Systems & Control Letters 59 (2010) 654–661

Contents lists available at ScienceDirect

Systems & Control Letters

journal homepage: www.elsevier.com/locate/sysconle

Local control strategy for moving-target-enclosing under dynamically changingnetwork topology✩

Jing Guo, Gangfeng Yan, Zhiyun Lin ∗

College of Electrical Engineering, Zhejiang University, 38 Zheda Road, Hangzhou, 310027, PR China

a r t i c l e i n f o

Article history:Received 9 November 2009Received in revised form16 July 2010Accepted 23 July 2010Available online 6 September 2010

Keywords:Cooperative controlMulti-agent systemsTarget enclosingFormation

a b s t r a c t

This paper studies themoving-target-enclosing problem for a group of autonomousmobile robots, whichcan be seen as the requirement of achieving a formation surrounding a moving target whose movementis not known a priori. A local information control law is proposed to solve the problem utilizing onlythe relative position information from the target and its neighbors. The derivation of the controller isbased on the idea of decoupling the task of target tracking and the task of inter-robot coordination. It isshown that the group of autonomous mobile robots collaboratively estimates the moving velocity of thetarget and asymptotically reaches a regular polygon formation to keep the moving target as its centroid.The neighbor topology may dynamically change as the system evolves. Hence, a non-smooth version ofLaSalle’s invariance principle is used to show the convergence.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Recent years have witnessed rapid progress in coordinatedcontrol of networked autonomous mobile agents. The problemof attaining a formation and orbiting around a point has beeninvestigated a lot (e.g., [1–4]). In [1,2], it has been shown thata circular motion of a group of unicycles is a result of cyclicpursuit while the vehicles are uniformly distributed on a circlein the steady state with the center of the circle dependent onthe initial state. In [3,4], a so-called splay state configuration isintroduced and an elegant control solution is given to achieve acollective circular motion around a point under either all-to-allcommunications [3] or limited communications [4]. However, inthese works, it lacks the capability of specifying the orbit center,which makes enclosing a (stationary or moving) target infeasiblewithout modifications.

For the target enclosing problem which can be used to entrapor attack a target [5] by reducing the escapewindows or protect anobject by blocking the intrusion ways of adversary agents [6,7], itrequires all the agents to surround a specific target which is eitherstationary ormoving. Moreover, an equal spacing formation on thecircular orbit is usually expected in order to provide the best overallcoverage of the target and its surroundings. For the stationarytarget case, the target enclosing problems are investigated

✩ The work was supported in part, by National Natural Science Foundation ofChina (60875074) and Qianjiang Talents Program (2009R10027).∗ Corresponding author. Tel.: +86 571 87951637; fax: +86 571 87952152.

E-mail addresses: linz@zju.edu.cn, zhiyun.lin@gmail.com (Z. Lin).

0167-6911/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.sysconle.2010.07.010

in [8–10]. If the robot model does not have nonholonomicconstraints like unicycles and the velocity of themoving target canbe known by the robots, the solutions for static target enclosingcan be easily extended for moving-target enclosing. That is, itcould be addressed in the same way as the static target caseby eliminating the translation of the whole group and studyingthe relative motion defined in the frame on the moving target.This idea is explored in [11–13] for the moving target enclosingproblem. Under the assumption that a group of robots can beidentified, cyclic pursuit-based algorithms are developed to solvethe target-enclosing problem in [11,12], respectively. In [11] eachagent is assumed to be a pointmasswith full actuation capabilities,while in [12] the model of each agent is a more general MIMOplant. By ideally knowing the velocity of the moving object, it isshown in [13] that the group of robots can eventually enclosethe object with a circular structure and the order among robotsdoes not change in the circular structure under the proposedcontrol law. However, assuming that the velocity of the targetis known is a strong (i.e., restrictive) assumption since usually itis not possible for the pursuers to know the current velocity ofthe target. To handle the difficulties caused by not knowing thevelocity of the moving target, [14] assumes that the upper boundof the velocity of the moving target is known and then developsan algorithm for capturing/intercepting a moving target based onthe sliding mode control method. In addition, a problem relatedto the target enclosing problem is the target pursuit problemwhich is historically studied in the field of non-cooperative games.Example works on one target/evader and one pursuer gameare [15,16]. Along the line of research, the pursuit–evasionprobleminvolving a single evader and multiple pursuers is investigatedrecently [17–19].

