Two points p1 and p2 are said to be mutual agent-neighbours always. This means

that two agents are always neighbours with each other. The reasons for introducing

the concept agent-neighbours are to separate it from the wall-neighbour concept and to

establish a terminology that can be used if the agent-neighbour definition is changed in

a later version of CEPABS.

Definition 3.5. Two points p1 and p2 are mutual wall-neighbours iff ‖p2 − p2‖ <

Rneigh, where Rneigh is a minimum distance for neighbour influence.

Rneigh is a parameter that could be based on experimental observations or tweaking. In

CEPABS it is taken to be 10 meters.

Two agents are always considered mutual neighbours in the agent-neighbour sense and

an agent is considered neighbour with a physical obstacle if agent and obstacle satisfy

the requirement of definition 3.5. The neighbour concepts could be extended in later

models.

3.2.2 Inter-agent forces

The forces fij acting between agents are

fij = [Aie(Rij−dij)/B +Kg(Rij − dij)]nij + κg(Rij − dij)∆vtjitij (3.2)

where the constants and variables are explained one by one. Rij is the sum of the radii

of agent i and j and dij is the distance between agent i and j. A is a constant modulating

the magnitude of the exponential force and B is is a constant affecting the steepness

of the exponential force as Rij varies. K is a Hooke’s law-like spring constant, κ is a

constant representing the size of the tangential frictional force between agents and g is

the function (rij −dij) ·H(rij −dij), where H is the Heaviside step function. ∆vtji is the

tangential component of the relative velocities of agents i and j, nij is the unit normal

vector from agent i to agent j and tij is the tangential unit direction defined as the unit

normal vector rotated −90 degrees.


The first term is an exponential force that models the will of an agent not to get too close

neighbouring agents and the second term is a spring type force that models the bodies

of agents as linearly compressible. The third term is a sliding friction term that slows

down agents that are in contact and moving tangentially to each other. The function g

works to make the spring type force and sliding friction force active only when agents

are in contact. The numerical values of the parameters are listed in section 3.2.4.

Figure 3.2 shows the vectors nij , tij , e0 and the tangential vector e0 to the desired

direction. Note that e0 is taken arbitrarily for the agents in the figure and is in general

not perpendicular to nij .

Figure 3.2: The vectors for the inter-agent forces.

3.2.3 Physical obstacle forces

The physical obstacle forces fiW from all fundamental physical objects W acting upon

agent i are the same as the inter-agent forces fij :

fiW = [Aie(RiW−diW )/B +Kg(RiW − diW )]niW + κg(RiW − diW )∆vtWitiW (3.3)

It should be noted however, that due to the definition of physical obstacles, diW = Ri

and ∆vtji = −vi. Figure 3.3 shows the vectors nij , tij , e0 and the tangential vector

e0 to the desired direction at two times as an agent moves around an obstacle. e0

is always directed towards the target. The individual fundamental physical obstacles

that compose the entire obstacle are clearly visible, and niW vector is seen to be the

(normalized) sum of normal vectors to each of the fundamental physical objects. Note


that the fundamental physical objects are given non-zero radii in order to be visible to

the reader.

Figure 3.3: The vectors for the obstacle forces.

3.2.4 Parameters used in the force model

The values of A, B, κ, K, τ and m are those that Helbing had used in his model. A is

set to 2000 N, B to 0.08 m, κ to 2.4× 105kgm−1s−1, K to 1.2× 105kgs−2 and τ to 0.5

seconds.

3.3 Forces added to Helbing’s model

The SFM of Helbing captures some behaviours well. There are however some situations

for which the model is not able to reproduce behaviour sought by the authors. Those

situations are for instance when an agent needs to move past a wall in order to reach a

target or when an agent should avoid other agents heading in the opposite direction. To

produce satisfactory behaviour in these scenarios, four forces are added to the model,

with constants fitted to the model rather than observed from real-world experiments.

