Circle formation algorithm for autonomous agents with local sensing

Theses

Thesis/Dissertation Collections

7-1-2004

Circle formation algorithm for autonomous agents

with local sensing

Andrew Mario Michael

Follow this and additional works at:

http://scholarworks.rit.edu/theses

This Thesis is brought to you for free and open access by the Thesis/Dissertation Collections at RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please [email protected].

Recommended Citation

By

ANDREW MARlO MICHAEL

Thesis submitted to the Faculty of Rochester Institute of technology

in

fulfillment of the requirements for the degree of

MASTER OF SCIENCE

IN

ELECTRICAL ENGINEERING

Approved by

Thesis Advisor

Thesis Committee

Thesis Committee

Department Head

Dr. Attimoottil Mathew

Dr. Ferat Sahin

Dr. Wayne Walter

Dr. Robert

Bowman

DEPARTMENT OF ELECTRICAL ENGINEERING, COLLEGE OF ENGINEERING

ROCHESTER INSTITUTE OF TECHNOLOGY, ROCHESTER, NEW YORK

Name of author:

=4

~

t':

e~~'~

Degree:

Ole

S

.

Program:

~l

ey

±

I

f.

V\g"

tl7

College:

Co "

flJ

oj

f=

"'-8

e

I understand that I must submit a print copy of my thesis or dissertation to the RIT Archives, per current RIT guidelines for the completion of my degree. I hereby grant to the Rochester Institute of Technology and its agents the non-exclusive license to archive and make accessible my thesis or dissertation in whole or in part in all forms of media in perpetuity. I retain all other ownership rights to the copyright of the thesis or dissertation. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation.

Print Reproduction Permission Granted:

I,

~

ol

{i.

h

0..."

L,

Technology to reproduce my print thesis or dissertation in whole or in part. Any reproduction will not be for commercial use or profit.

Signature of Author: Date:

O{lfi

3/01,

Print Reproduction Permission Denied:

I,

-f}

vi

w

ll.

M.

~(JI.e)

During my study at Rochester Institute of_Technology (RIT), New York, there

have _{been many} individuals who have helped me in numerous ways. Dr. Attimoottil

Mathew, myresearch _advisor, stands out_among them. He has tested me withhis sharp

questions that have made me think a great deal and helped me _{develop my} thesis. Dr.

Mathew has seldom declined to talk to me when I walk into his office without prior

appointment! I thank_{him for giving} me so much oftime inspite ofhis _busy schedule.

Working with _{Dr. Mathew has been very} challenging. It made me realize _my short

comingsandhelpedmeto_developas a researcher.ThankyouDr. Mathew.

I wish to thank the _{head, faculty} and staff of the department of Electrical

Engineering at RIT. The department funded most part of_my research and helped me

cover _my expenses at RIT. I thank the _{department for providing} me work space and

equipment for my research. The _faculty at RIT _{has been helpful in guiding} me and

sharingtheir expertise. Specialthanks to_mythesis committee members Dr. FeratSahin

andDr.Wayne Walter. Dr. Jayanti Venkataramanmetme acouple oftimes to talkabout

thepractical_{applicability}ofthealgorithm. Thankstoher. Ialso wishto thankMr. James

Stefano andMs. Patti Vicarifortheir_help in manyways.

Myheart felt gratitude goes outto _{my family} and fiancee for _being with me in

spirit even though we are oceans apart. _They have been my source ofinspiration and

encouragement attimes of need. Theirprayers havehelpedme survive_many difficulties

faced as an international student. Thank you so muchonce again to _my dearest _amma,

CIRCLE FORMATIONALGORITHM FORAUTONOMOUS AGENTS

WITH LOCAL SENSING

By

ANDREW MARIO MICHAEL

MasterofScience in Electrical_Engineering

Abstract

Researchon cooperativeroboticshas increased radicallyoverthepastdecade due

to its _simplicity and _{applicability in} a _variety of fields. Shape formation plays an

important role in such cooperative behavior. Our work deals with the formation of a

circle _by a _group of mobile agents _(robots) that _initially are _randomly spread and

randomly oriented in anunmapped terrain. The agents have simple characteristics and

limitedcapabilities. _Theyare_{autonomous,homogeneous, anonymous,}andmemory-less.

They do not communicate with each _other, but are able to measure the inter-agent

distances and angels. The agents follow the same distributed algorithm _{synchronously}

without _any central control. The existing algorithms make it necessary to scan all the

agents over the whole terrain. The main advantage of our algorithm is that each agent

makes use oflocal information collectedfrom two _neighboring partners. Our algorithm

also results in a _{regularly distributed} circle _{for any form} of initial distribution. _By

changing a parameter in_{the algorithm, the} circle can eitherbe made to grow or shrink

uniformly. Applications ofthis work can be made to a _variety of areas such as space

Table ofContents

Acknowledgments ii

Abstract iii

TableofContents iv

ListofFigures vii

ListofTables x

L Introduction 1

U Related Work 2

1.2 AdvantagesandDisadvantagesof past work 5

1.3 Our Contribution 7

1.4 Thesis Organization 10

2. Cooperative Robotics 13

2.1 AdvantagesofCooperative Robots 13

2.2 ExamplesofFonnationin Nature 15

2.2.1 SchoolsofFish 16

2.2.2 FlocksofBird 16

2.2.3 TermitesandArmyAnts 17

2.3 Importance ofSelf Organization in Cooperative Robots 17

2.4 Importance ofShape Formation in Cooperative Robotics 18

3.14.3 Irregular formation 63

3.14.4 Inappropriate DirectionofOrientation 67

3.14.5 _Verysmall span of scan 70

4. Simulation andResults 74

4.1 Coordinate Axis Change 75

4.2 Flow ChartoftheAlgorithm 78

4.3 Convergenceofthealgorithm 80

4.4 Effectof_steponFormation 84

4.5 Effectof span of scanonformation 90

4.6 Effectof span of scan and_step onfinalradius 96

4.7 Circle formation forvariousinitial distributions 97

4.8 _Shrinkingand _growingcircle 100

4.9 Circlewithaparticular radius 102

4.10 Algorithmcomparison 105

5. ConclusionandFuturework 108

5.1 Conclusions 108

5.2 Future Work 109

Reference 112

Appendix A 116

Appendix B 117

ListofFigures

Figure 1-1 Initial agentdistribution._{Circle showing}theposition oftheagentandthe lines

thedirection it isoriented 1

Figure 1-2 Apicturefromthe_{SydneyOlympics opening ceremony} 7

Figure 2-1 "SpiritofMarsrover",therobotthatexploredtheMartiansurface _[38] 15

Figure 3-1 A colonyofagents_{distributed randomly}with theirdirectionsof orientation

and spans ofscan 22

Figure 3-2 Aregularhexagon (n ₌₆₎is divided into 4_(n-2) triangles 28

Figure 3-3 Partnerselection_byagent_R; 29

Figure 3-4 Partners changingat each iteration 31

Figure 3-5 Nearestagents selected as partners 32

Figure3-6 Pathoffive agentsifthenearest agents are selected as partners 34

Figure 3-7 Pathoftheagentsifthesmallestangleagents are selected as partners 34

Figure 3-8 MeasurementofInter-Agentangle anddistance 36

Figure3-9 Twopossiblepositions_Rni and_Rn2witha = 6_{andequidistant}from_partners