J. Guo et al. / Systems & Control Letters 59 (2010) 654–661 655

This paper studies the formation control problem enclosing amoving target whose velocity is not known. It is expected thatas the target moves, the formation moves (or even more, rotatesaround the target) to keep the target as its centroid, maintains thedistance from the formation robots to the target, and distributesthe formation robots around the target spaced equally in theangle. The solution facilitates applications such as escorting orpatrollingmissions usingmultiple unmanned vehicles. The distinctfeature of the paper compared with the existing works is thatthe velocity of the target is not known by the robots. Hence,an adaptive scheme is designed in the paper so that the robotscan collaboratively reconstruct the target’s velocity with the errorbetween the estimate and the real velocity of the target convergingto zero. Then, the robots utilize the estimated velocity togetherwith the relative position information from the target and theirneighbors to coordinate theirmotion such that they asymptoticallyreach a regular polygon formation rotating around the target. Theadaptive control law thus assures that the robots are able to entrapthe target adaptively even though the target object may have anabrupt change on its velocity. Thederivation of the adaptive controllaw is based on the control Lyapunov function approach and theidea of decoupling the task of target tracking and the task of inter-robot coordination. The neighbor topology of the robots is definedaccording to their current positions in circular clockwise radialorder around the target. As the system evolves, the neighborsof each robot may change dynamically, which leads to a state-dependent switching system. Hence, to show the convergence ofthe system, a non-smooth version of LaSalle’s invariance principleis used.

Compared to the existing works [11–13] where the target’svelocity has to be available in order to solve the problem, ourapproach does not require to access the velocity informationof the moving target. Moreover, in contrast to the method in[11,12] where the interaction topology has to be fixed relying onthe labels of the robots, our approach does not require the robots tobe distinguishable and the interaction topology can be dynamicallychanging as the systemevolves. As a result, our approach is scalablein the sense that no labels of the robots is needed among therobots and can be applied to a large number of agents withoutre-designing the interaction topology. In addition, our approachis fully decentralized, while the pursuer strategy in [18] is partlydecentralized that involves role specialization in the form of leaderand followers. Finally, the control law proposed in the paperis practically implementable with only onboard sensors and therobots do not need to have a common sense of direction like someother formation control laws.

Throughout the paper, 1 represents the vector with all 1components, ‖ · ‖ represents the Euclidean norm, and ⊗ denotesthe Kronecker product.

2. Problem statement

Consider a group of n identical mobile robots which aremodeled as point masses freely moving in the plane, i.e.,

pi = ui, i = 1, . . . , n, (1)

where pi ∈ R2 is the position of robot i in the plane and ui ∈ R2

is the velocity control input of robot i. Suppose now that a targetmoves in the plane with a piecewise constant velocity. That is,

p0 = v0,

where p0 is its position in the plane and v0 is a piecewise constantvector.

Suppose that every robot carries an onboard sensor so that it isable to sense the relative positions of the target and its neighbors.However, the velocity of the target is not known a priori to anyrobot. With only local measured information, it is expected thatthe robots asymptotically form a formation enclosing the target.

1

4

3

2

target

Fig. 1. The pre-neighbor of robot 1 is robot 4; the pre-neighbor of robot 4 is robot2; the pre-neighbor of robot 2 is robot 3; and the pre-neighbor of robot 3 is robot 1.

As the target moves, the formation moves (or even more, rotatesaround the target) to keep the target as its centroid, maintains thedistance from the formation robots to the target, and distributesthe formation robots around the target spaced equally in the angle.The problem is called the moving-target-enclosing problem in thepaper and stated formally in the following. But we are going tointroduce the neighbor topology for the group of n robots first,which is defined according to their positions p1, . . . , pn in circularclockwise radial (CR)1 order around the target p0. In such a way,the one previous to robot i is called its pre-neighbor and the onenext to robot i is called its next-neighbor. Thus, if no two robotsare co-located, each robot has a unique pre-neighbor and a uniquenext-neighbor. The pre-neighbor of robot i is denoted by i+ andits next-neighbor is denoted by i−. The pre-neighbor and next-neighbormay change dynamically depending on p0, p1, . . . , pn. Letp = (p0, p1, . . . , pn) be the aggregate state and Ni(p) denotesthe set of robot i’s neighbors, which has two elements, namely,Ni(p) = {i+, i−}. An example is given in Fig. 1, in which the circularclockwise radial order of the four robots is 1, 3, 2, 4, 1, 3, 2, . . . .In the control law we study, each robot only uses the relativeposition information of the target and its neighbors. This leads tothe following definition.

