Distributed adaptive attitude synchronization for spacecraft formation flying with sampled-data information flows

Available online at www.sciencedirect.com

Journal of the Franklin Institute 352 (2015) 2796–2809 www.elsevier.com/locate/jfranklin

Distributed adaptive attitude synchronization for spacecraft formation flying with sampled-data information flows Chao Maa,n, Qingshuang Zenga, Xudong Zhaob,c a

Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin, China b College of Engineering, Bohai University, Jinzhou, China c Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, China Received 29 March 2014; received in revised form 13 January 2015; accepted 2 April 2015 Available online 5 May 2015

Abstract This paper investigates the adaptive attitude synchronization problem of spacecraft formation flying in the presence of unknown parameters and external disturbances. Specifically, by introducing the “sampled-data information flow” concept, a novel distributed control strategy is developed, which is more reliable and practical in the applications. Moreover, the velocity information is not required to be exchanged, such that the communication network load can be further reduced. Based on input-to-state stability, sufficient conditions are derived to guarantee that the attitude synchronization can be achieved under directed topology. An important and distinctive feature of the proposed method consists in the fact that it can effectively reduce the energy consumption for the spacecraft formation flying due to more efficient utilization of the communication network resources. Finally, a numerical example is presented to demonstrate the effectiveness and the benefit of our proposed method. & 2015 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction In the past decade, spacecraft formation flying has received increasing attention in the space industry. This is largely due to their potential applications in both civil and military uses, such as space n

Corresponding author. E-mail addresses: [email protected] (C. Ma), [email protected] (Q. Zeng), [email protected] (X. Zhao).

http://dx.doi.org/10.1016/j.jfranklin.2015.04.013 0016-0032/& 2015 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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interferometry [1], autonomous on-orbit assembly [2], synthetic-aperture imaging [3] and so on [4,5]. There are several advantages compared with the traditional control of an individual spacecraft. For instance, the functionality of large spacecraft can be distributed on some simpler, smaller and lessexpensive spacecrafts, such that redundancy is provided and launch costs may be reduced [1,5–8]. The attitude synchronization problem, as one of the fundamental issues of formation maneuvers, has been extensively addressed, which means that a group of spacecrafts can reach an agreement on the orientation based on local information exchanges. As a result, many effective approaches and synchronization criteria have been proposed in the literature [9–13], to name just a few. Note that the communication network characterizes the fundamental connectivity of the spacecrafts and plays a key role in the information exchanges. However, almost all of the existing control strategies are based on the assumption that the individual spacecraft can communicate with its neighbors continuously with time, which may be not appropriate for practical reasons. Normally, there are bandwidth restrictions in the communication network and limited communication resources are shared by the spacecrafts. In the continuous-time communication network, the communication channels are always occupied by the information flows and high communication capacity is required. These typical features may make it a challenging task to transfer continuous-time information flows. Furthermore, continuous-time information exchanges need continuously rapid progresses in sensing and communication hardware, which may increase the payload of the spacecrafts and additional launch costs. Therefore, it is of great importance to find a more reliable and practical way for information exchanges among the spacecrafts. Unfortunately, to the best of the authors' knowledge, few results concerning this problem have been reported in the literature, which is the first motivation of this paper. Another motivation of this study is the communication energy consumption problem of spacecraft formation flying, which has not been fully addressed in the existing literature. Since the energy supply of each spacecraft is limited, the energy consumption for information exchanges could not be negligible and special considerations are needed at early design stages. It is worth mentioning that even though the continuous-time communication network can be established, continuous-time sensing and transferring information will cost considerable energy and may lead to a decline in the space mission performance. From the energy perspective, energy efficiency strategies for information exchanges should be developed [14–18]. Notice that one possible way to solve this problem is reducing the amount of information needed for the formation flying, such that the communication load can be decreased. With the above discussions, a novel attitude synchronization approach for spacecraft formation flying is proposed by introducing the concept of sampled-data information flow. More specifically, the attitude information of each spacecraft is sampled and transferred to its neighbors in the communication topology according to a unified sampling sequence. In this way, the received information in each spacecraft is updated at each sampling instance and held constant during the sampling intervals. Hence, periodic communication sequences are obtained, which are represented in a discrete-time fashion. It is noted that the system parameters of the spacecrafts can be identified only approximately, such that adaptive control laws are provided to estimate the uncertain parameters. In addition, the proposed approach takes the external disturbances into consideration for high formation performance [19–22]. Compared with existing results, there are three main advantages of the proposed approach. Firstly, the spacecrafts are networked by the sampled-data information flows instead of continuous-time information flows, which is more practical and thereby facilitates implementation in the real world. Taking inspiration from the networked control system [23–28], the basic idea is to transfer sampled-data information reliably and efficiently. Secondly, the angular velocity information is not required to be exchanged for each spacecraft, which can simplify the sensing and communication devices and further reduce the communication network load. Finally, the proposed

