Formation Control of Robotic Swarm

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Hindawi Publishing CorporationTe Scienti c World JournalVolume , Art icle ID , pageshttp://dx.doi.org/ . / /

Research ArticleFormation Control of Robotic Swarm Using Bounded Artificial Forces

Long Qin, Yabing Zha, Quanjun Yin, and Yong Peng

College of Information System and Management, National University of Defense echnology, Hunan, Changsha , China

Correspondence should be addressed to Yabing Zha; zhayabing @ .com

Received September ; Accepted October

Academic Editors: S.-W. Lin and K.-C. Ying

Copyright Long Qin et al. Tis is an openaccess article distributed underthe CreativeCommons AttributionLicense, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Formation control o multirobot systems has drawn signi cant attention in the recent years. Tis paper presents a potential eldcontrol algorithm, navigating a swarm o robots into a prede ned D shape while avoiding intermember collisions. Te algorithmapplies in both stationary and moving targets ormation. We de ne the bounded arti cial orces in the orm o exponentialunctions, so that the behavior o the swarm drove by the orces can be adjusted via selecting proper control parameters. Tetheoretical analysis o the swarm behavior proves the stability and convergence properties o the algorithm. We urther makecertain modi cations upon the orces to improve the robustness o the swarm behavior in the presence o realistic implementationconsiderations. Te considerations include obstacle avoidance, local minima, and de ormation o the shape. Finally, detailedsimulation results validate the efficiency o the proposed algorithm, and the direction o possible utrue work is discussed in the

conclusions.

1. Introduction

In the past decades, multirobot systems have attracted lotso attention among researchers. With advances in commu-nication, networking, and computing, robotic swarms arebecoming more affordable in many given tasks that are toocomplex to be achieved by a singlerobotworking alone.So arthe possible applications range rom coordinated control o

UAVs [ ] to spacecraf [ ] andautomatic vehicles [ , ],and so orth.

Te ability to achieve, maintain, and change ormationis one o the undamental prerequisites or building aneffective multirobot system. Various approaches o ormationcontrol are investigated, which can be divided into threecategories: the behavior-basedapproach, the virtual structureapproach, and the potential eld approach. In the behavior-based approach, a set o behaviors are assigned to each singlerobot and the per ormance o the whole group is determinedby comparing the relative importance between the behaviors[ ]. Te virtual structure approach considers the entireormation as a rigid entity. Tereby the desired motion is

assigned to the rigid structure and the constraint unctionswhich relate the positions and orientations o the memberrobots can be de ned [ , ]. Te potential eld approachpresents arti cial orces between neighboring robots, sta-bilizing the system to the equilibrium mani old [ , ].In addition to the above classi cation standard, existingapproaches also vary rom leaderless to leader- ollower andcentralized to decentralized. In the leader- ollower control,a real or a virtual leader is de ned and its motion ollowsa desired trajectory. Te ollower robots take the leader asa re erence and adjust their motion to maintain the overallormation [ , ]. Te goal o the decentralized approachis to achieve a ormation while using only local in ormationconcerning positions and velocities, which is different romcentralized approach relying on global in ormation [ , ].

Due to the physical comparability, the potential eldapproach which ollows the law o gravitation is easier tounderstand. By correctly shaping potential elds, a desiredbehavior to a robotic swarm could be imposed [ ]. More-over, the potential eld approach is generally more adaptableor building decentralized ormation control algorithm [ ].


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Te Scienti c World Journal

Because o the intrinsic reliability o decentralized methods[ ], the potential eld approach is widely used in the eld o swarm behavior modeling. Tus in this paperwe ocuson theormation control based on arti cial potential orces.

Some researches ocused on swarm behavior model-ing [ ]. Ekanayake and Pathirana [ , ] proposed

a scalable decentralized control algorithm to navigate arobotic swarm into a stationary shape. Teir algorithmpossesses remarkable robustness under external disturbancesand works well in real world scenarios such as localizationerrors, communication range limitations, and boundednesso orces. Hengster-Movric et al. [ ] studied multiagent or-mation control based on bell-shaped potential unction. Tespecial eature o their potential unction is itsdependence ona control parameter that widens or, conversely, concentratesthe effective range o interactions. Tis dependence is usedto eliminate some potentially unwanted equilibrium states(e.g., local minima) through bi urcations [ , ]. Tey alsoproved that the proposed controller applies in moving targetsormation as well.