Definition 2.1. A controller ui is said to be a local informationcontroller if

vi = gi(p0 − pi), (pj − pi)|j∈Ni(p)

ui = fi

(p0 − pi), (pj − pi)|j∈Ni(p), vi

(2)

where vi is of suitable dimension and is called the state of thedynamic feedback controller.

Now we are ready to present the formal problem statement.

Problem 2.1 (Moving-target-enclosing). Find, if possible, a local in-formation controller (2) that only uses relative sensing information(p0 − pi) and (pj − pi), j ∈ Ni(p) such that

(i) limt→∞ ‖pi(t) − p0(t)‖ = d for i = 1, . . . , n, where d is theradius of the desired enclosing circle; (target following)

(ii) limt→∞ ‖pi(t) − pi+(t)‖ = limt→∞ ‖pj(t) − pj+(t)‖ for anyi and j. (even distribution).

A schematic picture of a desired relative configuration for targetenclosing is shown in Fig. 2.

Finally, let us recall some properties of Laplacian matrices.Represent the pre-neighbor relationship of the n robots by a

1 That is, starting from any point of p1, . . . , pn , the n points are sorted firstly in aclockwise order around the target point and sorted secondly in a radially decreasingorder if there are more than one points on the same ray.

656 J. Guo et al. / Systems & Control Letters 59 (2010) 654–661

target

v0

d

Fig. 2. A desired enclosing configuration.

directed graph G(p) with n nodes. From our definition for pre-neighbor, it is known that the directed graph is a unidirectional ringthough the edge setmay change dynamicallywhen p changes. Thatis, each node has a unique outgoing edge and a unique incomingedge, which form a ring structure no matter how the pre-neighborof each robot changes. Clearly, the directed graph is stronglyconnected and balanced [20].We denote the Laplacian of the graphG(p) by Lp. The following proposition regarding the properties of Lpfollows easily from the results of strongly connected and balancedgraphs [20,21].

Proposition 2.1. For any p such that none of p0, p1, . . . , pn are co-located, the following holds:

(i) Lp1 = 0 and 1T Lp = 0;(ii) LTp+Lp has a simple eigenvalue at 0with an associated eigenvector

1 and all other eigenvalues are positive.

3. Control synthesis for moving-target-enclosing

In this section, we develop a control strategy for the moving-target-enclosing problem and then provide rigorous analysis forconvergence.

As described in Problem 2.1, two specifications should beaccomplished. We consider a control law having two terms, thatis,

ui = u′

i + u′′

i .

Moreover, we consider the relative position of robot i in the localframe attached to the target, so we define zi = pi − p0. Thus, wehave

zi = u′

i + u′′

i − v0, i = 1, . . . , n. (3)

First, let us present a trivial observation in the following lemma,which will be used to design the first term u′

i in our controller.

Lemma 3.1. If zi = −zi(‖zi‖2− d2), then ‖zi(t)‖ tends to d as

t → ∞ for any zi(0) = 0.

Proof. Let ρi = ‖zi‖. It can be easily obtained that

ρi = −ρi(ρ2i − d2).

As this is a scalar differential equation, it is trivial to conclude thatρi(t) tends to d for any initial condition ρi(0) = 0. �

Lemma 3.1 suggests that when the target is stationary, wecan select ui = −zi(‖zi‖2

− d2) such that every robot caneventually attain the desired distance d to the target. Hence, forour moving-target-enclosing problem, we choose the first term u′

iin our controller as follows:

u′

i = −zi(‖zi‖2− d2), i = 1, . . . , n. (4)

v0

ui´

ui˝

i

–v0

Fig. 3. Illustration for control decoupling.

Second, considering the form of (3), we propose that u′′

i − v0should be orthogonal to u′

i in order not to affect u′

i in achieving thedesired distance d to the target (see Fig. 3). Now let us consider thepolar coordinates of zi =

zxi zyi

T . We define

ρi =

(zxi )2 + (zyi )2 and θi = arctan2(zyi , zxi ),

where arctan2 represents a two-argument arctangent functionreturning the angle of point zi as a numeric value between 0 and2π rad. Let

ri =

[cos θisin θi

]and si =

[− sin θicos θi

].