37

Figure 3-10 InternalandExternal agents 38

Figure3-11 A distributiontoexplainthe selection ofcorrect position 39

Figure 3-12 Position of partnersinthe localcoordinate system 41

Figure 3-13 Threepossiblewaysforpartnerstobewhen a<7t 42

Figure 3-14Threepossiblewaysforpartnerstobe for a > 7t 43

Figure 3-15 Newposition_Rnof_{Ri whena<7i,} case_(a) 43

Figure 3-17 Newposition_RNof_R when slope of_RlRr>0 48

Figure 3-18 Diagram showingthedistance Dto travelandthe angle_yto turn 51

Figure 3-19 Five agentsin formationof a regular pentagon_showingpreferreddirectionof

orientation 54

Figure 3-20 Theangle etobeadjusted_byRfor_(a) a< k and_(b)a > k 56

Figure 3-21 Aregular_{hexagon growing}or_{shrinkingdepending}on

6d

Figure 3-22 Relation betweeninter- agentdistance_(ird) andcircleradius_(RAD) 59

Figure 3-23 Directionof orientationfor_R;after_movinga_step 61

Figure 3-24 Twoagents _selectingthesame agents astheirpartner 62

Figure 3-25 Fouragentsin a rectangle_movingtonewlocationson another rectangle...64

Figure 3-26 Agents gettingtooclosetoeach other 65

Figure3-27 Simulation resultsfor irregularshapeformation 66

Figure 3-28 Simulationresults forchangeofinternalexternal agents 66

Figure 3-29 Aagent_{selecting inappropriate}partnerduetoits directionoforientation...68

Figure 3-30R7gettingstagnant atthe_wrongpositionintheformation 69

Figure 1-31 Twoagentstrappedinthe middle.Theirpath is also shown 70

Figure 3-32 Formationoftwo_shrinkingcircleduetosmall span of scan 72

Figure 3-1 Changeofaxiswith respectto theposition anddirectionof_Ri 75

Figure 4-2 Distributionofthe_{initial colony} 80

Figure 4-3 Convergencegraphs,n=50,with unlimited_stepandrange. Errorfor

iterations from_(a) 1-100_(b) 101-200_(c)Inter-agentangle _(d)final formation 81

Figure 4-4_(a)Errorvs.#ofiterations,n=_{49,withunlimited}

stepandrange, _(b) Final

formationafter300 iterations 83

Figure 4-5 Effectof_stepsize onformation.Theright column showstheformationafter

50iterations andtheleftcolumn showstheerror graph 87

Figure 4-6 Theerror graph andthefinalformationforan optimal_step 88

Figure 4-7 Numberof agentsthatbecomeexternal againstnumberof steps 89

Figure 4-8 The initial colonyshown in figure 4-2simulatedwith_step=0.5for first 10

iterationsand0.01 thereafter 90

Figure 4-9 Effectofspan of scan onformation forthe_colonyshownin figure 4-2 93

Figure 4-10Maximum,mean and minimumdistances fromthecentroidforthecase when

span ofscanis 0.2and0.3 94

Figure 4-11 Finalradius shown as a surface plot_{for different step}andspan of scan 96

Figure 4-12 Final formation _(right)forvariousinitial distributions _(left) 100

Figure 4-13 Radius ofthe_growingcirclefor differentvalues of50 101

Figure 4-14 Radiusofthe_shrinkingcirclefor differentvaluesof50 102

Figure 4-15 Mean distance fromthecentroid andfinal formation for differentrequired

radii 105

Figure 4-16 Distance fromthecentroid andthefinal formation in Suzuki'salgorithm. 105

Figure4-17 Distance fromthecentroid and thefinal formation inthealgorithm

developed inthisthesis 106

Figure 4-17 SimulationresultofSuzukiet al.'s algorithmwhentheinitial distribution is a

Table 3-1 Theoreticaland simulation valuesfortherequiredradius andthe_{corresponding}

inter-agentdistance 103

A groupof_{robots, autonomous, simple andtake steps as a result ofchanges inthe}

environmenttoachieve a common goal with_{very little human intervention}or supervision

can be used in applications such as space _{missions, agriculture, fire fighting,} search

operations, landmine _de-mining and _many others [1-3]. Over the years engineers and

roboticistshaveresearched_extensivelyon artificial intelligenceofan individualrobot. In

the recent past the attention has been diverted tocooperative _robotics, inspired_by social

insectcolonies or swarms.

If n number of autonomous robots are scattered and oriented _{randomly in} an

unmappedterrain (figure _1-1)orderingor_arrangingthembecomes anissue ofimportance

iftheserobots areto achieve a common task. Sincethe terrain is unmapped andthere is

no global coordinate system each robot will be _basically lost in the wilderness. The

distributionshowninthefigure needstoorganize andform ashapebeforetherobots start

toachieve atask.

Figure1-1 Initialagentdistribution. Circle showingthepositionoftheagentandthe

Research on _{forming shapes with cooperative roboticshas been ofinterest in the}

recent past. This serves as a _starting point for cooperative task achievement for

autonomous robots_randomlyspreadinan unknownterrain.

1.1 Related Work

The study of swarms and theircomplex task of _achieving a goal through simple

steps has been done for a _long period of time. Inspired _by the swarms, over the past

decaderesearchoncooperative robotics_{has radically increased due}to its applicationina

varietyoffields.

Thepioneers of shapeformations incooperative robotics wereSuzukiet al. [1-5].

In their research _they have developed algorithms for circle, line and simple polygon

formations. The algorithms are developed with the assumption that robots have global

sensing capacity. The robots need to know the positions of all the other robots in the

colony. Their circle formation algorithm is as follows. Each robot R monitors the

farthestrobot R andthenearestrobotR . If Dis thediameterofthecircletobeformed,

S a small positive constant anddthedistance between R and R , robotRthenmoves as

follows.

If d>D,thenRmovestowards R

If d<D-5,thenRmoves away from R

If D-S<d<D,thenRmoves_{away from R}

The algorithmintheir work is simple butthe formation is not precise. The circle

formed is an approximation of a circle and the robots are not distributed _evenly on the

circumference. _Further, for certain _{initial symmetric distributions their circle algorithm}

doesnot make a circlebuta shape calledReuleaux'striangle[1].

Suzuki et al. in _[3] give a_{formal discussion ofthe limitations oftheir algorithms}

under certain assumptions. In their later work _they have modified their algorithm to

represent a robot as adisc andhave alsoincludedcollision avoidance strategies [2].