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method can effectively reduce the communication energy consumption, since continuous-time communication is replaced by discrete-time communication. Moreover, it is noteworthy that the obtained attitude synchronization results in the context of spacecraft formation flying are also valid for other multiple rigid bodies which can be transformed to Lagrange expression. The remainder of this paper is organized as follows. Section 2 provides some preliminaries on the attitude dynamics and kinematics of rigid spacecraft and graph theory, and formulates the attitude synchronization problem. In Section 3, distributed controllers are designed based on local information exchanges among the spacecrafts. By input-to-state stability (ISS) analysis, sufficient conditions are further derived to ensure that the attitude synchronization can be achieved in the presence of external disturbances while the parameter estimates converging to their true values. Section 4 presents a numerical example to illustrate the effectiveness and the benefit of the proposed strategy and the paper is concluded in Section 5. Notation: The notation of this paper is fairly standard. Rn denotes the n dimensional Euclidean space. Rm
n represents the set of all m
n real matrices. I and 0 represent identity matrix and zero matrix, For any function f : R Z 0 -Rn , the L2 -norm is defined as R 1 respectively. 2 2 ‖f ‖2 ¼ 0 jf ðtÞj dt with the L2 space defined as the set ff : R Z 0 -Rn ∣‖f ‖2 o1g, while the L1 -norm is defined as ‖f ‖1 ¼ supt Z 0 jf ðtÞj with the L1 space defined as the set ff : R Z 0 -Rn ∣‖f ‖1 o1g. A continuous function α : ½0; a-½0; 1Þ is said to be of class K if it is strictly increasing and αð0Þ ¼ 0. The function is said to be of class K1 if a ¼ 1 and αðrÞ ¼ 1 when r-1. A function β : ½0; 1Þ2 -½0; 1Þ is said to be of class KL if, for each fixed t, the mapping βðs; tÞ is of class K and, for each fixed s, it is decreasing and βðs; tÞ-0 as t-1. A  B denotes the Kronecker product of the matrices A and B. 1N A RN is the column vector with all entries being 1. ReðÞ and ImðÞ denote the real and imaginary parts of a complex number, respectively. The matrix P40 means P is real symmetric and positive definite. λðPÞ denotes the eigenvalues of the matrix P. λmin ðPÞ and λmax ðPÞ denote the minimum and maximum eigenvalues of P, respectively. Finally, in symmetric block matrices, n is used as an ellipsis for the terms that are introduced by symmetry and diagf⋯g denotes a block-diagonal matrix. 2. Problem formulation and preliminaries 2.1. Rigid spacecraft attitude kinematics and dynamics Consider a group of N networked spacecrafts indexed by the set I ¼ f1; …; N g. By the Euler rotational equations, the angular velocity vector ωi A R3 of the ith spacecraft in its body-fixed frame can be described as [29] J i ω_ i  ðJ i ωi Þ
ωi ¼ ui þ di ; where J i A R3
3 represents the total moment of inertia, ui A R3 is the exerted torque and di A R3 denotes the external disturbance torque due to solar radiation, aerodynamics and magnetic fields. Given the slow time-varying nature of these parameters, it is assumed that di remains constant in the subsequent analysis. The orientations of the spacecrafts with respect to the inertial frame can be represented by the Modified Rodrigues Parameters (MRPs) [30]. Note that for MRPs, there is a geometric singularity problem at φi ¼ 72π for the three-dimensional attitude description. The MRP vector qi A R3 , iA I is defined as φ qi ¼ κ i tan i ;  2πoφi o2π; 4

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where κi A R3 and φi denote the eigenaxis unit vector and eigenangle corresponding to the given orientation of the i th spacecraft, respectively. Then, the attitude kinematics and dynamics of the ith spacecraft are given as q_ i ¼ Γðqi Þωi ; where Γðqi Þ is given by Γðqi Þ ¼ 12 ½Sðqi Þ þ qi qTi þ 12ð1  qTi qi ÞI 3 ; with Sðqi Þ A R3
3 being a skew-symmetric matrix function. Consequently, the attitude kinematics and dynamics of the ith spacecraft can be transformed to Lagrange expression as M i ðqi Þq€ i þ Ci ðqi ; q_ i Þq_ i ¼ τi þ τdi ;

iA I ;

