Decentralized attitude synchronization tracking control for multiple spacecraft under directed communication topology

Zheng Zhong,Xu Ying,Zhang Lisong,Song Shenmin

aCollege of Astronautics,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China

bScience and Technology on In formation System Engineering Laboratory,Nanjing 210016,China

cJiangxi Ship Sailing Instrument Co.Ltd,Jiujiang 332008,China

dCenter for Control Theory and Guidance Technology,Harbin Institute of Technology,Harbin 150001,China

Decentralized attitude synchronization tracking control for multiple spacecraft under directed communication topology

Zheng Zhonga,*,Xu Yingb,Zhang Lisongc,Song Shenmind

aCollege of Astronautics,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China

bScience and Technology on In formation System Engineering Laboratory,Nanjing 210016,China

cJiangxi Ship Sailing Instrument Co.Ltd,Jiujiang 332008,China

dCenter for Control Theory and Guidance Technology,Harbin Institute of Technology,Harbin 150001,China

This paper studies the attitude synchronization tracking control of spacecraft formation flying with a directed communication topology and presents three different controllers.By introducing a novel error variable associated with rotation matrix,a decentralized attitude synchronization controller,which could obtain almost global asymptotical stability of the closed-loop system,is developed.Then,considering model uncertainties and unknown external disturbances,we propose a robust adaptive attitude synchronization controller by designing adaptive laws to estimate the unknown parameters.After that,the third controller is proposed by extending this method to the case of time-varying communication delays via Lyapunov–Krasovskii analysis.The distinctivefeature of this work is to address attitude coordinated control with model uncertainties,unknown disturbances and time-varying delays in a decentralized framework,with a strongly connected directed in formation flow.It is shown that tracking and synchronization of an arbitrary desired attitude can be achieved when the stability condition is satis fied.Simulation results are provided to demonstrate the effectiveness of the proposed control schemes.

1.Introduction

The recent few years have witnessed the burgeoning interest in decentralized control of networked multi-agent systems.These systems include single-integrator system,double-integrator system,general linear system,Euler–Lagrange system and nonlinear system.Indeed,there are many potential advantages of decentralized control schemes of multi-agent systems,such as greater flexibility,fault tolerance and better performance.As an important branch of decentralized control,decentralized attitude synchronization tracking control of spacecraft formation flying(SFF)has attracted great interest during the past decades.Compared with the existing studies on multi-agent systems with linear or even integrator-type node dynamics,the attitude synchronization control of SFF is more challenging owing to the inherently nonlinear attitude dynamics of spacecraft.Deep space exploration,in-orbit servicing,Earth monitoring and military operations are involved in the potential applications of attitude synchronization control of SFF,1which indicates controlling a fleet of spacecraft,so that their orientations and angular velocities converge to equal asymptotically.

Various control schemes have been proposed to solve the attitude synchronization tracking control problems using local relative in formation.Leader–follower control approach was proposed for attitude synchronization.2However,the failure of the leader spacecraft would lead to the collapse of the entire SFF system.Virtual structure control strategy was developed for attitude synchronization in Ref.3,and the entire desired formation system is treated as a single entity.In Ref.4,behavior-based approaches for attitude coordination of SFF were presented with undirected communication topology.Passivity-based coordination approach was applied to the rigid body attitude coordination problem.5An output feedback control approach for attitude synchronization without angular velocity measurement was developed6and extended to the case of communication delays.7Refs.8–10proposed various decentralized sliding-mode control design for attitude synchronization and cooperative tracking.In Ref.11,attitude alignment approaches for multiple spacecraft were presented with local bidirectional interactions.In Ref.12,autonomous attitude coordinated controllers were designed using backstepping technology with input constraint,model uncertainties and external disturbances.Cooperative attitude synchronization and tracking control methods were studied13,14to tackle the problem that only a subgroup of spacecraft has access to the desired attitude in formation.Ref.15examined the robust attitude coordination control for SFF under actuator failures and obtained uniformly ultimately bounded stability of the closed-loop system.

An undirected communication topology is used in the most aforementioned works for attitude synchronization control3–15.Recently,cooperative control of multi-agent using a directed graph is increasing quickly.However,the attitude synchronization tracking control of SFF with a directed graph is not fully understood except Refs.16–18,especially when model uncertainties,external disturbances,and communication delays exist simultaneously.Ref.16investigated the problem of synchronization control for spacecraft formation using extended state observer over directed communication topology,but the external disturbances are assumed as constants.Ref.17studied the attitude synchronization control problem with model uncertainties,external disturbances,but the communication delays are not considered.Ref.18studied the attitude synchronization problem in the presence of communication delays,but the delays are assumed as constants and the disturbances are not considered.