Alongthelines ofSuzukiet al.'s _work,extensive workhas been done _byPrencipe

et al. [6-9]. In their research the robots are considered to be asynchronous and

formations _{resulting in finite} time. There is no common notion of time and a robot

observes the environment at unpredictable time instants. The formation problem of

cooperating and _coordinating a _group of independent robots is analyzed from a

computational point of view. The main problems analyzed in their work are: _arbitrary

pattern _{formation, gathering} and flocking. In the _arbitrary pattern formation the robots

have the _capacity of global _sensing, have a prior knowledge and agree upon a unit

distanceand a common direction. Therobots are also giventhecoordinates ofthepattern

tobeformedwith respectto theirlocalcoordinate system.Theproblemis_{mathematically}

analyzed. In the _gathering problem, the robots are supposedto gather at a certain point.

Prior knowledge of a common direction _by the use of a compass is exploited in the

developmentofthisalgorithm. The need of a compass arises_{only if}the robotshave local

sensing,but iftherobotshaveglobal _sensingthecompass_{is done away}with.

The flaws and imperfections in the work of Suzuki et al. has been modified in

[10] Asstated earlierin _[1] thecircle formation algorithmdoes not work wellforcertain

initial distributions and results in imperfect formations. In _[10] a different approach is

Thealgorithmis asfollows.

Robot R determines the furthest robot

Rf

closest robot Rc2. Computes the coordinates of the centroid _pm, of

Rf

,Rcl and Rc2.

Moves to point _pr which is r distance away from _pm on the line that passes through

currentposition andpm,whereristhedesiredradiusofthecircletobe formed.

In thisworktherobots are also assumedtobe _havingglobal_sensingcapacity. The

algorithmdoes not explain how each robot will move from its present location to reach

thecentroidpoint.

In _[11] the authors _try to make a circle from a _{randomly distributed colony} of

robots. The algorithm is based on Voronoi cells and Smallest _Enclosing Circle (SEC).

The smallest_enclosingcircle, as thename suggests, is thecirclewith the smallest radius

enclosing all the robots. Each robot determines the _boundary ofthe smallest _enclosing

circle and moves to that point. Here too the robots need to _{have sensing} of the whole

terrain. The authors state "Althoughwebelievethat thealgorithm could_{actually be} used

in_{practice, there} are severalimportant issuesthatmustbeaddressed". Eachrobot has to

scan the positions ofthe rest of the _colony and needs to make intensive computation to

determine theSEC. Also, how the SEC can be practically determined in real time is not

stated.

In _[12] the circle formation problem is rather trivialized _by the use of a beacon

around whichthecircleistobe formed. Iftherobotshavea priorknowledgeoftheradius

ofthe circle and_{by measuring} thedistance betweenthemselves and thebeaconand then

bymoving accordinglycircle can_{easily be formed. If}thereis acentralbeaconthe terrain

is nomore unknown._{The beacon itself}serves as theorigin of acoordinate system around

whichacircle ofdesiredradiusis formed.

From our literature _survey the above were the algorithms we found on circle

formation in cooperative robotics. There are other formation algorithms but do not

directlyrelatetocircleformation. We also listthoseworksbelow.

Balch et al. _[12-14], deal with the problem of_keeping aformation and _avoiding

obstacles while in motionrather than shapeformation of a_{randomly distributed} colony.

Keepingthe shapeofa_{fine, column,} diamondand wedgefora_groupofrobotsinmotion

is addressed.Moreoverthe robots arenotconsideredtobe homogeneous sinceeach robot

hasanidentificationnumber. Theglobal_positioningsystem_(GPS) is usedto transmit the

coordinates of the robot positions _making the system not simple. The use of world

coordinates withthe_help ofGPSmakesthe terrain_totallymapped.

Mataric et al. _{[16, 17]} have also worked on _{maintaining formations for} a small

group of robots. In their work the robots have local _{sensing but} through simple

communication _theyhave access to the global goal. The algorithm is developed _by each

robot_keeping a designated friend at a particular distance and angle _{by using} a_panning

camera.EachrobothasauniqueIDand a protocolforcommunicationpurposes.

1.2 AdvantagesandDisadvantagesof past work

Circle formation algorithms stated abovehave theirowndifferentadvantages and

disadvantages.

The advantages of the algorithms presented in _[1-5] and _[10] are that _they are

Themaindisadvantage of alltheabove algorithmsisthateachrobot needstoscan

the whole terrainfor the

functioning

to scan the positions of all the robots in the _colony, store the information collected and

make decisions _depending on the stored information. Global scanning is disadvantages

forthe

following

1. The _battery power of the robot will die off _faster, since _scanning the whole

terrain needs more energy. This will reduce the time period the robot can

function.

2. To attain global_{sensing it becomes necessary}to havemore powerfultransmitters

and sensors. Thismakes therobots sophisticated andnotsimple or weak.

3. Ifthenumber ofrobotsn is largetherobots needto havea_{memory array}to store

thepositions ofthe robots. Then therobots haveto sortthem to selecttheclosest

orthefarthestrobot.

4. Whenn is large scanningthewholeterrainis time_{consuming resulting in}a_delay

ateachiterativestep. Thisreducesthe_efficiencyofthe_colonyaswhole.

Anotherdraw backof pastworkis_they do not explainhoweach agentwill move

at eachiteration. The directionandthedistancea robotwill make ateachiteration are not

explained. In simulation it is easy to find the location of the new position of the robot

usingtheglobal coordinate system. In apractical situation of anunknown terrainthere is

no such global coordinate system. A robot has toknow how it will reach a new position

at each iteration. This decision making on how each robot has to travel is not _clearly

explained.

The other disadvantage is that the above algorithms do not always result in a

circle. The outcome is dependent ontheinitial distribution. The circleformed is also not

evenlydistributed.

1.3 OurContribution

Imagine the situation _{during Olympic opening} ceremonies (figure _1-2) and other

sportdisplays the performers make various forms and shapes in alarge unmarked field.

Anothersituationis when a_group oflargenumber of peopleis askedto makeacircle. In

both these situations the members are able to form shapes with no central control in an

unmarked terrain. _Essentially the members observe the positions of the _neighboring

members and positionthemselves_iterativelytilla_reasonably acceptable shapeis formed.

In this thesis we _try to make a circle in an unknown terrain with autonomous

mobile agents with _{only local sensing} capability. An agent observes thepositions ofthe

neighboring agents, oneto its leftand one to its right, computes thenext position it has

to be and moves there. The same iterative process of _observe, calculate and move is

simultaneously executed _by all the agents in the_colony till the_{colony forms} a circle. In

this thesis we present a circleformationalgorithmfor verysimpleagents.

The agents are autonomous. Human intervention is not needed after

initializingthecircleformation algorithm.

The agents in the _colony are all identical. There is no leader or even a

hierarchicalstructure.

The agents do not communicate with each other. _They do not have

distinctive IDnumbers.

The agents do not have knowledge of a global coordinate system since _they

are spread in an unknown terrain. There are no landmarks or beacons

aroundwhichthecirclehastobe formed.