ð1Þ

where M i ðqi Þ ¼ Γ  T ðqi ÞJ i Γ  1 ðqi Þ; _ i ÞΓ  1 ðqi Þ Ci ðqi ; q_ i Þ ¼  Γ  T ðqi ÞJ i Γ  1 ðqi ÞΓðq  Γ  T ðqi ÞSðJ i Γ  1 ðqi Þq_ i ÞΓ  1 ðqi Þ; T τi ¼ Γ ðqi Þui ; τdi ¼ Γ  T ðqi Þdi : Furthermore, the following properties regarding the systems (1) are given [31]: Property 1. The inertial matrix M i ðqi Þ is a symmetric positive-definite matrix and bounded such that 0oλmin ðM i ðqi ÞÞI 3 r M i ðqi Þr λmax ðM i ðqi ÞÞI 3 : Property 2. The matrix C i ðqi ; q_ i Þ is bounded such that ‖Ci ðx; yÞ‖ r kc ‖y‖ for all vectors x, y A R3 , where kc 40 is a constant. _ i ðqi Þ  2Ci ðqi ; q_ i Þ is skew symmetric such that for a given vector Property 3. The matrix M 3 r A R , it follows that _ i ðqi Þ 2Ci ðqi ; q_ i ÞÞr ¼ 0: r T ðM Property 4. Since di remains constant, the systems (1) can be linearly parameterized such that M i ðqi Þq€ i þ Ci ðqi ; q_ i Þq_ i  τdi ¼ Yðqi ; q_ i ; q€ i Þθi ; where θi denotes a physical parameter and Yðqi ; q_ i ; q€ i Þ represents the regressor matrix. 2.2. Graph theory The information flows among the spacecrafts can be conveniently represented in a graph. Let G ¼ fV; E; Ag be a directed graph indexed by the set I ¼ f1; …; N g, where VðGÞ ¼ fv1 ; …; vN g is the set of nodes, E denotes the set of edges with EDV
V, and A ¼ ½aij A RN
N is the weighted adjacency matrix. An edge of G is denoted by ðvi ; vj Þ and the self-connection is excluded. aij is defined by the rule that aij 40 if andPonly if ðvi ; vj Þ AE. Otherwise, aij ¼ 0. The Laplacian L ¼ ½lij A RN
N of G is defined as lii ¼ Nj¼ 1;j a i aij and lij ¼  aij ; ia j. The set of neighbors of node vi is the set of all nodes pointed to vi , denoted by N i ¼ fvj A VðGÞ : ðvi ; vj ÞA Eg. A directed path is a sequence of edges of the form ðv1 ; v2 Þ, ðv2 ; v3 Þ, …, where vi A V. A directed tree is a directed graph, where every node has exactly one parent except for one node, called the root, and

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the root has directed paths to every other node. A directed spanning tree of a directed graph is a direct tree that contains all nodes of the directed graph. If a directed graph has a directed spanning tree, L has a simple zero eigenvalue and all the other eigenvalues are in the open right half plane [32]. Let J A RN
N be the Jordan form of L, i.e., L ¼ UJU  1 , where U is a nonsingular matrix. It can be found that some eigenvalues of L may be complex and J ¼ diagfJ 1 ; …; J r g, where 2 3 λi

1

0

0

60 Ji ¼ 4 0

⋱ 0

⋱ ⋱

07 5 1

0

0

0

λi

;

i ¼ 1; …; r;

N i
N i

with λi denoting the eigenvalues of L and is valid: " # 0 0TN  1 1 J ¼ U LU ¼ : n J

Pr

i¼1

N i ¼ N. Moreover, the following decomposition

ð2Þ

2.3. Instrumental lemma Lemma 1 (Fridman et al. [33]). Define L2 ð½  r; 0; Rn Þ the space of all square integrable functions ϕ : ½ r; 0-Rn which are absolutely continuous on ½ r; 0 and have square integrable first-order derivatives denoted by W with the norm Z 0 1=2 2 _ ‖ϕ‖W ¼ max jϕðθÞj þ jϕðφÞj dφ : θ A ½  r;0

r

Consider the following system: x_ ðtÞ ¼ f ðxðtÞ; xðt  σðtÞÞ; wðtÞÞ;