With a strong practicability, the above algorithms haveovercame several issueswhich must be concerned in practicalapplications. However, when we attempted to apply them,we still ound some unresolved problems. First, the originalorces in the work o Ekanayake et al are de ned in the ormo upper unbounded unctions.As a result, i additional ruleso actuator limitations are absent, the orces will numerically over ow in certain cases. For example, when the distancebetween two robots is shortened signi cantly or one robotreaches the edge o an obstacle, the repulsion orce will turnextremely large. Second, local minima o the compositivepotential eld still appear nearby the boundary o largeobstacles. Such local minima are not taken into accountby their algorithms, so the robots moving in the obstacleenvironment usually linger at unwanted equilibriums. Tird,the ormation shape itsel , which is different rom the robots,is considered as insensitive to the arti cial orces. Tereby theshape is prede ned and incapable o de orming continuously to suit the potential change.

o resolve these problems, we rst de ne the arti cialorces in the orm o exponential unctionswhich possess two

types o control parameters: magnitude factors to determinethe orce magnitude and response factors to tune the reactingsensitivity. By care ully selecting the control parameterso these unctions, the distribution o the orces can beeffectively adjusted, bounding them within rational range.We urtherpresent modi cationsupon the control algorithmto suit realistic implementation considerations. Among thesemodi cations we specially de ne the repulsion orce roman obstacle which is ormed by two components: one avoidsthe robot rom collision, while the other pulls the robotto escape rom the local minima nearby large obstacles.In order to make a moving shape sensitive to potentialchange, we build a controller upon the shape contour whichprovides elastic orces to compress or stretch the shape,making it autode ormable to avoid collisions. Te detailedsimulation results show that our control architecture cansuccess ully resolve a orementioned problems and improvethe robustness o swam behavior. However, as discussed in

the conclusions, the undamental reason which produces thelocal minima remained unresolved, so we highlight it as ouruture work and state potential solution in the end.

Te outline o the paper is as ollows. Section presentsthe swarm model and analyzes the behavior o the swarmwith certain assumptions. Section discusses implementa-

tion considerations which are needed in real world scenarios.Section devises a controller acting on the shape contourso as to make the shape autode ormable, protecting it romcollision with obstacles. With the simulation case studiespresented in Section , we veri y the assertions proposed inSection and demonstrate the behavior o the swarm withconstraints and modi cations discussed in Sections and .Finally in Section we give some concluding remarks andstate directions to possible uture work in this area.

2. Swarm Model and Its Behavior Analysis

In Section . we introduce the dynamics o a swarm. In

order to reduce the system complexity and highlight thebasic control algorithm in the analysis phase, certain sim-pli cations are to be considered. With the simpli cationswe de ne a series o basic arti cial orces in Section . tonavigate the robots. Te behavior o the swarm drove by these orces is analyzed in Sections . and . . Te stability and convergence properties o the control algorithm are alsodemonstrated in the analysis.

. . Te Swarm Model. We assume that ( ) the robots arepoint masses which have no physical dimensions and ollow point mass dynamics; ( ) all the individuals in the sameswarm should be identical in physicalproperties such as mass

and mobility ( ) all the robots are capable o instant and errorree localization; and ( ) within the communication network o the robots, there is no delay in any in o transmission.

Te state vector o a swarm o robots is shown as = 10 + V 01, = 1,..., , ( )

where denotes the Kronecker product. Te entries o , V R 2 represent the position-like and velocity-like state variables. Tereby, the state vector o the whole swarm can bedenoted as = [1,...,]and it is determined by

= + , ( )where

= , = , ( )

with

= 0 10 , = 01. ( )In ( ), is the riction coefficient which can ensure theswarm a complete stop when the orces are balanced. is the

mass o a robot.


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. . Force De nition. Te control input in ( ) consists o = 1,...,, ( )

where arbitrary entry in the vector is described as ollows:

= , + , + , = R2

. ( )In our work all the components o are built in orm o exponential unctions which can be generally represented asollows:

, = , R +, ( )where , R + are constants. Our proposed orces de nedin such a orm o unction can be restricted in a rationalrange and the behavior o the swarm can be easily adjusted via selecting the parameters.

Te basic arti cial orces are de ned as ollows.