In this paper,decentralized attitude synchronization tracking control schemes of SFF are presented to track the desired attitude cooperatively,even in presence of model uncertainties,external disturbances,and time-varying communication delays.The rotation matrix,a 3×3 matrix that is orthogonal with determinant equal to one,is adopted to describe the attitude dynamics instead of unit quaternion in view of the highly undesirable attitude unwinding19with unit quaternion representation.A novel error variable containing attitude error and angular velocity error is proposed and the variable is exponentially stable under the designed controller without model uncertainties,external disturbances,or communication delays.Meanwhile,almost global asymptotical stability on the overall space except a set of measure zero is achieved using Lyapunov stability analysis method.When model uncertainties and external disturbances exist,a continuous robust adaptive attitude synchronization controller is designed and the corresponding Lyapunov stability is also proved.This method is further extended to the case of time-varying communication time delays.The stability condition of the closed-loop system is also derived by using the Lyapunov–Krasovskii function method.

This paper is organized as follows.Section 2 introduces the spacecraft attitude dynamics and reviews the mathematical preliminaries of graph theory.Section 3 proposes a novel continuous attitude synchronization controller to guarantee that each spacecraft approaches the desired time-varying attitude and angular velocity.Section 4 presents a decentralized robust adaptive attitude synchronization controller to ensure that the attitude errors and angular velocity errors reach zero even in the presence of inertia matrix uncertainties and external disturbances.Section 5 proposes a delayed decentralized robust adaptive controller with inertia matrix uncertainties,external disturbances and time-varying delays.Section 6 presents simulation results to illustrate the effectiveness of the proposed attitude synchronization methods.

2.Mathematical model and preliminaries

2.1.Spacecraft attitude model based on rotation matrix

The attitude dynamics and kinematics equations of the ith spacecraft are given as20,21

where i=1，2，...，n,and Ji∈ R3×3is the inertia matrix of the ith spacecraft;ωi∈ R3denotes the angular velocity resolved in the body frame;τi∈ R3and di∈ R3are control torque and disturbance torque,respectively;Ri∈ SO（3）is the rotation matrix from the body frame to the inertial frame.For a vector y= ［y1，y2，y3］T,the map × trans forms a vector to a 3 × 3 skew-symmetric matrix such that

The inverse of the map×is denoted by∨which trans forms a 3×3 skew-symmetric matrix to a three-dimensional vector.Several properties of the map∨are given as follows21:

where x ∈ R3,A ∈ R3×3,R ∈ SO（3）,and tr(·)is the trace of a matrix.

Denote the desired rotation matrix for all the spacecraft as Rd∈ SO（3）and the desired angular velocity as ωd∈ R3.So it follows that

where Rdis the rotation matrix that describes the orientation of the desired body frame relative to the inertial frame.The error rotation matrix and error angular velocity of the ith spacecraft are defined as~Ri=RTdRiand~ωi=ωi-~RTiωd,

respectively.From Eqs.(1),(2)and(6),we can write the error equations as

In this paper,we need to design attitude coordinated controllers to synchronize the attitudes of formation spacecraft,namely,Ri→ Rd，ωi→ ωdas t→ ∞.

2.2.Algebraic graph theory

Weighted graphs can be used to describe local in formation exchanges between spacecraft within a formation.11In this paper,we consider n spacecraft interconnected on strongly connected directed graphs.A weighted directed graph G= （ν，?，C）is composed of a node set ν= ｛1，2，...，n｝,an edge set ?? ν× ν,and a weighted adjacency matrix C.The node set ν represents all the agents of the spacecraft and the edge set ? denotes the in formation flow between the spacecraft.If there exists in formation transmission from the jth node to the ith node,then there is an edge from the jth node to the ith node,denoted as （i，j）∈ ?.The element of the adjacency matrix C= ［cij］is defined as cij＞ 0 if （i，j）∈ ?,and cij=0 otherwise.The Laplacian matrix L= ［lij］associated with the graph is defined as lij=∑nk=1cikifi=j,and lij=-cijotherwise.The directed path is a sequence of edges in the directed graph with the form （i1，i2），（i2，i3），...，（in-1，in）.A directed graph is called strongly connected if there exists a directed path between two arbitrary nodes.