ð3Þ

where xðtÞA Rn is the system state, wðtÞ A Rp is an exogenous signal, f : Rn
Rn
Rp -Rn is a continuously differentiable function, f ð0; 0; 0Þ ¼ 0 and σðtÞ is the unknown piecewise-continuous time delay that satisfies 0 r σðtÞr r. Let xt ðθÞ ¼ xðt þ θÞ, θ A ½  r; 0. Then the initial condition of the state xðtÞ is supplemented as xt ðθÞ ¼ ϕðθÞA W, θ A ½ r; 0. Given a measurable locally essentially bounded input wðtÞ, the system (3) has a unique solution. The system (3) is uniformly globally ISS if there exists a locally Lipschitz with respect to the second and the third arguments functional V : R
W
L2 ð½  r; 0; Rn Þ-Rþ , such that the function vðtÞ ¼ Vðt; xt ; x_ t Þ is absolutely continuous for the measurable essentially bounded wðtÞ. If additionally there exist functions α1, α2 of class K1 , and functions α3, θ of class K such that _ r α2 ð‖ϕ‖W Þ; α1 ðjϕð0ÞjÞ r Vðt; ϕ; ϕÞ _ _ _ V_ ðt; ϕ; ϕÞr  α3 ðVðt; ϕ; ϕÞÞ for Vðt; ϕ; ϕÞZ θðjwðtÞjÞ; where _ ¼ lim sup 1 ½Vðt þ h; xtþh ðt; ϕÞ; x_ tþh ðt; ϕÞÞ  Vðt; ϕ; ϕÞ: _ V_ ðt; ϕ; ϕÞ h-0 h

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2.4. Control objective Consider the rigid spacecrafts (1) with the preceding properties. The aim of this paper is to design distributed adaptive controllers such that for all physically realizable initial conditions, the attitudes of the spacecrafts can be synchronized, i.e., for all i; jA I , lim qi  qj ¼ 0;

ð4Þ

lim ωi ¼ 0:

ð5Þ

t-1

t-1

3. Main results In this section, the concept of “sampled-data information flow” is first introduced, then distributed adaptive controllers are designed for attitude synchronization under the directed topology. It is assumed that all the attitude information is sampled and transferred according to a unified sampling sequence: 0 ¼ t 0 ot 1 o⋯ot k o⋯, and t k -1 as t-1. The sampling interval is defined as h ¼ t k  t k  1 . In this way, the attitude information is measured in a discrete-time fashion, such that qi ðt k Þ is updated at each sampling instance and kept constant during the sampling intervals. Define the sampled-data attitude error between the ith and the jth spacecraft as eij ðt k Þ≔qiðt k Þ qj ðt k Þ;

8 iA I ; jA N i :

ð6Þ

By introducing the following auxiliary variable: X si ≔q_ i þ γ j A N eij ðt k Þ; 8 i AI ;

ð7Þ

i

the distributed input for the ith spacecraft can be given as X X ^ i ðqi Þ τ i ¼  K i si  γ M eij ðt k Þ  τ^ di ; e_ ij ðt k Þ γ C^ i ðqi ; q_ i Þ jANi

¼  K i si þ Yðqi ; q_ i ; 

X

jANi

e_ ij ðt k Þ; 

X

jANi

eij ðt k ÞÞθ^ i ;

ð8Þ

jANi

^ i ðqi Þ and C^ i ðqi ; q_ i Þ are the estimates of the respective matrices where γ40 is a constant, M available with time t, τ^ di is the estimated external disturbance, θ^ i is the time-varying estimate of the spacecraft's parameters given by θi. Substituting Eq. (8) into Eq. (1) yields X X M i ðqi Þ_s i þ Ci ðqi ; q_ i Þsi ¼  K i si  Yðqi ; q_ i ;  eij ðt k ÞÞθ~ i ; ð9Þ e_ ij ðt k Þ;  jANi

jANi

where θ~ i ≔θi  θ^ i is the estimation error of parameters. θ^ i evolves as 0 1 X X _ eij ðt k ÞAsi ; e_ ij ðt k Þ;  θ^ i ¼  Λi Y T @qi ; q_ i ;  jANi

jANi

ð10Þ

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with Λi being a symmetric positive-definite matrix. Note that e_ ij ðt k Þ ¼ 0, since qi ðt k Þ is kept constant during the sampling intervals. Eqs. (8) and (10) are rewritten as 0 1 X τi ¼  K i si þ Y @qi ; q_ i ; 0;  eij ðt k ÞAθ^ i ; 0 _ θ^ i ¼  Λi Y T @qi ; q_ i ; 0; 