, is an attraction orce on the th robot rom the shapewhich is denoted as , = 1 , 1

( )

, is the repulsion orce on the th robot rom the shape denoted by , = ,

. ( )In ( ) and ( ), R 2 denotes a point on the shape contour.We de ne that ( , ) = 1 when is inside the shape, and( , ) = 0 when it is outside. Te mutual exclusion ensuresthat , will vanish only i the robot is inside , and , will vanish only i the robot is outside .

, in ( ) re ers to the resultant repulsion orce actingon the th robot rom the remaining swarm robots. Tat is,

, = =1, =

. ( )

From ( ) to ( ), , , and adjust reacting sensitivity,, , and determine the magnitude o the orce, ( )/ , ( )/ , ( )/ R 2are unit vectors determining the orce directions.

Generally speaking, , and , are the orces navigatingthe robots towards the desired shape and evenly spreadingthem insidetheshape. , avoids the intermember collisions.

. . Analysis of the Swarm Behavior. As , and , aremutually exclusive and do not work simultaneously, theswarm behavior and the stability o the controller can beanalyzed in each instance separately. De nitions de ned in( ) will be employed by the analysis.

Consider

= =1 , = , = ( ), , and represent the center o mass o the swarm, the

center o mass o the shape contour, and length o the shapecontour.

. . . Motion of the Swarm and Its Members outside theFormation Shape. When all the robots are operating outsidethe target shape, the swarm can be viewed as one objectaffected by resultant orce described as ollows:

= + + . ( ) is the resultant intermember orce and equals to zero; that

is,

=

=1 =1, = = 0. ( ) is the attraction orce rom the target shape

= =1 1 . ( )

With de nitions in ( ), we can in er rom ( ) that:

< =1

1 1 = =1 = .

( )

From ( ), we know thatthe riction orce acting on the throbot is , = ( / )V , so the total riction orce acting onthe swarm can be calculated as ollows:

= . ( )We de ne

= ()with

= and

= . According

to ( ), the dynamics o the swarm possess a property asollows:

< + + + < 0. ( )

With ( ) we can conclude Teorem as ollows.

Teorem . Consider the swarm model whose dynamics of motion has property described as ( ) , the motion of the center of mass of the swarm ( ) is toward the center of mass of theshape contour ( ).


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Proof. aken Lyapunov unction = (1/2)( +), its differential coefficient on time . Tat is, canbe deducted as ollows: = + < + < 2.

( )

So < 0, or any =0, the only invariant point is ( = = 0), by using Lyapunovs method extended by Lasalle andLe schetz [ ], the Teorem is proved to be true.

Intuitively, Teorem argues that when all the robots areoutside the shape, the center o mass o the swarm will movetowards the center o mass o the target shape.

Te characteristic polynomial o system ( ) is

2 + + = 0, ( )and it is clear that, in order to archive damped motion whichensures the swarm is a smooth movement, the conditions >0, > 0 and 2 must be satis ed.

Teorem only states the dynamics o the swarm as oneobject, while the behavior o every single robot remainsuncertain. Next the behavior o an individual robot will be

investigated.For the th robot, itsdynamicso motioncanbe describedas ollows:

= , + , + , . ( )o the , de ned in ( ), where (1 ( , )) = 1, we have

, < 1 1

= , < .( )For , we de ne , = , with ( ) and the

riction orce de nition, the dynamics o motion o th robotis described as <

+ =1, =

. ( )

With de nition V = ( ), ( ) can be rewritten as + < +

=1, =

V + V + V , < 0.

( )

By de nition o = min , , and = }, we have, <

=1, =

= ( 1) ,( )

where

= =1, = 1 . ( )Given the de nition o and ,

= =1, = , = =1, =

1 = 1 1 < 1,( )

So rom ( ) we state that

, < ( 1) . ( )In order to orm the member robots into a coherent swarm,the arti cial orces in right hand side o ( ) must drive eachrobot toward the center o mass o the swarm, and this is justwhat Teorem concludes as ollows.

Teorem . Consider a robot staying outside the shape, if / > ( 1)/ is satis ed, then its motion is toward thecenter of the swarm .Proof. Choosing a Lyapunov unction as

= 12 V V + 12V V ( 1) > 0, ( )through taking derivatives, is bounded by

V 2 + , ( 1) V V , ( )


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and itis clear thatwith ( ), the second item o the right handside o ( ) is negative, so we have

V 2, ( )which proves the theorem.

Here with Teorems and we argue that with thearti cial orces de ned in Section . , the swarm stayingoutside the contour can move toward the target shape and atthe same time cohere the robots within certain interdistance.