3.3.1 Obstacle avoidance causing retarding

In order for an agent to avoid obstacles when moving towards them, an obstacle avoid-

ance force fobst.ret is introduced. Without this force agents tend to act more like particles

than humans, only noticing obstacles when they are in contact. This force is very similar

to the one Helbing used for his sliding friction force, the difference being that instead of


projecting the relative speed of the agents upon the tangential direction, it was projected

onto the normal. Due to the fact that the willingness of avoiding an obstacle increases as

the distance to it decreases, the force is multiplied by an exponential term as to mimic

the willingness of a human to avoid an obstacle. This force is only active for obstacles

that are in front of the agent, why it is multiplied by a factor k, which is equal to 1

should the current obstacle fall within the line of sight, and 0 otherwise.

fobst.ret = Aobst.retk∆vnWie

RiW−diW niW (3.4)

∆vnWi = −vi · niW

Aobst.ret is a scaling parameter, k is 1 if a wall target is visible and 0 otherwise. All other

constants and variables follow the definitions in section 3.2.3.

In addition, there is a force fobst.retclose that quickly retards the agents when they get

too close to a wall:

fobst.retclose = Aobst.retclose∆vnWie

(RiW+h−diW )/BniW (3.5)

∆vnWi = −vi · niW

Aobst.retclose is analogue to Aobst.ret and all other constants and variables are as in equa-

tion 3.4, with subscripts changed from ij to iW .

fobst.retclose is also applicable for avoiding other agents, but in that case the exponential

term is replaced with a function g′

that is equal to 1/(dij − Rij) whenever dij > Rij

and zero else. This gives the force fagent,ret.

fagent,ret = Aagent,retk∆vnjig

′nij (3.6)

∆vnji = (vj − vi) · nij

All variables other than g′

are as in equation 3.4.


3.3.2 Wall following

One of the previously mentioned scenarios is when an agent has been pushed in a different

direction by other agents, resulting in a diversion from the desired direction. If the

new direction intersects a wall, what happens is that the agent tries to move through

the intersecting wall rather than around it. Therefore, a wall following force fwall,foll

is introduced, similar to the exponential force Helbing introduced in his model. The

difference between this force and the one Helbing used is the change of the parameter A

to Awall.foll, that the combined radii is multiplied with a factor c = 1.07 as to get the

agent to begin following the wall earlier, and that it acts in tangential direction of the

wall. All other constants and variables are as in equation 3.2.

fwall,foll = Awall.folle(cRiW−diW )/BtiW (3.7)

3.3.3 Speed dampening

In Helbing’s model, are no forces damping an agent at a high speed which is expected

due to falling or ”friction” with the floor. Simulations show how agents that have been

compressed bounce off at high speeds due to the compression force. To counteract this,

a speed dampening force fspeed.dampening is introduced, similar to the speed dampening

force seen in the analysis of spring-mass systems, viscosity in fluids and air drag models.

fspeed.dampening = Aspeed.dampening2√||vi||vi (3.8)

Aspeed.dampening is a scaling constant and vi is the velocity of agent i.

3.3.4 Obstacle avoidance

An agent should not only slow down when approaching an obstacle; it should also try

to avoid it. To realise this, an obstacle avoidance force fobst.av is introduced. The factor

k mentioned earlier is also used here, in order to limit the influence from the forces from

obstacles that are in front of the agent. In a first assumption the obstacle avoidance

force is dependent on both the speed and direction of the agent as well as the distance

from the agent to the obstacle. The force contains as scaling constant, but otherwise


uses constants and variables from 3.6 with the difference that the force now acts in the

tangential direction.

fobst.av = Aobst.avk∆vnWig

′iWe

0i,tang (3.9)

∆vnWi = −vi · niW , g′iW =

1diW−riW , diW > RiW + δ0, diW < RiW + δ

The force is fobst.av also applicable for avoiding other agents, but then uses subscrips ij

instead of iW and is denoted fagent.av:

fagent.av = Aagent.avk∆vnjig

′tij (3.10)

3.3.5 Desired direction of an agent

The desired direction e0i of agent i, not under influence from any leaders, is calculated as

in 3.11. Note that this direction is also part of Helbing’s original model, but is written

here for ease of comparison. ri is the position of an agent and rtargeti its target point.

e0i =rtargeti − ri||rtargeti − ri||

(3.11)

In order to account for situations where there are leaders in a crowd, Helbing calculates

the desired the direction of agent i as the desired direction of that agent plus the mean of

the desired directions of all the neighbouring agents j. This is weighted with a parameter

p that indicates the tendency to follow others.

e0i =(1− p)e0i + p〈e0j 〉||(1− p)e0i + p〈e0j 〉||

(3.12)

Instead of adopting Helbing’s model, another approach is taken for the leader-follower

mechanic. It mimics the behaviour of ducklings following their mother. The duckling

model starts by finding all the neighbouring leaders of an agent and setting the closest

one as its current leader. After that the target of the following agent will is set to be the

current position of the leader. This approach results in a natural behaviour according

to the authors, but it can certainly be refined. It should be mentioned that the duckling


approach (and simile) is inspired by [17]. This leader-follower model is further described

in section 3.13.2.