It can be noted that si is formed by counterclockwise rotating ri ofπ/2 rad and so it is orthogonal to ri.With the polar coordinates andthese new notations of ri and si, the formula of u′

i can be writtenas

u′

i = −ρi(ρ2i − d2)ri.

On the other hand, in order to make u′′

i −v0 orthogonal to u′

i and inorder to achieve a uniform distribution while enclosing the target,we expect to have

u′′

i − v0 = ρi(θi+ − θi + ζi)si, (5)

where

ζi =

0 if θi+ − θi ≥ 0,2π if θi+ − θi < 0.

Intuitively, the control effort (5) suggests that each robot rotatesaround the target with the speed proportional to the angulardistance to its pre-neighbor. In other words, it rotates fast to chaseits pre-neighbor when it is far away from the pre-neighbor androtates slowly when it is close to its pre-neighbor. Thus, they willeventually be evenly spaced and rotate around the target with thesame speed. Nevertheless, notice that to satisfy (5), the secondterm u′′

i should be equal to v0+ρi(θi+ −θi+ζi)si, whichmeans eachrobot should access the velocity of the target. However, the target’svelocity v0 is unknown, so we introduce a new variable vi for eachrobot i and construct an adaptive control so that vi can adaptivelyconverge to v0. Thus, we take u′′

i of the form

u′′

i = ρi(θi+ − θi + ζi)si + vi, i = 1, . . . , n, (6)

where the dynamics of vi will be designed later.Considering u′

i in (4) and u′′

i in (6), we can write the dynamicsof ρi and θi as

ρi = −ρi(ρ2i − d2) + rTi (vi − v0),

θi = (θi+ − θi + ζi) +1ρi

sTi (vi − v0).(7)

Let

ρ = [ρ1, . . . , ρn]T

∈ Rn,

θ = [θ1, . . . , θn]T

∈ Tn= [0, 2π)n,

v = [vT1 , . . . , v

Tn ]

T∈ R2n.

J. Guo et al. / Systems & Control Letters 59 (2010) 654–661 657

Moreover, we define

D =

ρ1 · · · 0...

. . ....

0 · · · ρn

, h(ρ) =

ρ21 − d2

...

ρ2n − d2

,

R =

rT1 · · · 0...

. . ....

0 · · · rTn

∈ Rn×2n,

and

S =

sT1 · · · 0...

. . ....

0 · · · sTn

∈ Rn×2n.

Finally, we let e = v − 1⊗ v0 and let ζ = [ζ1, . . . , ζn]T . According

to our setup and the definition of ζi, we know that there is only oneζi equal to 2π and all others are 0, which means 1T ζ = 2π . Thenwe write system (7) in a vector form, which is

ρ = −Dh(ρ) + Re,θ = −Lpθ + ζ + D−1Se. (8)

In the following, we will write Lp as L(ρ,θ), meaning that theLaplacian matrix depends on the state (ρ, θ).

Next, we adopt the controlled Lyapunov function approach tofind the dynamics for vi. Let us consider the following Lyapunovfunction candidate:

V =14h(ρ)Th(ρ) +

12(−L(ρ,θ)θ + ζ )T (−L(ρ,θ)θ + ζ ) +

12eT e. (9)

From the definition,we know that the functionV is continuous, butnot continuously differentiable everywhere due to possible topol-ogy switching. However, it is piecewise continuously differentiablealong the solution. Hence, in what follows, we use the right upperDini derivative D+V for our analysis. More details can be found inthe Appendix. Along the solution of system (8), one obtains

D+V =12h(ρ)T

∂h∂ρ

ρ − (−L(ρ,θ)θ + ζ )T L(ρ,θ)θ + eT e

= h(ρ)TD[−Dh(ρ) + Re] + eT e − (−L(ρ,θ)θ + ζ )T

× [L(ρ,θ)(−L(ρ,θ)θ + ζ + D−1Se)]

= −h(ρ)TD2h(ρ) −12(−L(ρ,θ)θ + ζ )T (L(ρ,θ) + LT(ρ,θ))

× (−L(ρ,θ)θ + ζ ) +h(ρ)TDR − (−L(ρ,θ)θ + ζ )T

× L(ρ,θ)D−1S + eTe.