. . . Motion of the Swarm and Its Members inside theFormation Shape. In this section we proo that when a robotis inside a shape contour which is symmetrical over two ormore straight lines and i it is controlled by , de ned in( ), then the intersection point o those lines will be theequilibrium point where the robot will stay still afer a certaintime span.

Te lemma we used is given without proo . Readers whoare interested in this can nd more detail in [ ]. Lemma use odd and even unction properties to derive a property o a complex unction which is symmetrical over real axis o thecomplex plane, as described below.

Lemma . Fora closed contour ( )and functions 1( ) R + ,2( ) for [0,2 ] , with the following properties:

() = (2 ), 1() = 1(2 ),2() = () , R . ( )

We de ne I

( ) the imaginary component o a complex

number; then the ollowing statement holds:

I2

01()2() = 0. ( )

As shown in ( ), , =(( )/ ) consists o twomajor components, ( )/ de nes the directiono the orce, and determines the magnitude o theorce. Te expression or , is the contour integral o

=1

, where , ( )or

, when , 1 = +1 . ( )We de ne ( , ) = (1/ ) . Using polarcoordinate representation or complex plane, = ( ) and

, can be rewritten as

, = 20 (), () () . ( )

I the contour ( ) with [0,2 ) is symmetrical overthe real axis and the th robot is staying along on the real axis(i.e., ( ) = (2 ) and I ( ) = 0); then we have

1() = (), ()= (2 ), (2 ),

2() = ().( )

Tus using Lemma , I ( , ) = 0 is proved, which is used toprove Teorem as described below.Teorem . Given a robot inside a shape contour ( )with the following properties:

( ) ( ) = (2 ) ,( ) I

( ) = 0.

Ten I ( , ) = 0.Te above assertion can also be extended to determine

the behavior o the whole swarm when all the robots areinside the shape, through viewing the swarm as one object,the center o mass o the swarm will nally move to theequilibrium point, and the whole swarm will evenly bedistributed inside the contour, as described in Teorem .

Teorem . Given a shape contour ( ) with two or moresymmetric axes. Ten for a robot i or the center of mass of aswarm located at the intersection of those axes, , = 0.Proof. Consider ( ) has symmetric axes (1,...,)orming [0,2 ) angles to the real axis, rom Teorem westatethat whena memberor the centero masso a swarmis locatedon theintersection o those axes, therepulsion orce

, is in the orm o

, = 1 1 = = , ( )where 1,..., R , because 1 =2,..., = ; this implies

1 = .. . = = 0, which means that , = 0.. . Discussion on ransitional State. In this section we ocuson the transitional state in which a part o the swarm isinside the contour, while the rest o them stay outside. FromTeorem , when increase,the condition / > ( 1)/ will be satis ed and the attraction orce , acting onmembers who stay outside the contour will be dominating.Tus, the remaining part can still move toward the targetshape.However, when a robot is staying closerbut outside thecontour and thecontour is not large enough to accommodateall the robots; then , turns larger than , , which will leadthe robot in an unwantedequilibrium state.Tisproblem canbe eliminated by care ully tuning and to suit the swarmsize or, as described in the next section, to modi y , to getuni orm distribute o the orces near by the contour.


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3. Implementation Considerations

When implementing the proposed control algorithm in areal world scenario, the simpli cations assumed in Section will no longer hold. So in Section . we discuss certainmodi cations upon the proposed basic orces, which ensurethe robustness o the swarm behavior in the presence o realistic implementation considerations. In Section . weinvestigate the rules o selecting proper control parametersso as to suit certain applications. Moreover, we show inSection . that by certain coordinate trans ormations thecontroller can be applied to moving targets ormation as well.

. . Controller Modi cation for Practical Implementation

. . . Physical Dimensions of Robots. Here the robots are nolonger considered as point masses. For general case, a robotwith physical dimensions possesses length ( ) and width( ), where > (see Figure ).Te operational diameter o the robot is then de ned as = , where > 1. is the sa e actor or collisionavoidance. By selecting the distribution o the repulsionorce can be adjusted to suit the maximum speed o the

motion. Tereby the modi ed intermember repulsion orcecan be stated as ollows:

, = =1, =

( ). ( )

Te comparison between , and , is shown inFigure (a) . Additionally we give , and , de ned in [ ]as counterpart in Figure (b) . It is shown that the magnitudeo

, and , de ned in our work is bounded, whichcan avoid numerical over ow by selecting proper controlparameters.