Lemma 122.If the Laplace matrix is associated with a strongly connected directed graph,there existsa positive vector η = ［η1，η2，...，ηn］T(η1，η2，...，ηn＞ 0)such that ηTL=0.

3.Attitude synchronization with a directed topology

In this section,we present a solution to the attitude coordination problem of SFF with a directed communication topology,which means that the spacecraft can receive the state in formation from the neighbors and does not necessarily send its own in formation to the neighbors.The ideal case is considered for the sake of simplicity,where the disturbance diis zero and the inertia matrix Jiof the ith spacecraft is known.Besides,thefollowing assumptions are made about the spacecraft formation system.

Assumption 1.The communication graph is directed,strongly connected,and fixed.

Assumption 2.The desired angular velocity ωdand its derivative˙ωdare assumed to be bounded and known to each spacecraft.

Remark 1.In practice,ωdandmay be available only to a subset of the team spacecraft.In this case,we can use the idea of sliding mode estimator to estimate ωdand.Letandbe the estimation of ωdand,respectively,andwhere constant β ＞ 0,sgn(·)is sign function.From the analysis of Ref.23,after a finite time T0.So every spacecraft can obtain the desired angular velocity ωdand its derivativeafter a finite time

The attitude error of the ith spacecraft is defined as

We can see that 0≤ei≤4,and ei=0 only when Ri=Rd.From Eqs.(3)and(7),the derivative of eiis

Using Eqs.(3)–(5),the derivative of φ（~Ri）can be calculated as

where αi＞ 0 is a constant.We can conclude the following lemma:

Lemma 2.If the error variable si∈L2,and si，˙si∈L∞,then

~ωi→0 and φ（~Ri）→0 as t→∞.

Proof .From Eq.(10),we can obtain that

So~ωi∈L2and φ（~Ri）∈L2.Additionally,we conclude that~ωi∈L∞fromsi∈L∞andφ（~Ri）∈L∞.Consequently,˙φ（~Ri）∈L∞from Eq.(11),and˙~ωi∈L∞from˙si∈L∞.Now using the corollary of Barbalat’s Lemma,22we can conclude that~ωi→0 and φ（~Ri）→0 as t→∞.□To develop the controller,the following equation is derived from Eqs.(8)and(12):

An ideal decentralized attitude synchronization tracking controller is designed as

where ki＞0;cijis the element of the adjacency matrix C with the communication graph.

Theorem 1.Consider the spacecraft formation attitude tracking dynamics described by Eqs.(7)and(8)with the ideal decentralized attitude synchronization controller Eq.(16).DefinethesetΩi=｛（si，~Ri）si=0，｜tr（~Ri）=-1｝.Ifthe Assumptions 1 and 2 are valid,then

(1)The error variable siis exponentially stable to zero.

(2) Ωiis a forward invariant set and the point of Ωiis an unstable saddle point.

(3)For almost all the initial states satisfy （si（0），~Ri（0））?Ωi,the trajectory converges to the set Qi=｛（si，~Ri）si=0，｜~Ri=I3｝,i.e.,Ri→Rdand~ωi→0 as t→∞ for any 1≤i≤n.

Proof .Consider the following candidate Lyapunov function:

With Eqs.(15)and(16)and Lemma 1,differentiating V1relative to time yields

Using unitary decomposition of the rotation matrix~Ri,it is obtained that φ（~Ri）=0 means that Ri=Rdor tr （~Ri）=-1.25We can see that the attitude error eihas a minimum value at the expected equilibrium point Ri=Rdand a maximum value at the equilibrium point tr （~Ri）=-1.Now we consider the trajectory from the point of the set Ωi.The dynamics with controller Eq.(16)can be given as

We can see that si（t）=0 andω~i（0）=-αi（R~i（0）-R~Ti（0））=0.By using the uniqueness results of solutions for differential equations,the solution of Eq.(19)is given as

The rotation matrix corresponding to the quaternion q is R（q）=I3+2q1.The error quaternion is defined as~qi=°qiwhereis the conjugated quaternion of qd,qithe attitude of the ith spacecraft,and°the multiplication operator of quaternions.In the set Pi,φ（）=4qq=0 implies

We can see that the variable qi1is not stable due to αi＞ 0 and siconverges to zero.As a result,siis stable and qi1is not stable,and hence the point of Ωiis an unstable saddle point.