Theagents donot usetheworldcoordinate systemwiththe_helpofGPS.

Theagentsdo nothavea sense of commondirection or use acompass.

The main advantage ofourcircle formationalgorithm is that the agents have

limited scanningpower.

Evenwith local scanningthe agents do not scan alltheagents withinits span

of scan. An agent selects and collects information only from the two

neighboring partners. This makes _{it unnecessary} to store information of all

theagentsinthecolony.

The agents are _{memory less. They} do not store information about their past

path pointsorabouttheirpast partners.

A novel idea of _using the inter-agent angle information is exploited for the

formation algorithm. The inter-agent angle and distance information is used with

properties ofregulargeometricfigures intheformation.

The agents move in iteration synchronously. Their functions at a particular

iteration are:_{Move, Scan,}Measureand Compute. Inouralgorithm wehaveused ahigher

computing power for the agents than other algorithms. An agent can perform algebraic

and trigonometric calculations to obtain the angle and distance it has to turn and move

respectivelyateachiteration.

We give a formal explanation _{mathematically} and with the _help of diagrams on

how each agent has to move at _{every iteration. We mathematically derive} equations for

the distance and angle an agent has to take to reach the new position. The higher

computing power is used to obtain these distances _{by substituting in} the equations we

derived.

Oursimulationdoesnot usetheglobal coordinates.Eachagent scans with respect

toits localcoordinate systemwithitsposition asits origin and directionof orientation as

thepositive x-axis.Simulationsare madetorepresenthow in arealsituation an agent will

observethecolony, select_{the partners,} and calculate the distanceandangle it hasto take

In _[6] the author _quotes, 'Little is known about the _solvability of other

geometrical problems like _spreading and exploration...'. In our work we present

extensions ofthe circle _algorithm, such as _flocking or _gatheringat one pointor _foraging

in the formof a _growingcircle. These arethe extensions ofthe samecircle algorithm but

areachievable_byalteringa single parameter.

Finally our algorithm works well for all type of initial distributions resulting in

evenly distributedcircles.

1.4 Thesis Organization

Chapter 1

In this chapter we _briefly explained the area of shape formation in cooperative

robotics and its importance. Past work in shape formation and their advantages and

disadvantages were noted. _Finally we gave details ofthe contribution we have made in

our circleformationalgorithm.

Chapter 2

In chapter 2 we give an _{understanding} of cooperative robots. Examples of

cooperative robotsinnature are stated. Advantagesof cooperativerobotics are given.The

importance of self organization and shape formation in cooperative robotics is brought

forward.

Chapter3

Chapter 3 isdedicatedforthedevelopmentofthecirclealgorithm.We startwith a

description of the _colony and the characteristics and capabilities of its robots. A

mathematical nomenclature of the _{colony is} given. The mathematical idea behind the

circle algorithm is explained. Then we _develop the _{understanding} and some definitions

neededforthe algorithm. Howeach robot calculates thepositions ofother robots andthe

coordinates ofits new position with respectto its local coordinate system is worked out.

A mathematical derivation of the _distance, the angle and the direction of orientation

adjustment a robot has to take at each iteration is given. We then explain how this

algorithmcanbealtered tomake a _growingor _shrinkingcircle._Finally somedraw backs

ofthe algorithmand possible solutions onhowtoovercomethem are addressed.

Chapter 4

In this chapter we obtain simulation results of the algorithm we developed. In

simulationeachrobot usesits localcoordinate systeminsteadof_makinguseoftheglobal

coordinates. The required coordinate axis change is explained. A flow chart of the

algorithmis thengiven. Theconvergenceofthealgorithmtoa_{regularly distributed}circle

is verified _{by plotting}the inter-robot distances. The effect of _step size and span of scan

on the final circle formation is analyzed. Then we simulate the results _{for growing} and

shrinking circle. _Forming a circle with a particular radius is also simulated. _Finally we

Chapter 5

Chapter five is for conclusion and future work. Issues involved in practical

implicationwillbestated.

Reference

Appendix

Swarm based robotics relies on a_group of simple robots that are able toperform

tasks without explicit representation ofthe environment and of the other robots [4]. In

this approach extensive _{initial planning}of_achieving atask is minimized_{by allowing} the

robots to react to changes in the environment. Social insects are autonomous and

communicate with each other through the environment. Indirect communication _among

insects through modifications of the environment was coined _{stigmergy by Grasse,} an

entomologist[4]. _{Stigmergy theory}states thatstepstaken_by a_{colony is} regulated_bythe

effect oftheprevious steps. It also states how activity can be regulated_{using only local}

perception and indirect communication through the environment for _coordinating

distributed behaviortoachieve a globaltask.

Social insects are _very simple and ineffective as an _{individual entity but} as a

colony they cooperate to attain global needs. In autonomous cooperative robotics the

sameidea is imitated_byhavingseveral simplehomogeneousrobotsthat worktogether to

achieve a user defined goal. Unlike the artificial intelligence of a single robot that is

expensive, complicated, tailored to specific _problems, cooperative robotics uses a

differentapproach of_usingteamsof_{simple, interacting}robotstoperform a wide range of

tasks [5].

2.1 AdvantagesofCooperative Robots

The fact thatresearchers are yet to invent a_highly autonomous robot capable of

functioning in a _changing environment has lead them to propose the organization of

robotics has several advantages over individual robots with artificial intelligence. If a

specific task is to be performed _byan individual robot, it increasesthe robot _complexity

making it difficult to design and fabricate. On the _contrary cooperative robots have

elementaryfeatures makingthemeasiertodesignand manufacture.

Due to the _simplicity and the increased number of robots _made, the cost of

manufacturingcooperative robotsis _highlycost effectivethan thatof anindividual robot.

A group of cooperative robots _{is comparatively} more fault tolerant than an

individual robot. Sincethere are a number of_robots, ifoneofthemmalfunctions therest

may carry on with the task. In our model we assume all the agents to be homogeneous

and with no hierarchy. This makes the _{colony further} more fault tolerant. If there is a

hierarchy and there is a malfunction at the _top of the_hierarchy several or all the robots

may beaffected.

In cooperative robots the algorithm_by which _they are driven plays an important

role intheir _{functioning. It is comparatively}easier to change thebehavior ofthe_colony

by changingthe algorithm than tochange the performance of a single robot designedto

meet specific goals. Thismakes cooperative robots moreflexible.

Whena_groupofrobots are engaged_{in achieving}thesame_{task, efficiency}oftask

completion increases. In applications such as space _mission, search operation, lawn

movingor_harvesting, moreareacouldbecovered_bythe_colonythan theareacovered_by

anindividualrobot.

In the recent space missionsby NASA toexplore the Martian surface individual

robots are _being employed. One such robotis shown in figure 2-1. In these single robot

missions therecould be numerable problems and thepossibilities of a mission failure is

fairly

Martian surface, a high degree of fault tolerance _{may be} achieved. Moreover the

efficiencyof exploration could alsobe_{significantly}increased.