. . . Obstacle Avoidance. Obstacle avoidance is one o themost important aspects in practical implementation o thealgorithm [ ]. In general the repulsion orces romobstacles only effect within a nite distance; that is, thesensing radius o a member and its magnitude should belimited too. So repulsion orce acting on th member romobstacle is presented as ollows:

, = , () , ( ), =

, when 0,,0, when > ,

( )

where denotes the weight determining the magnitude o the orce. is the obstacles simply-closed contourand R 2is an arbitrary point on that contour.

However, only using ( ) to determinetherepulsion orcemay leadto localminimanearby the boundaryo anobstacle,

d ij

w m

l m

d

F : Car-like robots with turning diameter.

because , and , may get balanced. Figure (a) showsthat three robots lingered near by the obstacle boundary. Tereason is that, as Figure (b) shows, when the robots move totheedge o theobstacle, , (represented by red line) togetherwith , (represented by blue dot line) drive the robot intoan unwanted impasse, where , was counteracted by

, , so

there will be no actuator to navigate the robot heading or thedestination.

o resolve this problem we decompose , into twocomponents, pointing at directions ( )/ and 0 1

1 0 (( )/ ), respectively, so , is rewritten asollows (where 22 denotes identity matrix):

, =12 22 + 0 11 0

,when 0,,

0, when > .

( )

Te part multiplied by 0 11 0 (( )/ ) in ( )generates a orce pulling robot to move along the boundary and escape them rom the impasse. Figures (c) and (d) arecounterparts o thesame scene but using ( ) to calculate , .Tese two guresshow that thewhole swarm can success ully escape the local minima and get into the shape contour.

. . . Actuator Limitations. A robot has its limitation onacceleration and velocity, so the orce de nitions must bemodi ed to eliminate unrealistic accelerations. Considering

the mass o the th member is , and its upper threshold o acceleration is ; then the controller is shown as = , ( )where

= {, i < , , i . ( )We use velocity limiting unction introduced in [ ] as,

V = {V , i V < V max ,V

max , i V < V max , ( )


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0 5 10 15 20

0

5

10

Actual size of the agent

for agents viewed as point masses

10

5

F i,m

F i,m

Coordinate X (m)

C o o r

d i n a t e

Y ( m )

for agents with turning diameter d t = 4 m

d t = 4 m, the turning diameter of the agent

(a)

0 5 10 15 20

0

5

10

Actual size of the agent

5

10

for agents viewed as point massesF

i,m

F i,m

Coordinate X (m)

C o o r

d i n a t e

Y ( m )

for agents with turning diameter d t = 4 m

d t = 4 m, the turning diameter of the agent

(b)

F : Comparison o , and , .

the term V max represents maximum velocity that the thmember can achieve. V is the desired speed.. . . Chattering Effect. A drawback o the basic controlleris the discontinuous nature o the basic arti cial orces (i.e.,

, and , de ned in Section ). As shown in Figure (a) ,the orces have a signi cant variation at the boundary. I theshape area is relatively small, the repulsion orce rom otherrobots which are already inside the contour will prevent theothers rom passing through theedge. o tackle this problem,the orce de ned in ( ) is modi ed to add a term which cancompensate , when the robot is getting closer to the shapecontour. Te modi ed , is as ollows:

, = 1 ,

1 + .

( )

In ( ), determine the magnitude o the additionalorce. By tuning parameter , the operating range o thisterm can be limited in a short distance rom the contour.Figure showsthat compared with original , (Figure (a) ),modi ed , (Figure (b) ) haspre erable uni ormity in eitherside o theshape contour. For thesake o simplicity, thevalueso the parameters are set as = 1, = 1, = 1, = 1,and = 1.. . Adaptation of Control Parameters. Te behavior o theorces de ned in this paper ismainly determined by thevalue

o the control parameters which can be classi ed into two


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100

Obstacle contour

Shape contour

150

100

50

0

50

150

200

250

300400 300 200 100 0

Coordinate X (m)

C o o r

d i n a t e Y

( m )

(a)

0

200

Time (s)

F o r c e m a g n

i t u d e

( N )

shape contour

obstacle contour

Agent ID =obstacle contour at t = 8 .1 s

1 moves to the180

160

140

120

100

80

60

40

20

0 10 20 305 15 25 35

F r repulsion force from

Fa attraction force from

(b)