jANi

X

1

eij ðt k ÞAsi :

jANi

To this end, by defining qðϑÞ ¼ φðϑÞ A L2 ð½  h; 0; R3N Þ and qt ðϑÞ ¼ qðt þ ϑÞ, ϑ A ½ h; 0, where q denotes the column stack vector of qi , the following theorem is presented to summarize the main results. Theorem 1. Consider a group of N networked spacecrafts (1) with the sampled-data information flow (6), where the communication topology has a directed spanning tree. The attitude synchronization of the spacecraft formation flying can be achieved under the distributed control inputs (8), if there exist positive symmetric matrices P, Q A R6ðN  1Þ
6ðN  1Þ and positive constants γ, α3, μ, such that Πo0, where " # Π1 Π2 Π≔ ; n Π3 " #  2γPJ~ þ ðα3 þ μÞP γPJ~ ; Π1≔ n  ðQ  hðα3 þ μÞQÞ=h h i T Π 2 a P  γ J~ T Q0γ J~ Q ; " # Q  μI 6ðN  1Þ Π3≔ ; n  Q=h " # RmðJ  I 3 Þ  ImðJ  I 3 Þ J~ ≔ : ImðJ  I 3 Þ RmðJ  I 3 Þ Proof. Firstly, consider the following Lyapunov function: VðtÞ ¼

N N 1X 1X T sTi M i ðqi Þsi þ θ~ Λ  1 θ~ i : 2i¼1 2i¼1 i i

By Properties 1 and 2, the derivative of VðtÞ along the trajectory of Eq. (9) is given by V_ ðtÞ ¼

N X i¼1

¼

sTi M i ðqi Þ_s i þ

XN 1 XN T _ i ðqi Þsi þ sTi M θ~ Λ  1 θ~_ i i ¼ 1 i¼1 i i 2

XN

sT K s : i¼1 i i i

Since K i 40; it can be obtained that V_ ðtÞ r 0; which implies that si A L2 \ L1 . Next, by utilizing the input-delay approach, Eq. (7) can be rewritten as q_ ¼  γðL  I 3 Þqðt  ηðtÞÞ þ s;

ð11Þ

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where 0 r ηðtÞ≔t  t k r h and s denotes the column stack vector of si . Note that if the communication topology has a directed spanning tree, Eq. (11) can be further decoupled as y_ ¼  γðJ  I 3 Þyðt  ηðtÞÞ þ s~ 1 ;

ð12Þ

x_ ¼ s~ 2 ;

ð13Þ

where y and x denote the first 3ðN  1Þ rows and last 3 rows of ðU  1  I 3 Þq, s~ 1 and s~ 2 represent the first 3ðN  1Þ rows and last 3 rows of ðU  1  I 3 Þs, respectively. By similar analysis in [34], it can be verified that the attitude synchronization can be reached, if y-0 as t-1 and x converges to an equilibrium point as t-1. Separating the real and imaginary parts of Eq. (12) yields Rmð_y Þ ¼  γðRmðJ Þ  I 3 ÞRmðyðt  ηðtÞÞÞ þ γðImðJ Þ  I 3 ÞImðyðt  ηðtÞÞÞ þ s~ 1 ; Imð_y Þ ¼  γðRmðJ Þ  I 3 ÞImðyðt  ηðtÞÞÞ  γðImðJ Þ  I 3 ÞRmðyðt  ηðtÞÞÞ; which can be further rewritten in a compact form as ξ_ ¼  γ J~ ξðt  ηðtÞÞ þ sξ ; Z t _ ¼  γ J~ ξ þ γ J~ ξðφÞ dφ þ sξ ;

ð14Þ

t  ηðtÞ

where ξ≔½RmðyÞT ; ImðyÞT T and sξ ≔½~s T1 ; 0T . In order to analyze the ISS of Eq. (14), the following Lyapunov–Krasovskii function is chosen V~ ðt; ξt ; ξ_ t Þ ¼ V~ 1 ðt; ξt ; ξ_ t Þ þ V~ 2 ðt; ξt ; ξ_ t Þ; where V~ 1 ðt; ξt ; ξ_ t Þ≔ξT ðtÞPξðtÞ; Z t Z t T _ _ ~ V 2 ðt; ξt ; ξ t Þ≔ ξ_ ðϕÞQξðϕÞ dϕ dφ: th