Spirrt Mars Exploration Rover

Figure 2-1 "SpiritofMars_{rover", the} robotthatexploredtheMartiansurface

Forrobotsto functionas a_{group in}suchautonomoustasksinunknownterrainsit

becomes a_necessitythat _theyare capable of"Self

-Organizing"

or workin formations.

Social insects and animals that inspired cooperative robotics work as a _{group in}

formationsorinanorganized manner_{[16, 17,} and 18].

2.2 ExamplesofFormation inNature

Socialinsectsare distributedsystems inwhich _{colony level behavior}emerges out

of interactions among individuals [19]. Intricate phenomena such as _foraging, nest

capabilities. There are other social animals in nature which thrive as a _{group but} as an

individual entity may notbe able to survive. In many such colonies there is no_hierarchy

or central control. Local information _by neighbor observation is used to obtain global

goals.We examinefewsuchnatural systems.

2.2.1 SchoolsofFish

Partridgeexamines how schools offishmove incertain formationstoincreasethe

effectiveness ofthe school [20]. Forexample Tunaschools form a parabolic shape with

concave side forward and swim parallel to its axis. A prey reactingto the curved school

will be driven to the focus oftheparabola which makes the capture easier. A school of

fish moving in a certainformation has ahigherchance of_detectinga predatorthan afish

swiniming individually. On the contrary predators also move in formations to increase

thesearch area.Thishelpstosharethefood found_byaparticular member[21]. This idea

canbeappliedinsearchrobots for planetaryand_military applications.

Formations were maintained in these schools _by individual fishes maintaining a

particular angle and distance with the _{neighboring fishes. This idea is} utilized in the

algorithmofthis thesis.

2.2.2 FlocksofBird

Birds also _flyin formations toincrease the scanningarea like schools offish. Air

force fighterpilotsuse this technique todirecttheirvisualandradar search _dependingon

their position in a formation [22]. Three mechanisms: collision avoidance, velocity

matchingwith _{nearby flock-mates} and flock centering in attemptto _stayclose to _nearby

flockmates are utilized [23].

2.2.3 Termitesand_ArmyAnts

Ants and termites are some ofthe most organized social insects. Construction of

nest _{structures, finding the shortest path between two}

points, _foraging efficiently as a

swarm, _bringingback the food to the nest are some examples of self organization [24].

They use minimal communication between the members ofthe _colony through trails of

pheromone. Inthese insectsformation is intheformofestablishedtrails.

2.3 ImportanceofSelfOrganizationinCooperativeRobots

Whena_group of autonomous robots aredropped inan unknown_terrain,the_group

has to organize itself before it proceeds with task achieving. A _{self-organizing}system is

defined_{by Farley} et al. _[25] as a system that changes its basic structure as a function of

its experience and environment. In self _organizing system a change in the environment

may influence the same system to generate a _{different task,} without _any change in the

behavioral characteristics. _Anysmall differences in individual behaviorcan influencethe

collectivebehaviorofthe system.

Self-Organizing autonomousrobotshave afew more advantagesthancooperative

robots with central control. In a _{large group} of robots communication overhead is

prohibitively high to collect all relevant information to a central location. It is also

computationally infeasible fora centralcontrolto generate aschedulefortheentire setof

robots inreal time [1]. Hencetheneed of autonomous robotswith distributed computing

In military applications, the whole _army of robots is destabilized if the central

control or the leader is destroyed. On the _{contrary if} the robots are autonomous and

homogeneousthemission continues evenifafewrobots aredestroyed.

There are afew disadvantages inautonomous cooperativerobotics.Dueto lack of

coordinationthere couldbe stagnation [16].A_group ofrobots could findthemselves in a

dead lock detrimental to the global task. The other disadvantage is that the

miniaturizationofrobots_{severely limits}their_sensingand_computingpowers.

2.4 ImportanceofShape Formation in Cooperative Robotics

1. Iftheagents arerandomly spread andrandomlyoriented each agent will goabout

doing a task without the coordination of the rest of the colony. The behavior

would be haphazard, chaotic and will not be directed towards a particular goal

achievement.

2. Thisproblemis important, because itprovides a_{way for}the agentsto agree on a

commonorigin and a common unit _distance,for instance_{by forming}a circle [1].

A flock of agents can converge to a point and use that point as the origin as a

startingpointtoachieve varioustasks.

3. Formations can be effectively utilized _{in exploring} an unknown terrain. If the

exploration is done in an unorganized manner it would be less efficient and

inconclusive.

4. Formations _helpto increase the _scanningrange ofa _group of agents. In military

applications where sensor assets are _limited, formations _help to cover a wider

areaif individual members concentrate acrossa portion oftheenvironment while

theirpartners covertherest [13].

5. Can be used_{in military}operations: toformabarricade in theshape ofacircle or

This chapter is dedicated to the development of the circle formation algorithm.

The chapter begins with an introduction of the _colony and a description of the

characteristics and capabilities of the agents. _{The colony} and the _{way it functions} are

defined_bya mathematical nomenclature.

The basic mathematical idea behind thealgorithm is explained. Then we proceed

to showhoweach agent will moveto a new position_using this mathematical ideasothat

the_colonyas awhole will performthecircleformation algorithm.

Recallthatwearenot_usinga globalcoordinate system. Thismakesit compulsory

to scan the terrain with respect to alocal coordinate system. The current position of an

agentis selectedas the origin andthe current direction oforientation as the direction of

positive x-axis ofthelocalcoordinate system.

An agent selects two partners, measures the inter-agent distances and the angles

and moves to a new position based on these information. For this an agent has to first

selecttwo partners. Partnerselection and _whyan agent selectspartners in such amanner

is described in detail. The two parameters to be measured from the partners: the

inter-agentdistanceandtheinter-agentangleare defined.

Forthe agent to move to anew position it has to compute the distance _(D) it has

to travel from its current position and the angle _(y) it has to turn from its current

direction of orientation. We mathematically derive formulae for these two physical

quantitiesintermsoftheknownparameters.Aftertheagent movestoitsnew positionthe

direction of orientationoftheagenthas tobeadjusted forappropriate partner selectionat

how it is altered is explained. _Finally the chapter deals with the short comings of the

circle algorithm and abriefexplanation onhow toover comethemaregiven.

3.1 The_ColonyofAgents

The colony has n agents that _initially are _{randomly distributed in} an unmapped

and unknown terrain. There is neither a global coordinate system nor landmarks that

couldbe identified _byan agent. Theagents inthe_colony are allidentical in physical and

functional terms. _They all start _{functioning synchronously} in terms of _scanning,

measuring, computingandmoving.

Initially the agents are _randomly spread and _randomlyoriented. The direction of

orientation of an agent is the direction from which it starts its scanning to its left and

right. The agentshave alimitedspan _ofscan within which_they candetect the positions

ofother agents. _{By moving}together_{synchronously in}each iterationcycle, the final goal

istoformaregular polygonthatis onthecircumference of a circle.