Note that v = e. So we set

v = −RTDh(ρ) + STD−1LT(ρ,θ)(−L(ρ,θ)θ + ζ )

so that h(ρ)TDR − (−L(ρ,θ)θ + ζ )T L(ρ,θ)D−1S + eT = 0. In otherwords,

vi = −ρi(ρ2i − d2)ri −

1ρi

ϕisi,

where ϕi = (θi − θi− + ζi−) − (θi+ − θi + ζi).Thus, the complete adaptive control law for each robot i is given

asvi = −ρi(ρ2i − d2)ri −

1ρi

ϕisi,

ui = −ρi(ρ2i − d2)ri + ρi(θi+ − θi + ζi)si + vi.

(10)

Fig. 4. Locally implementable control.

Remark 3.1. Note that in (10), ρi is the distance from the robot ito the target, (θi+ − θi + ζi) is the angle between the robot i andits pre-neighbor, (θi − θi− + ζi−) is the angle between the robot iand its next-neighbor, ri is the direction towards the target and siis the perpendicular direction (see Fig. 4 for an illustration). All theinformation can be measured by an onboard sensor (e.g., camera)attached to the robot i. Therefore, the control law (10) is locallyimplementable. Also, the group of robots do not require to have acommon sense of the direction.

Consider any equal spacing configuration of the n robots withthe order i1, . . . , in on the circle around the target and denote theset of corresponding states for the system (8) with the control law(10) by χi, that is,

χi =(ρ, θ, v)|ρi = d, vi = v0, θik+1 − θik + ζik = 2π/n,

k = 1, . . . , n − 1.

Denoteχj the set of states (ρ, θ, v) corresponding to another equalspacing configuration with a different order j1, j2, . . . , jn of the nrobots on the circle around the target. It is noticed that the set χiand the set χj in the state space are not connected.

Next, we present our first main result of stability.

Theorem 3.1. The equilibrium set χi is stable.

Proof. First, it is observed that χi is an equilibrium set for thesystem (8) with the control law (10). Consider the Lyapunovfunction V as in Eq. (9). For the closed-loop system, it is obtainedthat

D+V = −h(ρ)TD2h(ρ) −12(−L(ρ,θ)θ + ζ )T

× (L(ρ,θ) + LT(ρ,θ))(−L(ρ,θ)θ + ζ ).

Notice that (L(ρ,θ) + LT(ρ,θ)) is positive semi-definite fromProposition 2.1. Hence, it follows that

D+V ≤ 0.

Hence, the equilibrium set is stable. �

Remark 3.2. From the proof of Theorem 3.1, we can see that ifthe initial state lies in a small neighborhood of χi, i.e., V (0) ≤

ϵ for some small ϵ > 0, then V (t) ≤ ϵ for all t ≥ 0 dueto D+V (t) ≤ 0, which in turn implies that the angle betweenany two neighbor robots remains close to the value 2π/n. Thus,no switching could occur since the topology switching happensonly when the angle between two neighbor robots becomeszero at some time. However, when the state is not in the smallneighborhood of any equilibrium set χi, the interaction topologymay change depending on the evolution of the state.

658 J. Guo et al. / Systems & Control Letters 59 (2010) 654–661

Finally, we present the convergence result showing that thecontrol law (10) solves the moving-target-enclosing problemasymptotically.

Theorem 3.2. For a group of n robots with the control law (10), ifnone of p0(0), p1(0), . . . , pn(0) are co-located, then

(i) limt→∞ ‖pi(t) − p0(t)‖ = d for i = 1, . . . , n;(ii) limt→∞ ‖pi(t) − pi+(t)‖ = limt→∞ ‖pj(t) − pj+(t)‖ for any i

and j.

Proof. Consider the same Lyapunov function V as in the proof ofTheorem 3.1 and we have D+V ≤ 0. Let E := {(ρ, θ, v)|D+V =

0}. Recall that the kernel of (L(ρ,θ) + LT(ρ,θ)) is span{1} (see alsoProposition 2.1), so we obtain

E =(ρ, θ, v)|ρi = 0 or d, −L(ρ,θ)θ + ζ ∈ span{1}

.

It can be checked further that the largest invariant set in E is (referto the Appendix for the details)

M = {(ρ, θ, v)|ρi = 0 or d, −L(ρ,θ)θ + ζ ∈ span{1}, v = 1 ⊗ v0}.