Obstacle contour

Shape contour200

100

150

100

50

0

50

150

200

250

300400 300 200 100 0 100

Coordinate X (m)

C o o r

d i n a t e

Y ( m )

(c)

0

50

100

150

200

250

Time (s)

F o r c e m a g n

i t u

d e

( N )

into the shape

from the obstacle

to the obstacle

contour at t = 32 s

Agent ID = 1 move

= 1

Agent ID = 1 move

boundary at t = 9 .3 s

Agent ID =

boundary at t = 22 .1 s

from obstacle contour

from shape contour

the repulsion force

the attraction force

0 10 20 30 405 15 25 35 45

Fa

Fo

escape

(d)

F : Comparison on original and modi ed , on impasse resolution.

0 105 15 200

0.5

1

1.5

Distance from center (m)

F o r c e m a g n

i t u d e

( N )

F i,r

F i,a

(a) Original , and ,

F i,rF i,a

0 105 15 200

0.5

1

1.5

Distance from center (m)

F o r c e m a g n

i t u d e

( N )

(b) Modi ed , and ,

F : Distribution o original and modi ed

, and

, nearby the shape contour.


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types: magnitude factors ( , , , , and ) and response factors( , , , ,and ). Te basic rules o determining values o these parametersare summarized in this subsection.

. . . Magnitude Factors. For the determination on magni-tude actors

,

,

, and

, an uni orm representation is

de ned as ollows:

= upper = upper , ( )

where upper R + is thetotal orcethata shape orobstaclecangenerate. From the de nitions o , , , , and it is obviousthat , where the symbol can be replaced by , , and.

For we state that = per, ( )where per R + is the unit repulsion orce and is the counto member in the swarm.

With ( )and( ), the proposed orces can be limited by certain upper bound.

. . . Response Factors. Te de nitions o the arti cial orcesshow that the response actors can adjust the orcesreacting sensitivity. o set the value we have

= 1, ( )where the symbol can be replaced by , , , , and R +to represent the constant o the robots sensing radius.. . Moving argets Formation. When a target shape is exe-cuting maneuver, there is a rame o re erence tightly connecting with it. Te direction o the swarms velocity coincides with the -axis and points on the shape contourare there ore xed in such a rame. At time , the position-like state vector o the th member under the stationary (global) rame o re erence is ( ) = [ , ], by adoptingdisplacement and rotation trans ormation as shown in ( ), ( ) under can be calculated by

() = cos + sin + sin + cos + , ( )where denotes the angle o rotation o on and[ , ]is the displacement vector pointing rom origin to .Consider that the mode o a D vector is invariant undercoordinate trans ormations, , and , under rame can be calculated via replacing , in ( ) and ( ) with ( ), .Proof. Since

() = cos + sin sin + cos () = () .( )

So the scalar quantity o , and , is invariant undercoordinates trans ormation de ned by ( ).Here the orces generated rom a moving ormation was

denoted as , and , are rewritten as, =

cos

00 sin () () 1 ( )

, ( )

, = cos 00 sin () () ( )

.( )

By replacing original orce de nitions presented in ( )and( )with ( ) and ( ) respectively, we can analogously analyzethe swarm behavior and prove that the relative assertions

proposed in Section also hold or , and , .4. Analysis of the Controller Acting onthe Shape Contour

Most o the ormation control strategies that are based onarti cial potential eld only control the overall geometry,while recently some novel strategies have been put orwardto control the exact shape o the ormation [ ].

When a moving target shape is about to pass through anarrow channel or collide with an obstacle, it is expected thattheshape canbe autode ormableto suit potential eld changeandthere ore prevent therobotswithinit rom collisions withobstacles. Moreover, in order to accommodate enough spaceto the whole swarm, it must be ensured that the area o theshape will not change during the de ormation phase.

o achieve this purpose, in this section we present acontroller on the shape contour which is sensitive to thepotential change. Te controller generates two orces actingon the shape contour: the pressure rom obstacles and thetension generated by the contour itsel . We analyze thebehavioro this controller via a case in which a robotic swarmis passing through a narrow channel, as shown in Figure .

We de ne as radius o the smallest enclosing circle o the shape. When the circle is compressed into an ellipse; thatis,

2

2 + 22 = 2, ( )where we have

= 00 , ( )the area o the resulting ellipse is = 2. It is clearthat i = 1, the area o the circle will be invariant underthe compression. Teorem states that the shape within thecircle also keeps its area invariant during the compression.