Figure 3-1 shows the positions of a _colony of agents. The agents, Ri, R2..., are

represented _by a circular disc. The line with arrow denotes the direction of orientation.

The agents are _looking towards various different directions as shown _by the lines with

arrow. The Dottedcircles arethelimits of spans of scanofthe agents. Each agent selects

twopartners within its span of scan and moves toanewlocation. This process continues

R6

.

V

Rn

^^Spanofscan

\

/\Ri

R,

R8

Z

z^

R4

I

R7

R9V

Figure 3-1 A colonyof agentsdistributed randomlywiththeirdirectionsof orientation and spansofscan

3.2 Agent CharacteristicsandCapabilities

The agents in the _colony are considered to be _having simple characteristics with

weakorlimitedcapabilities._Theyare,

Characteristics: Autonomous, Homogeneous, Anonymous, _Memory less, Synchronous,

UncommunicativeandMyopic.

Capabilities: Scan, Measure,ComputeandMove

Autonomous- This is

one ofthe main features and advantages ofthe agents in

thecolony. _{The colony istotally}independentof_anycentralcontrol afterinitialization _by

theuser. Theagents areallinitialized atthesame timeinstant. Thenthe agentsfollow an

algorithm_iterativelyuntil auser specifiedtermination.

Homogeneous- The

agentshave thesame limitedfeatures and areidentical in all

sense. There are no physical or _{functional differences. They use the same distributed}

algorithmtodeterminetheirnext position at_{every iteration.}

Anonymous

-As a result of_homogeneityit is impossible to have a_hierarchy in

the _colony or _{individual identification numbers for each agent. When an agent scans the}

terrainall theother agents appearthesame.

Memory less - The

agents do not have any form of memory. _They do not

remembertheirpast locations in the terrain. _They also do not remembertheirpartners in

thepreviousiterationand ateach _steptherest ofthe_colonyappearsdifferent.

Uncommunicative- The

agents donotcommunicate with each other. Interagent

communication becomes a _difficulty due to bandwidth limitations, especially when the

number of agents is large. Communication is through the environment in an implicit

manner rather thanexplicit communication between the agents. The agents_only observe

thepositions oftheirneighbors withrespectto theirpositionanddirectionof orientation.

Myopic- The

agents are limitedwith_{only local scanning facility. The} agents are

ableto_{detect only}theagentsthatfallwithinits limitedspan _ofscan.

Synchronous

-The colony has a global clock which is launched at _colony

initialization. Thereafter all theagents act_{synchronously} with respect toglobaltime and

also _stop atthe same time. The global _timingplays a _{very important}role in the iterative

procedure of circle formation. The agents have to _{be stationary} to measure inter-agent

angle anddistance. Iftheagentsarenotsynchronous and some ofthemarein motionitis

almost impossible to measure these quantities. The necessity of _{synchronicity} could

Theagentshavethe_followinglimited capabilities.

Scan- While

the_colony of agentsis_stationary, each agentcanscan_alonga plane

fora range of2k radians. It is ableto detectother agents in the _colony while scanning.

Anagent scans toits leftand right from its currentdirection of orientation.The instant it

detectsan agenttoits left it stops_scanningto theleftanddoesthesametoits right.

Measure

-After_detectingthe agents to its leftandright it is abletomeasure two

physical quantities: inter-agent distance and the inter-agentangle of the left and right

neighboring agents. More ofhow and _whythe _neighboring agents are selected in such a

manner will be explained in section 3.5. The definitions of agent angle and

inter-agentdistancearegivenin section 3.6.

Compute

-Using the distances and the angles _{measured, the} agent is able to

compute _y the angle_bywhich it has to turnfrom its currentdirection of_{orientation, the}

distance D it has to travel from its current position and the angle adjustment to be

made attheendof movement. The equationsfor_{D, y} and will be derived in sections

3.10 3.12 respectively. The agents are capable of _{performing basic} arithmetic and

trigonometriccalculationstoobtain_{D, y,} and .

Move- The_agentsinthe_{proposed algorithm are able}to traversein any direction

in a two dimensional horizontal plane. _They first turn an angle_y andthen move a user

definedmaximumdistanceof_steporless inaunittime.

The agents in the _{colony synchronously do} the above foursteps at each iteration

cycle.Then thecycle starts again with_scanningthen _measuringand so onand movestoa

new position.

3.3 DefinitionsandNotation

Thenotations wehave used relateto the definitionsgiven _{in [3]. The colony has}n

agents.Let C denote theset of agentsin the_colony and_R..R2...

.,R, ... Rn bethe agents

inthecolony. _R;denotesthe/'*

agentofthecolony.

c={*,|i<,<}

The synchronous formation ofthe _{colony is} timed _by the global clock. Afteruser

initialization, theagents_{scan, measure,}compute atdiscretetime instances.

t=_kT

(3.2)

Here k is a positive integer, and T is the time period ofone iteration cyclein the

algorithm. One iteration cycle consist offour steps of _{scanning, measuring, computing}

and moving. Let us denote these time intervals for these steps as _{Ts, Tm, Tc} and Tm.

respectively.

T^+T^+T^+T^

The time consumed for each of these steps _by all the agents is the same to

maintain synchronicity. _{If synchronicity is lost} the implementation of the algorithm

becomes difficult. For instance, _during the _{scanning step} the whole _{colony has} to be

stationary. Ifsome of theagents are in motion it willbe difficult to detect and measure

thedistancesand angles oftheagentsinmotion.

Let _pi

(t)

we canwrite_P(t), the setof positionsofallthe agents_as,

P(t) is the set of positions of all the agentsin Cat various discrete time instances.

Each agent_R; observes_{P(t)differently} unless the _{distribution is perfectly} symmetric and

the

agents'

orientations are directed in a uniform manner. Ifthe agents in the_colonyare

myopic or with limited span of scanthen _R, cannot observe P(t). It will _{only be} able to

observe a subsetofP(t). Letus denote _Vj(t) tobethesetofagentpositions observable_by

Rattimeinstantt._Then,

^(OcP(f) (3.5)

Vi(t)=

{pv(t)\l<v<n}

Since theagents in the _colony are myopic vis not equal ton. v is the number of

agents detectable_{by R;} within its span ofscan. Ifagents have global _{sensing capability}

then v=_n

.Ifthespan ofscanis small orifan agentis lost inthe terrain, itcannot sense

any other agent, then v=0

. Vrft) is different for all the agents unless in a very special

case ofa_perfectlysymmetric distributionwith

agents'

orientations directeduniformly. A

formation in the shape of a regular polygon where the

agents'

orientation are directed

uniformly and_{radially inwards} or outwardsis one suchdistributionwe can thinkof. The

ultimateaim ofthecircleformationalgorithmis tomake_Vtft)of alltheagents same.

The step an agent needsto movedepends _totallyontheprevious configuration of

the colony. _Therefore, in a global sense _P(t)directly relates toP(f-l). We could also

define _P(t)in an alternative manner. _P(t)is also a function of Vt(t). A step each agent

takes depends _solely on how it observes part of the _colony at that time instant.