Note that there will be trajectories that converge to Ωialong the stable center manifold associated with the stable variable si.24,25However,it is known that such trajectories are no more than measure zero in the overall state space.Besides,notice that the unstable equilibrium set Ωiis measure zero in the overall space R3×SO（3） □

Remark 2.Theorem 1 gives local stabilization results on the attitude coordination control problem for the multiple equilibrium points of the closed-loop system.However,as we all know,continuous time-invariant feedback of attitude control scheme cannot achieve global attitude stability.19,20In other words,the results of Theorem 1 is almost globally stable except a set of measure zero,which is an preferable result when continuous time-invariant feedback is adopted in this problem.Remark 3.For the rotation matrix~Ri=RTdRi,its trace is bounded by-1 ≤ tr（~Ri）=1+2cos σi≤ 3,20,21where σiis the eigenangle from Rito Rdalong an axis.From the results of Theorem 1,the eigenangle σiof the undesired equilibrium point is±π for any 1≤i≤n,which means that the spacecraft points to the opposite directions of the desired attitude.

4.Adaptive robust attitude synchronization control

In this section,we present a solution to the attitude synchronization tracking problem with a directed communication topology in the presence of model uncertainties and unknown external disturbances.We also provide a stability analysis of the resulting decentralized adaptive robust controllers.The following assumptions are made about the spacecraft formation system in this section.

Assumption 3.The inertia matrix Jiis an unknown positive definite matrix,and the disturbance torque diis bounded with‖di‖∞≤ˉdi,whereˉdi＞0 is an unknown constant.

Lemma 326.For all real scalars x and all nonzero real scalars y,it follows

where α is a positive constant with the minimum value α*=x*（1-tanhx*） for x*satisfying e-2x*+1-2x*=0.

Denote the inertia matrix of the ith spacecraft as

The adaptive parameter of the ith spacecraft is introduced as

The decentralized adaptive robust attitude synchronization controllers are designed as

where^θiis the estimated value of the parameter θi;^diis the estimated value of the parameterˉdiwith^di（0）＞0;Γiis a 6×6 symmetric positive definite matrix;γd＞ 0;piis time-varying subject to pi（0）＞ 0;α is defined in Lemma 3.

Remark 4.It should be noted that the controller Eq.(28)has an explicit structure.Thefirst term is the adaptive compensation item;the second term is the attitude tracking feedback;the third term is attitude coordination with its neighbor spacecraft;the last one is the disturbance attenuation item.The explicit structure is convenient in selecting the parameters of the controller when we emphasize the formation tracking,coordination or disturbance attenuation in the spacecraft formation control.

Proof .

The inequality(22)in Lemma 3 can be rewritten as

where~θi= θi-^θi.We can see that the Lyapunov function is positive-definite.From Eqs.(28)–(31),the derivative of V2is given as

Substituting Eq.(33)into Eq.(35)yields

Since the communication graph is strongly connected,we obtain thefollowing results from Lemma 1.

Remark 5.In this study,we design a continuous decentralized controller to realize the attitude synchronization of the spacecraft formation,and the controller is robust to unknown disturbances and model uncertainties.Compared with traditional non-continuous control schemes such as sliding-mode control,8–10the proposed continuous controller is more applicable when flywheels or control moment gyroscopes are adopted as the actuators of the spacecraft.

5.Adaptive robust attitude synchronization control with communication delays

Next,we consider the coupling communication delays between the formation spacecraft in the attitude synchronization tracking control of SFF.And then the real available in formation of the ith spacecraft is the delayed in formation sj（t-Tij）in the decentralized controller,which is formulated as

where Tij（t）≥ 0 is time-varying and it is not desired that Tij=Tji.Assume that the precise value of Tijis not available,but the bound of its derivative is known with˙Tij≤hijwhere hij＜1 is a constant.In the following,we demonstrate that the controller Eq.(38)is still applicable in the case of timevarying delays if the parameters of controller Eq.(38)are selected properly.

Theorem 3.Consider the spacecraft formation attitude tracking dynamics described by Eqs.(7)and(8)with the decentralized adaptive robust attitude synchronization controllers Eq.(38)and Eqs.(29)–(31).If the Assumptions 1–3 are valid and the controller parameter satisfies

for any 1 ≤ i，j≤ n,where ρ ＞ 1 is a constant,the conclusion of Theorem 2 still holds.