3.3.6 Moving followers out of the way for leaders

One problem with the duckling approach is that the leaders get swarmed by follow-

ers, hindering the leaders from moving. Therefore another force fshield is added which

emulates how followers make way for an authority. This behaviour is implemented by

introducing a virtual obstacle in front of a leader that only affects non-leaders, similar to

a shield. The term Rvirt.obst is the combined radii of the agent and the virtual obstacle.

Here, the radius of the virtual obstacle was set to be 2 m.

fshield = AshieldeRvirt.obst−di,virt.obstni,virt.obst (3.13)

Ashield is a scaling parameter, di,virt.obst is the distance between agent i and the virtual

obstacle and ni,virt.obst is defined as the agent-obstacle normal as for all other forces.

3.3.7 Parameters of the added forces

Reasonable values of the parameters in the added forces were found to be Aobst.ret =

20 kgms−1, Aobst.retclose = 200 kgms−1, h = 0.05 m, Aagent,ret = 24 kgm

2s−1, Awall,foll

= 200 N, Aspeed,dampening = 19 kgm1/2s−1/2, Aobst.av = 12 kgm

2s−1, Aagent.av = 12

kgms−1, δ = 0.15 m and Ashield = 20 N.

3.4 Geometrical representation of a building

This section deals with the geometrical representation of a building which agents are to

be evacuated from. The geometrical representation affects both computational costs and

which mathematical tools are available for describing exit strategies of agents. Before

describing the model in depth a few definitions regarding the geometrical representation

are given.

Definition 3.6. A building is defined as a connected subset of R2 such as there is only

one building in R2.


The building is in this thesis denoted B.

Definition 3.7. The outside is defined as the complement of B in R2.

The outside is in this thesis denoted O.

Definition 3.8. A room is an element in a partition of R2 such that O is one of the

elements in the partition. Furthermore, every room must have at least one door point (as

defined in definition 3.9) and door points have to be placed such that O can be reached

from any point in B by a curve changing rooms only when traversing door points.

Following from the definition all rooms can be assigned a unique identifier called room

index. The room index will in the remainder of this thesis be represented by the room

index i, where i is an integer. Furthermore, in the remainder of this thesis room i will

refer to the ”room with room index i”.

Definition 3.9. A door point is a point that lies on the boundary of two rooms and

not on a visual obstacle.

Having defined the outside, a definition of when an agent is evacuated can be given.

Definition 3.10. An agent is defined to be evacuated if r ∈ O for for at least one

previous time step.

Definition 3.10 implies that once the agent has set foot outside of the building, it is

considered evacuated (even if it would for a moment return to the building).

Lastly, for convenience of communication the set of all visual obstacles in a building is

denoted WB and the set of all visual obstacles in room i is denoted Wi.

The above definitions define the mathematical space in which agents are positioned.

They aid in describing the model and guarantee that a point-like agent can reach the

outside from any point in the building and become evacuated.

3.5 Target points

Target points are the set of points that agents may choose to be their target point rtarget

as described in sections 3.12 and 3.13. In CEPABS there are four kinds of target points:


door target points of door targets, viewpoint target points, auxiliary target points and

terminal points that will be described in the following sections.

3.5.1 Door targets

A door target contains two elements; a point which is situated at door points (as defined

by definition 3.9) and a direction designating which direction leads directly out of a

room. Consequently there are always two door targets at every door point with opposite

directions. This concept is formalised in definition 3.11.

Definition 3.11. A door target consists of a point and a direction. The point must be

a door point such that there is exactly one door target point with the same point but

opposite direction in an adjacent room. The direction is the outward unit normal to the

boundary of the room at the point of the door target.

Following from the definition all door target points in a room can be assigned a unique

identifier called local door index. Furthermore, set of points of all door targets in room

i is denoted PD,i.