Consequently,P(t)is dependenton_Vj(t), V2(t),..., Vt(t),..., Vn(t). Wecan_saythat _P(t)is a

functionofthe set,

V{t)=

{Vi{t)\l<i<n)

Forthecircleformationalgorithm letusdenotethis functionas C.

TO=

C(Wn)

The function C foreach agent _{Ris calculating}the angle _{y it has}to turn from its

current _orientation, the distance D it has to move from its current position and the

angle adjustmentneeded at the end ofeach step. D and _{y depends} on _{Vj(t), particularly}

theinter-agent distances and theinter-agent angles ofthe partners selected _by R;. Let us

see onwhat mathematicalidea Dand _y are computed.

3.4 Mathematical Model

The mathematical idea behind our circle formation algorithm is simple and

straightforward. Formation of a circle with n agents can be considered as _forming an n

sided regular polygon (n-gon). In other words it can be proved that the vertices of a

regularn-gon lieonthe perimeter of acircle. Ifthenumber ofagents, n, is largethen the

polygon would appearto_{be distributed uniformly}onthecircumference ofacircle. Inour

algorithm we are _trying to make a regular polygon which _necessarily is on the

circumference of a circle.

The main idea behind the algorithm is to maintain a certain angle between the

neighboringagents. Ifthe agents are atthevertices ofaregular polygon this angle is the

same for all agents. For a regular polygon this angle can _{be easily found using basic}

Figure 3-2 Aregularhexagon (n=6)is divided into 4(n-2)triangles

Figure 3-2 shows ahexagon divided into fourtriangles. We can generalize itto a

polygon with n sidesto state that the polygon canbe divided into

(n-2)

sum ofthe interior angles of atriangle is K radians. Ifa polygon with n sides can be

divided into

(n-2)

given_by (n 2)7t radians.Aswe discussedearlier, since we areinterested in_formingan

n sided regularpolygon, the interior angles would all be the same forsuch a polygon. If

wedenotetheinteriorangle of ann sidedregular polygon_by 6, itcanbe givenby,

(n-2)n

e=- _(3.8)

An algorithm that _{progressively} makes all the interior vertex angles of the

formation to be 6 and the _neighbouring sides _equal, in a _colony of n _randomly

distributed agents willleadto a circle formation. Inother wordswecan _saythatifall the

agents _try to make the internal angles between their _neighbouring agents to be 6. then

gradually the formation will become a circle. To achieve this _R should first find two

partners and move to a position where the angle will be 6 between the partners and

equidistantfromthem. Inthenext section we shall seehow _Rjselectsitspartners.

3.5 Partner Selection

An agent has to _identify two appropriate partners and move to anew position at

every iteration. The new position has to be equidistant from its partners _making the

interiorangletobe 6.Toselectpartners anagenthastoscanthe terrainwithinitsspanof

scan. An agent can scan an angle of27t radians within its span of scan. Our algorithm

makes it unnecessary to scan this whole range. Ifeach agent scans to its left and right

from its current direction oforientation and collects information about its partners it is

sufficient for the _functioning of our algorithm. Here we will explain how the partner

agents are selected.

Spanofscan

Figure 3-3 shows a possible distribution of the positions of a few agents. Here

agent _R is the agent of concern. The line with arrow is the direction ofR's orientation.

Agents_{Rl, Rr, Ri} and_R4,are all withinRj's span of scan and _R2 and_R3 are not._R scans

to its left andright as indicated in figure 3-3. The first agentit detects while _scanningto

its leftsideis itspartnerto theleftandthefirstagentsit detectswhile_scanningtoits right

is its partner to the right. Let us name these partners as _Rl and _RR respectively. Even

though agent R. is closerto _R,it is not selected as a partner. The partner selected to the

right of _Rj is _Rr since _RR makes the smallest angle from R;'s direction of orientation.

Similarly RL is selected as Rj's partner to the left. Agents _R2 and _R3make the smallest

angle withthedirectionoforientation of_Ri,butsince_they are outofthe span of_scan,R

will notbeabletodetectthem.

From the above explanation we could define the partners of_Rj asthe agents that

are within the span of scanof_Rj andthose that make the smallestangles to Rj's leftand

right with respect to Rj's current direction of orientation. Our algorithm functions with

theinformationgatheredfromthesepartneragents.

An agent is unable to select the same two agents as its partners for the entire

formation course. There are three reasons _why an agent cannot _carry on with the same

partners. The firstreason is that since the agents are allhomogeneous thereis no _wayto

physically identify a previous partner. _Recalling a partner is further made impossible

since the agents are _memory less. The second reason is that the agents are

uncommunicative and as a result there is no _way an agent could send a signal to its

partner _{regarding its} current location. The main reason is that the _{colony is changing} at

each iteration. A partner at a particular _{iteration may} not be so in the next. The partner

may haveselectedtwoother agents asitspartners and moved_awayto anotherlocation.

R₂ R3N

R

|i

/

Rs

3

R4

o

RlN

Figure 3-4 Partners changingat each iteration

Figure 3-4 shows apossible distribution offive _{agents, Ri, R2,...,} R5. Letus see

howthe partner selection will _{vary for} agents _Ri and R3. The partner selection depends

solely in thedirectionoforientation ofthe agents. The arrowed linesshow the directions

oforientationfor_Ri andR3. _According to thedefinition ofpartners_R3 selects _R2and_Ri,

since_they makethe smallestanglestoR3's left andrightrespectively. Ri does not select

R3butitselects_R4andR5. The newpositions of_R3 and_Ri are _R3Nand_Rinrespectively.

In the next iteration partners selected _{by Ri} and _R3 will depend on their direction of

orientation at that time. We will discuss in section 3.12 why and how the directions of

orientation are alteredaftertheend ofmotionat each iteration. It is intuitivethatafterthe

agents have reached the perimeter of the circle _they should select the same partners to

maintainthe circle. Ina closedcircular_path,where allthe agents are onthe _periphery of

formation will collapse. In section 3.12 we will see _{how, by altering} the direction of

orientation, we can make sure _up to a certain _extent, that the agents select the same partners_theyhadselectedinthepreviousiteration.

3.5.1 _Whysmall angle partner selection

The agents in the _colony as stated are simple and _{memory less. By selecting} the

firstagents detectedwhen _Rj scans to its leftand rightmakes _scanningthe whole terrain

withinits span of scan unnecessary.Suppose we _say 'selectthe twoclosest agents asRj's

partners', then_Rhasto scanthe full 2n radianswithin its span ofscan. Moreover it has to store the distances and the angles ofeach agent it detected and sortto select the two

shortest distances and the respective angles. This procedure of scanning, storing and

sortingthedata ismade_{unnecessary if}thefirstagentsdetectedon eitherside are selected aspartners.This increases the_efficiencyandreducesthe _simplicityof agents.