Proof .The candidate Lyapunov function is chosen as

From the proof of Theorem 2 and the controller Eq.(38),the derivative of V3can be written as

Notice that

Substituting Eqs.(42)and(43)into Eq.(41)leads to

This indicates that V3≥0 and˙V3≤0 hold for any time,and thus∫∞0˙V3（τ）dτ exists and is finite.Meanwhile,∈L∞from thefact that˙V3≤0.Integrating both sides of inequality(45)results in

Fig.1 Communication topology.

Fig.2 Rotation angle with controller Eq.(16).

Fig.3 Angular velocity errors with controller Eq.(16).

Consequently,it is obtained that si∈L2.According to Eqs.(21)and(22),it follows that˙si∈L∞.By applying Lemma 2,we can infer that~ωi→0 and φ（~Ri）→0 as t→∞.From the proof of Theorem 1,we can also conclude that the angular velocity error and attitude error will converge to zero in the overall space except a set of measure zero,i.e.,Ri→Rdand~ωi→0 as t→∞ for any 1≤i≤n.□

Fig.4 Control torque with controller Eq.(16).

Fig.5 Rotation angle with controller Eq.(28).

Remark 6.The communication delays are considered in the attitude coordinated control of SFF.7,9However,cooperative attitude synchronization tracking control with time-varying delays and directed topology has not been studied to the best of our knowledge.Meanwhile,the disturbance and model uncertainty are considered in the delayed decentralized attitude coordinated controller,which is also a merit of the proposed controller.

6.Numerical simulations

In this section,numerical simulations are presented to validate the effectiveness of the proposed formation control strategies.A scenario with four spacecraft is considered in the simulations.The directed communication topology that describes the in formation flow of spacecraft is shown in Fig.1,in which node Si(i=1,2,3,4)represents the ith spacecraft.

We set the disturbance di=0 when the controller Eq.(16)is implemented.And gravity-gradient torque dgiand the periodic disturbance torque d0are considered with the controller Eqs.(28)and(38),i.e.,di=dgi+d0.

Fig.6 Angular velocity errors with controller Eq.(28).

Fig.7 Control torque with controller Eq.(28).

We assume that the inertia matrices are known with the controller Eq.(16)and unknown with the controller Eqs.(28)and(38).And the nominal inertia matrices of the spacecraft are assumed to be

The initial angular velocity of each spacecraft is chosen to be zero,and the initial value of rotation matrix for each spacecraft is chosen as

The initial value of the desired rotation matrix of the formation spacecraft is chosen as

The time-varying desired angular velocity of the spacecraft is identical as

When the proposed three controllers are implemented,the weighted adjacency matrix C associated with the communication topology is chosen as

The parameters of the controller Eq.(16)are chosen as αi=3 and ki=1 for i=1,2,3,4.The parameters of the controller Eq.(28)are chosen as αi=2 and ki=1.The gains of the adaptive law Eqs.(29),(30)and(31)with the controller Eq.(28)are chosen as Γi=0.1I6, γd=0.00005 and α =1,respectively.The initial value of ^θiin Eq.(29)is chosen from the nominal inertia matrices of the spacecraft;the initial value of ^diin Eq.(30)and p2iin Eq.(31)are chosen as^di（0）=0.001 and（0）=1,respectively.The parameters of the controller Eq.(38)and the adaptive law Eqs.(29)–(31)are chosen as the same as those of the controller Eqs.(28)–(31).The timevarying communication time delay Tijis assumed to be

Fig.10 Angular velocity errors with controller Eq.(38).

Obviously,the inequality Eq.(39)is satisfied with the selected parameters.

To investigate the performance of the proposed controllers,the scalar measure of attitude error is given by the rotation angle σialong the eigenaxis,which is given as20,21

The rotation angle σi,angular velocity error and control torque with the controller Eq.(16)are shown in Figs.2–4,respectively.From Fig.2,it is observed that the attitude tracking errors of each spacecraft converge to zero eventually,which shows that the cooperative attitude tracking of the SFF system is achieved.The responses of the angular velocity errors of spacecraft are shown in Fig.3.It can befound that the angular velocity errors decay quickly and the final values are small enough.Fig.4 demonstrates the control torque of the spacecraft within the formation.