Since V (t) is continuous andD+V ≤ 0, then V (t) is non-increasing.Therefore V is bounded. From the definition of V , it is obtained thath(ρ), −L(ρ,θ)θ + ζ and e are bounded. Also, we know that θ ∈

[0, 2π)n. Thus, from the non-smooth version of LaSalle’s InvariancePrinciple [22], every trajectory approachesM as t → ∞.

For the states inM , we have−L(ρ,θ)θ+ζ ∈ span{1}, so it followsthat −L(ρ,θ)θ + ζ = a1 for some constant a. Recall that

1T (−L(ρ,θ)θ + ζ ) = 2π.

As a result, 1Ta1 = 2π , which implies

a = 2π/n. (11)

Thus, it is clear that χi ∈ M and χi ∪χj ∪ · · · = M \ {(ρ, θ, v)|ρi =

0}. Notice that for any initial state, the trajectory approaches Mas t → ∞. Moreover, since the equilibrium set χi and χj arenot connected, hence, the trajectory has to approach one of theseequilibrium sets if no ρi(t) tends to 0.

Next we show that ρi(t) does not converge to 0 by contradic-tion. Suppose that there is a solution ρi(t) → 0 for ρi(0) = 0.From the above proof, we know that vi(t) tends to v0. Also, fromLemma 3.1, it is known that every solution ξ(t) of the systemξ = −ξ(ξ 2

− d2) approaches d for any initial condition ξ(0) = 0.Combining these three observations, it follows that for any arbi-trary small ε > 0 and δ > 0, there exists a T such that for allt > T ,

d − ε < ξ(t) < d + ε, (12)ρi(t) < ε, (13)

|rTi (vi − v0)| < δ. (14)

From (12) and (13), one obtains

|ρi(t) − ξ(t)| ≥ d − 2ε for all t ≥ T . (15)

On the other hand, we treat the dynamics of ρi as the perturbedsystem of ξ = −ξ(ξ 2

− d2) with its initial time at T . From theperturbation theory (Theorem 9.1 in [23]), it follows that for allt > T ,

|ρi(t) − ξ(t)| < ke−γ (t−T )|ρi(T ) − ξ(T )| + βδ,

where k, γ and β are positive constants independent of thechoices of ε and δ. Since ke−γ (t−T )

|ρi(T ) − ξ(T )| converges to 0exponentially, for any sufficiently small ε′ > 0, there exists aT ′ > T such that for all t > T ′,

|ρi(t) − ξ(t)| < ε′+ βδ. (16)

–200

–180

–160

–140

–120

–100

–80

–60

–40

–20

0

20

40

60

80

100

120

140

160

180

200

–200 –180 –160 –140 –120 –100 –80 –60 –40 –20 0 20 40 60 80 100 120 140 160 180 200

Fig. 5. The simulated trajectories of the eight robots enclosing a moving target.

Note that both ε′ and δ can be selected arbitrarily and we choosethem so that ε′

+ βδ < d − 2ε. Thus, a contradiction is reachedbetween (15) and (16).

Therefore, ρi → d, (θi+ − θi + ζi) → 2π/n, and vi → v0 as thetime t → ∞, which proves the theorem. �

4. Simulations

In this section, we present several simulations to illustrate ourresults. Firstly, we simulate eight robots tracking and enclosing amoving target where the velocity v0 of the target is a piecewiseconstant signal changing its value at t = 30 s. The desired distancebetween every robot and the target is d = 50. In the simulation,the initial positions of the eight robots are randomly generatedin the plane. The simulated trajectories of the eight robots underour control law are given in Fig. 5. It can be seen that each ofthem converges and maintains the desired distance to the movingtarget. Moreover, they are uniformly distributed when enclosingthe target inmotion. After the abrupt change of the target’s velocityat t = 30 s, the eight robots with our control law are capableof recovering to the desired circle formation. The evolutions ofthe distances between all the robots and the target (namely ρi =

‖pi − p0‖, i = 1, . . . , 8) are plotted in Figs. 6 and 7, whereFig. 6 shows the first part for the time interval [0, 30) and Fig. 7shows the rest starting at t = 30 s. From Fig. 7, we observe thatthe distances between the robots and the target are perturbedaway from the desired distance because of an abrupt change fromthe target, but our control law enables the robots to resume theenclosing motion and to adapt to the new target’s velocity after ashort transient period. In addition, the evolution of the separationangles between any two neighboring robots is depicted in Fig. 8.From the figure, we can see that the separation angles between anytwo neighboring robots converge to a common value 2π/8, whichvalidates our theoretical result.