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R

aR

bR

FC

FC FT

FT

F : Te de ormingprocedure o theshape contour in the caseo passing through a narrow channel.

Teorem . Given a shape presented as a convex polygon with 3 vertex, if a compression satis es condition = 1on its smallest enclosing circle, then the area of is invariant.Proof. can be divided into - triangles, or each one o them (denotes as ) with vertexes V 1,V 2,V 3},

= 12V 1

V 2 1V 2 V 3 1V 3 V 3 1 = 12V 1

V 2 1V 2 V 3 1V 3 V 3 1 = , ( )where and are the area afer and be ore compression,respectively. I = 1, then = . So

= 2=1

= 2=1

= = , ( )which supports the assertion.

During the compression phase, the smallest enclosing

circle itsel will generate a tensile orce which tends topull the shape contour back to its original state, which isrepresented as = , ( )

where denotes the compression ratio, and R + therange.We de ne the width o the channel is , the diameter o the smallest enclosing circle is ; then the compression orcecan be calculated by ( ) as

=

, i

< ,0, i . ( )

Consider state vector

1

2 = 1 1 , ( )then the dynamics o motion o [1, 2]will be

1 2 = 0 1 12 + 01 , ( )

where

= 4 + 1 is the riction actor.

From ( ) and ( ), we can get the trans er unction o asecond-order system as ollows:

() = /2 + + = /2 + 4 + 1 + , ( )itsdamping ratio is = (1/2) 4 + (1/) > 1,soitisanoverdamping system with adjustingtime = 8/ 4 + 1,andthecompression matrix is

00 = 1 1 00 11 1 . ( )

So an arbitrary point on theshape contour willbe de ormedinto a new position as = 00 and the attraction andrepulsion orces generated by the shape can be recomputedand updated to drive the robots.

5. Simulation Results and DiscussionIn this section we rst test the proposed control algo-rithm and the swarm behavior by three basic cases: orma-tion convergence (in Section . ), ormation transition (inSection . ), and obstacle avoidance (in Section . ). Withinthese cases the basic swarm model proposed in Section andthemodi ed one investigated in Section are both taken intoaccount, which by contrast highlights the improvement o thelatter when it is working under certain restrictions in the realworld scenario.

In Section . we applythe control algorithm to navigateaswarm to trace and nally orm certain moving target shapes,which demonstrate the assertion studied in Section . .Additionally, we put a narrow channel between the swarmand the destination. When the swarm is passing through thechannel, theshape controller devised in Section can de ormthe shape to suit the channel and there ore take robots out o the channel.

. . Formation Convergence. Figure shows the motion o a swarm with robots when they are trying to convergeinto our types o static shape. For this case, we select theparameters as magnitude actors = 1, = 2, = 5, = 15, = 1, and = 10; maximal accelerationand velocity o the robot are m/s 2 and m/s; reacting rangecontrol parameters

= 0.1,

= 0.1,

= 0.1,

= 0.0and = 0.05.Figures (a) and (b) use the basic attraction orcede nition as described in ( ). It is clear that the robotsare lingering back and orth near by the contour. Tisphenomenon illustrates that with the basic de nition o ,and relatively small shape area, some robots usually ailed tostep into the shape. Tis is because , decreases then theresultant repulsion orcethat generated rom therobotsin thecontour repel them.

Figures (c) and (d) use the modi ed , de ned in( ). Te whole swarm can avoid chatter effect and convergeinto the prede ned ormation shape. Te reason is that themodi ed , can keep theuni ormity o the orcedistribution


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F : Formation convergence in different shapes. In the sub gures, red dots represent initial positions and black crosses represent nal

positions, while the thin lines denote the trajectories. Te initial positions are uni ormly distributed.

on both side o the boundary, so the attraction orce will holdon till the robots get through into the shape; thereafer, ,continuingly navigates them to stay within the shape.. . Formation ransition. In real world implementations,ormation transitions are very usual. Figure shows thesimulation results about the behavior o a swarm with =10

robots. Te swarm is subjected to a sudden ormation

transition. Te parameters used here are thesame as the onesused in Section . .

We test ormation transitions in both stationary andmoving target shape scenarios. As is seen in Figures (a) and(b) the ormation typetransitsin interval o seconds romtriangle via pentagon to square.