3.5.2 Viewpoint target points

Viewpoints are points that agents can move to in order to get a better view of their

surroundings so that they potentially can see door target points. Before giving the

definition of a viewpoint target point, a definition of a room corner is given.

Definition 3.12. A room corner is a point where two fundamental visual targets that

lie entirely on the border of adjacent rooms intersect.

From definition 3.12 a room corner is shared by a minimum of two rooms, and in practice

it is what would constitute a corner of (at least) two walls in a real-world building.

Definition 3.13. All viewpoints target points in B are defined with an algorithm: Find

all room corners in B. Pick a room corner and pair it with the closest corner point

that isn’t a member of the same room as the first. If there is more than one point at a

distance shorter than all other points, do the next step for both points. The midpoint

of the paired corner points is a viewpoint target point. Start with a new room corner

and repeat the process until all room corners have been chosen.


The set of viewpoint target points in room i is denoted PV,i.

3.5.3 Auxiliary target points

In order to facilitate realistic movement of agents and the transition between rooms,

there are points called auxiliary points that are placed at a fixed distance from door

target points. Once an agent has reached a door target point, it should pursue the

auxiliary point that is associated with that door target point. This mechanism makes

sure that agents really make it out of a room before picking a new target point.

Definition 3.14. An auxiliary target point is a target point at a certain offset daux from

the point of a door target in the direction of that door target.

daux is a parameter that should be tweaked. In CEPABS it was taken to be 3 meters. It

is mentioned in section that every auxiliary target point has an associated zone, which

explains why 3 meters is not as large as it might seem. The set of all auxiliary points in

room i is denoted PA,i.

3.5.4 Terminal points

Definition 3.15. A terminal point is a point p ∈ O at an offset distance dterminal in

the direction of each door target in B.

dterminal is in CEPABS taken to be 10 meters, although it is not likely to be sensitive

to small changes. The set of all terminal points is called PF (F for final) and is should

be non-empty.

When an agent has reached a terminal point it does no longer pursue any other targets

and is considered fully evacuated. It should be noted that an agent that has reached

the outside O (that is considered evacuated) it will still search for terminal points and

is not inactivated until it has reached one such point. The motivation for using terminal

points is that otherwise evacuated agents would block the door that led them outside,

which the authors do not consider realistic.


3.5.5 Target zones

Since it is close to impossible within numerical float position to make an agent reach its

target point, each target point is assigned an associated target zone.

Definition 3.16. A target zone is a rectangle with width wzone and height hzone that

is associated with a target point. If the target point is a door target, the target zone it

is shifted a non-zero distance dzone in the direction of the door target point. Otherwise

it is centred about the target point.

wzone and hzone are both taken to be 2 meters in CEPABS and have been subject to

tweaking. dzone is taken to be 2 meters for door targets, placing the zone just in front of

the door. For auxiliary points dzone is taken to be 3 meters, placing one zone boundary

1 meter from its associated door. When an agent is within a target zone, it is considered

to have reached the target point associated with that zone.

3.5.6 Exit signs

Exit signs is an active evacuation measure that could aid agents in finding an efficient

(short distance) route out of a building. Exit signs are modelled to be situated right

above doorways and illuminated such that they are assumed to be always visible if

within the sight range of an agent. Exit signs can have three colours: Green, Red or

Plain. Door target points with Green signs are considered by agents to be a preferable

before Plain or Red signs and Plain signs are considered preferable before Red signs.

The influence of Exit signs on exit choice is further explained in section 3.6. Adding the

sign property to the door target point concept urges the extension of the definition of a

door target point.

Definition 3.17. An extended door target is a door target which in addition to its

elements point and direction has en element colour, where colour ∈ {Green, P lain,Red}.

Since an extended door target with colour ”Plain” is equivalent in the Point Search

Algorithm (described in section 3.6) to a door target, the more general concept extended

door target will be used in the remainder of this thesis. For ease of communication the

two concepts will be used synonymously.


3.6 Point Search Algorithm

The ability of agents to spot target points is an integral part of the Exit Search Al-

gorithm. In this section a point search algorithm (referred to as the Point Search

Algorithm) for determining whether a target is visible to an agent or not is described.

The algorithm draws inspiration from the one described in [4]. Before continuing, a few

things needed to describe the Point Search Algorithm are reiterated.