Secondly, we simulate four robots for the moving-target-enclosing problem to show that the neighbor topology indeedchanges depending on the evolution of the system state. Ourapproach does not require to label the robots. However, in order toexplain more clearly the dynamically changing topology, we labelthe robots in the simulation. The initial distribution is depicted inFig. 9 (left), fromwhich we can see that the neighbor topology is asfollows according to our definition: the pre-neighbor of robot 1 is

J. Guo et al. / Systems & Control Letters 59 (2010) 654–661 659

t

ρ i

25

30

35

40

45

50

55

60

65

70

75

0 5 10 15 20 25 30

Fig. 6. The distances between the eight robots and the target for the time interval [0, 30].

t

ρ i

45

46

47

48

49

50

51

52

53

54

55

30 35 40 45 50 55 60

Fig. 7. The distances between the eight robots and the target for the time inverval [30, 60].

0 10 20 30 40 50 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

t

i+ –

i +

i

Fig. 8. The separation angles between any two neighboring robots.

robot 2; the pre-neighbor of robot 2 is robot 3; the pre-neighbor ofrobot 3 is robot 4; and finally the pre-neighbor of robot 4 is robot 1.

After applying our proposed control law, the final enclosingdistribution is depicted in Fig. 9 (right). At thatmoment, we can see

660 J. Guo et al. / Systems & Control Letters 59 (2010) 654–661

4

1

target

2

3

–200

–180

–160

–140

–120

–100

–80

–60

–40

–20

0

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–200–180–160–140–120–100–80–60 –40–20 0 20 40 60 80 100 120 140 160 180 200

2

1

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1

–200

–180

–160

–140

–120

–100

–80

–60

–40

–20

0

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–200–180–160–140–120 –100 –80 –60 –40 –20 0 20 40 60 80 100 120 140 160 180 200

Fig. 9. Initial distribution (left) and final distribution (right).

that the pre-neighbor of robot 1 is robot 3 and the pre-neighbor ofrobot 2 is robot 1 instead, whichmeans that the neighbor topologychanges at some time instant depending on the evolution of thestate.

5. Conclusion

In this paper, we have discussed a moving-target-enclosingproblem. Without knowing the target’s velocity, each robot onlyuses the local relative position information from the target andits neighbors. With the setup proposed, target tracking and inter-robot coordination are essentially decoupled. One part of thecontrol law amounts to ensuring the convergence of the distancebetween the robots and the target to a constant. The other partof the control law is used to achieve uniform distribution whenenclosing the target in motion. Besides, an adaptive scheme isproposed to estimate the velocity of the target. Lyapunov-basedtechniques and graph theory are brought together for controlsynthesis, which allows for changing topologies. The convergenceand stability properties are investigated in detail using a non-smooth version of LaSalle’s invariance principle. Our controlstrategy is practically implementable by only onboard sensors. Ourresults extend existing work in two aspects: allowing for a movingtarget with an unknown velocity and tolerating with neighborchanging which removes the requirement of labels for the robots.

Appendix

The Dini derivatives

The upper right-hand Dini derivative D+V (t) is defined as

D+V (t) = lim suph→0+

V (t + h) − V (t)h

where lim sup is the limit superior of a sequence of real numbers.

Calculating the largest invariant set M in E

From the dynamics (7) and (10), we obtainri = −ρi(ρ

2i − d2) + rTi (vi − v0),

θi = (θi+ − θi + ζi) +1ρi

sTi (vi − v0),

vi = −ρi(ρ2i − d2)ri −

1ρi

ϕisi.

When the states are in E, from (11) one hasri = rTi (vi − v0),

θi =2πn

+1ρi

sTi (vi − v0),

vi = 0.

To calculate the largest invariant set in E, let ri = 0 and θi+ − θi = 0for any i = 1, . . . , n. Then we haverTisTi

(vi − v0) = 0.

Since [risi] is invertible, it follows that vi − v0 = 0. Therefore, thelargest invariant set in E isM = {(ρ, θ, v)|ρi = 0 or d, −L(ρ,θ)θ + ζ ∈ span{1}, v = 1 ⊗ v0}.

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