Figure (b) demonstrates that the control algorithm orstationary shape is also adaptable in moving ormation con-trol. By using theoretical assertions proposed in Section . ,


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(b)F : Obstacle avoidance in the complex terrain, using modi ed , .

the shape can ollow arbitrary trajectory and the orcesrom the shape are calculated by ( ) and ( ), which is

subjected to a relative re erence ramework moving with theshape.

. . Obstacle Avoidance. In this section, the velocity andactuator limitations are taken into account. As shown in

Figure (a) , there is a swarm o robots in a line ormation(represented as red dots) and the target shape (a triangleormation) is located ar away rom the line. Tere are several

obstacles between the start line and the target shape. Te bluedots denotes the paths o the robots and the black crosses aretheir nal positions. Te velocity andaccelerate are limited tom/s and m/s 2 respectively. is set as , = 0.01, and = m.


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Figure (a) shows that all the robots that success ully

escape avoid these obstacles and orm the desired ormationwithin s. By using the modi ed , de ned in ( ) togetherwith ( ), the robot getting closer to the obstacle boundary isdragged by a orce parallel the boundary, so it can pass by theobstacles and avoid local minima during the whole process.

Figure (b) shows the change o , (the red curve) and, (the green curve) acting on robot ID = when it gets

through the ormation contour boundary. We can gureout that with the modi ed , de ned in ( ), the orceuni ormity between , and , at time = . s is ensured.. . Shape Contour Controller. In this part the validity o thecontour controller is demonstrated via the case o passing

narrow channels. In order to make the swarm a damping sys-

tem that can make the ormation contour de orm smoothly,the value o the parameters should be = 15, = 4 + 1 = 7.8102, = (1/2) 4 + (1/) = 1.008,and = 8/ 4 + 1 = 1.0243.Figure (a) shows the motion o a swarm o robotsmoving in a triangle ormation. When there is no channel toget through, the controller keeps compression orce = 0and there ore the shape contour is invariant.

In Figures (b) and (c) , a swarm o robots ollowsa square and a triangle ormation respectively. When theswarm reaches the entrance o the channel, the ormation,contour begins to suffer pressure whose value is deter-minedby( ), which de orms thecontour into a compact one


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and navigates the whole swarm through the channel. Afergetting out o the channel, the tensile orce de ned in ( )stretches the contour back to its original state.

Figure (d) shows the change o the compression ratiosde ned in ( ). It is clear that the value o (representedas blue stars) always equal , which means that the area o the ormation shape is invariant. When the swarm is passingthe channel, the behavior o the compression matrix which isdescribed in ( ) will ollow the dynamic model de ned in( ).

6. Conclusions

Tis paper presented a potential eld-based approach orormation control o robotic swarms. We de ned arti -

cial orces in the orm o exponential unctions; there ore,the magnitude o the resultant orce is constrained withinreasonable range. Te behavior o the swarm along withsingle robot was both analyzed. Realistic implementation

considerationsand adaptation o the control parameters havebeen investigated to enhance the robustness o the controlalgorithm. We additionally built a controller acting on theshape contour, making it autode ormable to prevent themember robots rom collisions. Simulation results validatethe efficiency o the proposed algorithm.

In this paper the potential eld is generated by severalwave rontexpansion procedures starting at thecontour o theormation shape or edges o obstacles. By ollowing the ow o the negated gradient rom the target shape, a trajectory isobtained to navigate each member o the swarm. Using suchkind o resultant potential eld is the undamental reasonthat produces local minima near by the obstacle boundary [ ]. o reduce the risk o such an impasse and to enlargethe maneuvering space o the robot, we highlight our utureresearch on nding novel algorithms to generate potentialeld that is more exible and less likely to produce localminima. One o the most promising trends is to make use o topological properties o the underlying environments. Forinstance, we are now working on the construction o sparsebut adequate roadmaps such as Generalized Voronoi Dia-grams (GVDs) and Equidistance Maps (EM). Tese spatialrepresentation can be adopted to generate potential eldsavoiding most local minima and providing reduced searchspace or navigation tasks.

AcknowledgmentsTe authors appreciate ruit ul discussion with the Simgroup: Xiaocheng Liu, Shiguang Yue, Lin Sun, Qi Zhang, andLiang Zhu. Te authors also thank Pro essor Li Yao and Dr.Shan Mei or their proo reading andconstructivecomments.Te authors appreciate eedback rom our reviewers.

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