From the introduction of this chapter an agent has a position r, a target point ptarget

and a sight range rsight. From the conclusions of definition 3.8 an agent is by necessity

situated in one room with room index i, and the set of all visual obstacles in a room

with room index i is called Wi.

The set of all potential targets in room i is now introduced.

Definition 3.18. The set of potential targets in room i is PD,i∪PV,i∪PF if the agent’s

position r ∈ O. Otherwise it is PD,i ∪ PV,i.

The set of potential targets is denoted Ppot. Ppot is in other words the set of targets that

an agent will look for during evacuation.

Before continuing, mutual visibility of two points is defined. It is restated that visual

obstacles are composed of straight line segments, each which can be described as a subset

of a line.

Definition 3.19. Two points p1, p2 ∈ R2 are defined to be mutually visible in the

presence of a set of visual obstacles Wvisual if the straight line segment L defined by p1

and p2 does not intersect any of the line segments in Wvisual that are non-parallel with

L and the length of L is less than a positive scalar Rmax,vis.

Rmax,vis is a scalar representing a sight range.

A motivation for excluding all line segments inWvisual that are parallel to L is that such

a line segment does not obstruct line of sight, in accordance with the ”visibility” concept

in real life. Furthermore, excluding parallel lines enables the use of linear algebra to find

any intersection points between L and a line in Wvisual.

Definition 3.19 has a special interpretation when one of the points is an agent’s current

position, the other point a target point in Ppot, Wvisual is Wi and Rmax,vis is Rsight of


the agent. Then the definition tells us whether a target point is visible to an agent or

not.

Below, an algorithm for finding the visibility of a target for an agent is described. It

can make use of a sparse matrix and is thus beneficial when efficiency is desired. The

algorithm is used in CEPABS.

1. Consider only the set Wi,non−parallel ∈ Wi of obstacles in Wi that are not parallel

to the line L between r and rtarget.

2. Pose one linear system of equations for each line overlapping a line segment of

Wi,non−parallel and L.

3. Solve each system of equations to get the set of intersection points Pintersect.

4. Discard all points in Pintersect that do not lie on the line segment corresponding

to the considered line (those points do not obstruct line of sight).

5. Consider the rectangle Arect with diagonal points r and rtarget and for each point

pi ∈ Pintersect consider the distance di between r and pi.

6. If no point pi satisfies (di < rsight) ∧ (pi ∈ Arect), then that target is visible

according to definition 3.19 (no proof given).

3.7 Mental map

It was shown in the previous section that each agent has a set of visible target points

Pvisible for every position. This does however make an agent unable to consider a target

point as soon as it falls out of line of sight. In order to better reflect the way a human

being remembers targets, the sets of visited and known targets are introduced.

Definition 3.20. Every agent has a set of known target points, which consists of all

target points that have at at least one time step since the start of the simulation been

part of the set of visible points Pvisible of the agent.

The set of known target points is denoted Pknown. Some subsets of Pknown are Pknown,i,

which is the set of known target points in a room with room index i, the set of known


door taregt points Pknown,D, the set of known viewpoint target points Pknown,V and the

set of known terminal points Pknown,F .

Similarly, an agent keeps track of all target points that has been visited at least once.

Definition 3.21. Every agent has a set of visited target points, which consists of all

target points that have at at least one time step since the start of the simulation had

the agent’s r in its target zone.

The set of visited target points is denoted Pvisited. Pvisited has subsets that are defined

analogue to those of Pknown.

Definition 3.22. Every agent has a set of invalidated target points, which consists of

all target points that are not considered to be candidate target points by an agent.

The set of invalidated target points is denoted Pinvalid. Reasons for invalidating a point

could be that there is a perceived imminent danger at the door or that the agent has

grown impatient of pursuing that door (see section 3.8). Pinvalid has subsets that are

defined similarly to those of Pknown and Pvisited.

The mental map of an agent denoted S is now defined as S = {Pknown,Pvisited,Pinvalid},

and intuitively contains information about all doors that an agent has been made aware

of up until a certain time. It should be noted that [12] uses a similar concept called

cognitive map, but that a different name has been used as to not cause confusion between

the concepts.

The cognition and remembrance of target points plays a key role in the Exit Choice

Algorithm described in section 3.13.4.