The main reason for not _selecting the closest two agents is because the _colony thenwould convergeto a single pointinsteadof_formingacircle. Letus_trytounderstand

this phenomenon of_converging to a point with the _help offive agents _trying to make a

regularpentagon.

,A Ri

R2 *C

R3

Figure3-5Nearestagents selected aspartners

Figure 3-5 shows apossible distribution offive agents _Ri, R2,...R5. Ri will select

R2and _R3as itspartners since _they are the closest agents. It wouldthen move toposition

C, equidistantfrom _R2 and _{R3 making} angel 0 _SimilarlyR2'spartners will be R. and_R3

andit would moveto B. _R3 will moveto _{A selecting}Ri and_R2 as partners. R4will move

to C after_selectingR2 and_R3 as partners. _R5will select_Ri and_R2and moveto A. In the

above explanationletus assumethepartners selected arethe closesttwoagents. Afterthe

first iteration the five agents wouldhave converged to three points. Herewe assume that

the _step size is large enough for the agents to reach their final destination without

stoppingafter _moving aunitstep. Forthe second iteration agents at A will select agents

at B and C and move to a point between B and C. _Similarly C will move to a point

between A andB andB between A and C. As we could see the inter-agent distances are

progressively getting smaller and_ultimatelywillconvergeto a single point.

We simulated the sameinitial distributions offive agents and observed theirpath

forthe twodifferentcases. Inthe firstcase thenearesttwoagents areselected as partners

andinthe second case partners are selected _accordingto our definition given in 3.5. The

agents in the simulation move a unit_step after_selecting the partners without _moving to

thefinalposition. Figure 3-6 shows thepaths offive agentswhere thenearest agents are

selected as partners and figure 3-7 shows the paths of the same initial distribution but

here smallest angle neighbours are selected as partners. The circle mark denotes the

initial position ofthe agents and the asterisk denotes the final position. As we can see

fromthe _diagrams, when nearest agents are selected as partners the agents _gradually get

closer and closerand_finallyconvergetoa singlepoint.In thesecond case wheresmallest

Figure 3-6Pathsoffiveagentsifthe nearest agents are selected as partners

Figure 3-7Pathsoftheagents ifthe smallest angle agents are selected as partners

After the appropriate partners are selected _R has to measure the inter-agent

distances and the inter-agent angle between the partners. In the next section we will

define and explainthesetwoparameters.

3.6 Inter-Agentangle andInter-AgentDistance

Figure 3-8 showsthe positions of_Rj and its partners. The thicklines with arrows

show two possible orientations of R,. Let O be the direction of orientation ofR. After

scanningtoits left andright_startingfrom its currentdirection oforientation_R detects its

left partner _RL and right partner RR. The distances _RlR and _RRR are the inter-agent

distances _R will measure. These distances aredenoted_{by dL} and _dR respectively. Hence

dLand_dRarethe distances between R'scurrentposition andto theleft and right partners

respectively.

The angles RlRjO and RrRjOare the angles between Rj's directionof orientation

and to the left and right partners respectively. The angle to the left partneris denoted as

at andto therightpartner_asaR.

aL=^RlR;0 _and

aR =jCRrR,0

Letus _denote,

a=

aL+aR _(3.9)

ais theinter-agentangle made_bythe two_partnering agentswith theposition of

Ri which includes the direction of orientation of _Rj It could be less than n radians or

greater _depending on the direction of orientation of Rj. The orientation can be in any

direction,butit is ofimportance toknow ifthedirectionoforientation is intheconvexor

(b)

(a)

Figure 3-8MeasurementofInter-Agentangle anddistance

(a)when_a^+_<r<n and _(b)when_ol+_<*r>n

Figure _3-8(a) and figure _3-8(b) show three agents at the same locations but

differing inter-agent angle a for R. a is different due to its _differing directions of

orientation of R,. In figure 3-8(a) the direction of orientation is in the convex _region,

thereforea<7t. In figure 3-8(b) direction of orientation is in the concave region and

a>n.

Thevalue of a, i.e. whether a > n or a < n is ofimportance forR to move to

therightposition. In thesubsequent section we shall explain the importanceofthe value

of a. We shall usethevalueof a todefine internaland externalagents.

3.7 InternalandExternal Agents

The goal of_Rjat eachiteration istomaketheinter-agent angle a equal to 6 and

dL=dR=d. d is not auser defined parameter but depends on the inter-agent distances

and the inter-agent angle between the partner agents. After _Rj detects its partners, there

are two positions _Rj could move _to, so that the inter-agent angle is 6 and is equidistant

from itspartners.

Figure3-9Twopossible positions_RNi and_RN2with a=₀and equidistantfrom

partners

In figure 3-9 the old position of_Rj is denoted _{by R<> Ri}detects_Rl and _Rr as its

partners and measures_{du d^ aL}and _aR. The two possible positions R could move are

shown in the figure. Let us name these positions as _Rni andRn2- These positions are in

theperpendicularbisectorof_RiRrand atboththese positions,

RLRN1

RRRN1 =

RLRN2

RRRN2 =d _3nd

^RLRN1RR

^RLRN2RR

Onepossible position is on the same side of_Ro, with respect to the line_joining

thisline. Tochoose which position _{R, should select we make}use oftheinter-agent angle.

Tounderstandthis _{decision makingwe need} tointroduce two newterms:Internal Agent

andExternal Agent.

i i i i i

\

\ \

Figure 3-10 InternalandExternalagents

In a _{randomly distributed colony} of agents we could define Internal agent and

External agent as follows. Imagine of _connecting the agents with lines and _making

polygons so that all the agents in the _colony are inside the polygon. We could form

various polygons with different number of sides. Such a polygon with the smallest

number of sides is of our interest. The agents at the vertices of the polygon with the

smallest number of sides and that encompasses the rest ofthe _colonycan be defined as

the External agents. The agents that are contained _by this polygon are the Internal

agents. In figure 3-10 the agents that are shaded inside are the external agents and the

ones thatare plain aretheinternalagents.

Theobjective ofthe_colonyofagents would be forall agentsto become external.

In other wordswe could _saythat theinternal agents havetomovetowards the_periphery

of the _encompassing polygon while the external agents position themselves so that the

polygonbecomesregular. Letus look intoadistributionand_trytounderstandhowagents

would cometoknow if_they areinternal agents or external agents.

Figure3-11 A distributiontoexplaintheselection of correct position

Figure 3-11 shows an agent distribution of six agents. Ifthese six agents are to

make a regular hexagon letus examine how agents _Ri and _R5 should move so that the

formation leads toward a hexagon. Let us assume that the directions oforientationsfor

these agents are as shown_by the arrows. Agent _Ri will detect_R2and _R6 as its partners.

For the formation to get closer to a hexagon _Ri has to be on the same side of its old

positionwithrespecttoline R2R^- hithecase of_R5 itwill select_R4and_R^asitspartners.

Again forthe formation tobecomearegularhexagon R5has to cross overline _R4R6 and

selec