The simulation results of the controller Eq.(28)with model uncertainties and disturbances are shown in Figs.5–8.Fig.5 presents the responses of the rotation angle σi.It is shown that the angle σiof the ith spacecraft converges to zero,which shows that the attitude synchronization tracking of the SFF system is achieved eventually.The responses of the angular velocity errors of the spacecraft are shown in Fig.6.It can befound that the angular velocity errors decay to zero quickly and a high tracking accuracy is achieved.Fig.7 demonstrates the responses of the control torque of spacecraft and the final control torque is less than 0.2 N·m.Fig.8 gives the estimations of the inertia parameters of each spacecraft,and the estimations all converge to constant values.

Fig.11 Control torque with controller Eq.(38).

Fig.12 Estimations of inertia parameters with controller Eq.(38).

The simulation results of controller Eq.(38)with timevarying communication delays,model uncertainties and disturbances are illustrated in Figs.9–12.From Fig.9,it can befound that the rotation angle σiof the ith spacecraft converges to zero,which shows that the cooperative attitude synchronization tracking of the SFF system is achieved eventually.The responses of the angular velocity errors of each spacecraft are shown in Fig.10.It can be observed that in this case,the angular velocity errors are well convergent eventually despite of model uncertainties, external disturbances and timevarying communication delay network.Fig.11 demonstrates the responses of the control torque of each spacecraft and thefinal control torque is less than 0.2 N·m.Fig.12 gives the estimations of the inertia parameters of each spacecraft with controller Eq.(38).

We use rotation matrix to design the attitude synchronization controllers rather than quaternions or modified Rodrigues parameters.The reason is that the controllers based on quaternions or modified Rodrigues parameters may lead to attitude unwinding phenomenon.Such phenomenon is highly undesirable from the viewpoint of fuel consumption and vibration suppression.Fig.13 gives the quaternions and rotation angle under the controller in Ref.4based on quaternions with the same initial conditions as controller Eqs.(16),(28)and(38),where δq0iis the scalar part of the error quaternions,and σithe rotation angle.From Fig.13,we can see that the rotation angle of each spacecraft in attitude maneuvering is beyond 300°,which shows that the controller based on quaternions leads to attitude unwinding.In contrast,our controllers do not lead to attitude unwinding as shown in Figs.2,5 and 9.This result shows the advantages of the proposed control methods in preventing attitude undwinding.

Fig.13 Quaternions and rotation angle with controller in Ref.4.

7.Conclusions

(1)The main contribution of this study lies in the development and stability analysis of the continuous decentralized attitude synchronization tracking control schemes for multiple spacecraft formation based on the rotation matrix representation.The continuous controllers are almost globally asymptotically stable except a set of measure zero,which is preferable to some extent when continuous time-invariant feedback is adopted.

(2)The cooperative attitude synchronization tracking is achieved for the formation system using directed interspacecraft communication link.When disturbances are zero and inertia matrix is known,it is proved that the error variable including attitude error and angular velocity error is exponentially stable to zero.Although the multiple equilibrium points appear with the rotation matrix description,the undesired equilibrium points are unstable and the region of attraction with the desired equilibrium point is the overall space except a set of measure zero.When it comes with the model uncertainties and disturbances,adaptive laws to estimate the inertia matrix of spacecraft and the bound of disturbances are designed.Almost global asymptotical stability with the robust adaptive controller is also obtained by using Barbalat’s Lemma.Then this approach is extended to time-varying communication delays,and a proper Lyapunov–Krasovskii function is chosen to prove that the controller can make the formation system stable if the stability condition is satisfied.

(3)Simulation results demonstrate the favorable performance of the proposed controllers,and all the spacecraft converge to the desired attitude and angular velocity eventually.

Acknowledgements

This study was supported by the National Natural Science Foundation of China(Nos.61573115 and 61333003).

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Zheng Zhong received his Ph.D.degree in control science and engineering from School of Astronautics,Harbin Institute of Technology.Currently,he is a lecturer in College of Astronautics,Nanjing University of Aeronautics and Astronautics.His main research interests include spacecraft formation control and nonlinear control.

1 September 2015;revised 18 December 2015;accepted 7 March 2016

Available online 22 June 2016

Attitude synchronization;

Directed topology;

Lyapunov stability;

Spacecraft formation;

Time delays

?2016 Production and hosting by Elsevier Ltd.on behalf of Chinese Society of Aeronautics and Astronautics.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-ncnd/4.0/).

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E-mail address:zhengzhong8610@126.com(Z.Zheng).

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1000-9361?2016 Production and hosting by Elsevier Ltd.on behalf of Chinese Society of Aeronautics and Astronautics.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).