3.8 Patience

Another aspect of the human mind that has been modelled in CEPABS is that of pa-

tience, drawing inspiration from [6]. Patience is the concept of an agent wanting to

change its rtarget if all of two criteria (and possibly a third optional criterion) are ful-

filled:


1. There should be at least one available target point that is not the current target

point.

2. The patience factor ppatience exceeds a threshold value ppatience,min.

3. (Optional) The current time step should be an update time.

An available target point (in criterion 1) is a target point that is currently in Ppot,i ∩

P∗invalid. This criterion functions such that if there is only one available door, the agent

will still pursue that door and not get impatient with it. ppatience and ppatience,min in

criterion 2 are described below. The third optional criterion primarily gives computa-

tional benefits of a lower update frequency, but could also allow for an agent to build

up speed after having become impatient as to not be impatient directly afterwards.

When an agent has become impatient, it adds its current target point to the set of

invalidated targets Pinvalid,i and sets out to find a new target. Once the agent has

reached a target point in the current room, all doors in the set of invalidated points

Pinvalid,i that have been invalidated due to impatience are removed from that set.

Definition 3.23. The patience factor ppatience of an agent is defined as

ppatience = 1− 〈v〉v0

(3.14)

where 〈v〉 is the mean value of the agent’s speed over the npatience last time steps and

v0 is the desired speed of the agent. The lower the averaged speed over npatience time

steps in relation to the desired speed, the higher is ppatience. In other words, the patience

factor is larger the lower the speed of an agent in relation to desired speed. npatience is

in CEPABS set to four time steps, with time step 0.025. If npatience is set too small,

temporary peaks or dips of the speed could have large influence on 〈v〉, while a too large

npatience could result in an agent that is unresponsive to low speeds, which counteracts

the purpose of the patience factor.

The minimum patience factor ppatience,min required in order to fulfil the first criterion of

impatience is in CEPABS set to 0.85, producing reasonable results.


3.9 The leader concept

One active intervention measure is the concept of a leader-follower mechanism. In this

mechanism there are some agents that are followers and some agents that are leaders.

As described in section 2.1.3 some authors divide agents into distinct classes of leaders

and followers while others let agents have degrees of leader or follower mentality. In

CEPABS, the scheme of [12] and [7] is adopted, having a distinct class of leaders and

one class of followers.

Definition 3.24. A leader is an agent that does not consider patience, does not follow

other agents, for which Pknown = PD and the only exit strategy is the exit search strategy

for leaders (see section 3.12).

That an agent knows know the complete floor plan allows it to pick an optimal route

through doors for finding its way out of a building. The leader-follower mechanism and

exit search strategies are described in the sections 3.12 and 3.13.

3.10 Pathfinding Algorithm

A pathfinding algorithm is an algorithm used by an agent for finding a path to a sought

target along a set of allowed waypoints. This kind of algorithm can be used by leaders

in order to find the shortest way out of a building, or for an agent to find a path to a

target point that it remembers. Here, an algorithm is presented in which the allowed

waypoints are door target points PD. The problem of finding the shortest path to a

target point through a set of waypoints lends itself well to be described using a graph.

Definition 3.25. A door target graph GD is an undirected graph in which all door

target points in the set PD are nodes, and in which weighted edges exist between nodes

(door target points) that are mutually visible (according to definition 3.19), excluding

all obstacles that are not walls from the set of fundamental visible targets. The weight

of an edge is the Euclidean distance between the two nodes for which the edge is defined,

except if the coordinate of two nodes is the same, in which case the edge between them

is given a weight that is negligible compared to all edges that connect nodes that do not

share coordinate.


For a building B and outside O, there is only one GD. Edges between nodes (door

target points) represent routes between door target points that may be considered by

agents. The distance weights of the edges represent the penalty of moving along that

edge. The reason for adding a negligible weight to nodes that share coordinate is that

an undirected zero-cost edge could be traversed an arbitrary number of times without

affecting the cost of a route that contains that edge. What is a negligible distance in a

building will have to be decided by the one who runs the simulation. Visual obstacles

that are not walls are not considered, because it is viable that such obstacles should be

easily passable. What constitutes an impassable object is not well defined in this thesis

and until a strict definition has been made it is up to the one who runs a simulation to

decide that. In order to assure that every node can be reached from every other node in

GD, GD needs to be connected. This is not ensured in the current definition of GD and

is discussed in chapter 5.

There exists many algorithms for finding the shortest path from one point to another

point in an undirected weighted graph. The algorithm used in this thesis is Dijkstra’s

Algorithm, provided by MATLAB’s TMbuilt-in function graphshortestpath.

3.11 Pathfollowing Algorithm

The pathfollowing algorithm presented here is the algorithm used by an agent following

a path found using the Pathfinding Algorithm described in section 3.10. The algorithm

is as follows:

Given a path defined as a sequence of nodes:

1. Set the target point rtarget of the agent to be the first node in the path (which is

a door target point).

2. If the zone of the current target point rtarget is reached, if rtarget is a door target

point go to 2a, else if rtarget is an auxiliary target point go to 2b, else if rtarget is

a terminal target point go to 2c.

(a) rtarget to be the auxiliary point associated with the just reached target point.


(b) Set rtarget to be the next node in the path.

(c) Terminate the Pathfinding Algorithm.

3. Repeat 2.

3.12 Exit search strategy for leaders

Leaders, knowing the complete floor plan of a building are modelled as being able to

find the shortest way out of it. The algorithm for doing this is the following:

1. Find the door target point with the closest distance to the leader’s r and pick that

door target point as the starting point in the Pathfinding Algorithm described in

section 3.10.

2. Find the terminal target point with the closest distance to the leader’s r and pick

that door target point as the final point in the Pathfinding Algorithm.

3. Get the nodes that define the shortest path from the starting point to the final

point and if two nodes share coordinate, discard the point that appears first in the

path.

4. Follow the pathfollowing algorithm (defined in 3.11) until the final target is reached

and the leader is evacuated.

Leaders do not consider other strategies for evacuation in the current model.

3.13 Exit search strategy for non-leaders

The exit search algorithm for non-leader agents (agents that are not leaders according to

definition 3.24) is comprised of three strategies: Leader Following, Single Target Pursuit

and (non-leader) Pathfollowing. In addition, patience may interfere with the strategies

Single Target Pursuit and Pathfollowing. First the three strategies are described, fol-

lowed by the Exit Search Algorithm which determines when the respective strategies are

triggered. Since the algorithm for prioritizing target points is central, it is dedicated a

subsection of its own.


3.13.1 Prioritising of target points

When an agent sets out to find a new target point in the Single Target Pursuit strategy,

it makes use of the Point Search Algorithm described in 3.6 to find target points to

include in its Pvisible. From that set the agent needs to choose one target point to be

its rtarget. In doing so the agent needs to prioritise between points in the sets PD,visible,iand PV,visible,i and take into account whether a target point has been visited or not,

the distance to the target point, the colour of if the target point is a door target point

door and whether the target point has been invalidated. In the current model only door

target points can be invalidated. The priorities for targets are given as a list, where

invalidated and visited target points can not be chosen at all:

1. Terminal target points

2. Green door target points

3. Plain door target points

4. Viewpoint target points

5. Red door target points

If there are no target points in Pvisible,i ∩P∗visited,i ∩P∗invalid, then the agent shall pursue

the strategy Pathfollowing (after first having acquired a path with the Pathfinding Al-

gorithm). The motivation for excluding invalidated target points is that the invalidation

of a target implies that it should be avoided at all costs. The motivation for excluding

visited targets is that agents should prefer to explore new targets rather than targets

they have already visited.

The priority list for targets is based solely on heuristics and trial during simulations. It

should be critically assessed and investigated in extensions to the model CEPABS.

3.13.2 Leader Following

Non-leaders can follow a leader (see definition 3.24) given that two criteria are fulfilled:

1. The agent and leader are in rooms with the same room index.


2. The distance between the leader and agents positions is less than Rfollow,max.

Rfollow,max is a maximum distance for which the agent is under sufficient influence from

the leader to follow it. In CEPABS, Rfollow,max is set to 10 meters. This value is chosen

from trial simulations but should be critically assessed in an extension to CEPABS.

The following definition defines what it means to follow an agent:

Definition 3.26. An agent that is following a leader sets its rtarget to the leader’s

position rleader at every time step.

The leader following in definition 3.26 produces duckling-like behaviour, meaning that

an agent strives to acquire the leader’s position at every time step without trying to

predict a leader’s movement.

3.13.3 Single Target Pursuit

Single Target Pursuit is a strategy for finding the way out of a building. The strategy

consists of that an agent pursues one target, and

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