φ

The following inequality holds: Z t T _ ðφ  t þ hÞξ_ ðφÞQξðφÞ dφ V~ 2 ðt; ξt ; ξ_ t Þ ¼ th Z t 2 _ r hλmax ðQÞ jξðφÞj dφ: th

Since λmin ðPÞjξj r V~ 1 ðt; ξt ; ξ_ t Þr λmax ðPÞjξj2 ; it follows that Z t 2 _ λmin ðPÞjξj2 r V~ ðt; ξt ; ξ_ t Þr λmax ðPÞjξj2 þ hλmax ðQÞ jξðφÞj dφ; 2

th

which means that α1 ðjξt ð0ÞjÞ r V~ ðt; ξt ; ξ_ t Þr α2 ð‖ξt ‖W Þ; where α1 ðχÞ ¼ λmin ðPÞjχj2 and α2 ðχÞ ¼ λmax ðPÞjχj2 þ hλmax ðQÞχ 2 are all of class K1 . The following S-procedure is applied to satisfy V~_ ðt; ξt ; ξ_ t Þ þ α3 V~ ðt; ξt ; ξ_ t Þo0;

ð15Þ

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for V~ ðt; ξt ; ξ_ t ÞZ jsξ j2 , where α3 40 is a constant. It can be obtained that Eq. (15) holds if there exists a constant μ40 such that Φ≔V~_ ðt; ξt ; ξ_ t Þ þ α3 V~ ðt; ξt ; ξ_ t Þ þ μðV~ ðt; ξt ; ξ_ t Þ  jsξ j2 Þo0: ð16Þ The derivative of V~ ðt; ξt ; ξ_ t Þ along the trajectory of Eq. (14) is given as   Z t _ T _ _ ~ ~ ~ V ðt; ξt ; ξ t Þ ¼ 2ξ P  γ J ξ þ γ J ξðφÞ dφ þ sξ t  ηðtÞ Z t T T _ þhξ_ Qξ_  dφ: ξ_ ðφÞQξðφÞ

ð17Þ

th

Consequently, it follows that   Z t _ Φ r 2ξT P  γ J~ ξ þ γ J~ ξðφÞ dφ þ sξ t  ηðtÞ Z t T T _ þhξ_ Qξ_  dφ þ ðα3 þ μÞξT Pξ ξ_ ðφÞQξðφÞ th Z t T _ dφ  μsTξ sξ : ξ_ ðφÞQξðφÞ þhðα3 þ μÞ th

By Jensen's inequality [35], it holds that Z t T _ dφ ξ_ ðφÞQð1  hðα3 þ μÞÞξðφÞ  th Z t Z t 1 T _ r ξ_ ðφÞ dφQð1 hðα3 þ μÞÞ ξðφÞ dφ: h t  ηðtÞ t  ηðtÞ T _ where It can be verified that Φr ςT Π ς þ hξ_ Qξ, 2 3 Π 11 γPJ~ P 6 7  ðQ  hðα3 þ μÞQÞ=h 0 Π ≔4 n 5; n n  μI 6ðN  1Þ

Π 11 ≔ 2γPJ~ þ ðα3 þ μÞP;  Z t T T T T _ ς≔ ξ ; ξ ðφÞ dφ; sξ : t  ηðtÞ

By Schur complement [36], it follows that if Πo0 holds, Φo0. By Lemma 1, it can be obtained that Eq. (14) is input-to-state stable with respect to the input sξ and the state ξ, which together ^ ^ _ eij ðt k ÞA with Eq. (13) implies that q; q; P L1 . Moreover, since M i ðqi Þ and C i ðqi ; q_ i Þ are bounded, it can be concluded that Yðqi ; q_ i ; 0;  j A N i eij ðt k ÞÞ is bounded. Then, it can be obtained from Eq. (9) that s_ i A L1 , such that V€ ðtÞ A L1 . By Barbalat's Lemma [37], it follows that V_ ðtÞ-0 as t-1, such that si -0 as t-1. As aresult, by ISS property, it can be verified that ξ-0 as t-1, which clearly implies that y-0 as t-1. For Eq. (13), since x_ -0 as t-1, it can be obtained that x can converge to an equilibrium point as t - 1. Finally, it follows from Eq. (7) that qi - qj as t-1, qi -0 as t-1, which implies that ωi -0 as t-1, for all i; j A I . This completes the proof.□ Remark 1. In some applications, a desired relative attitude is required, i.e., limt-1 qi  qj ¼ ρij and limt-1 ωi ¼ 0, for all i; jA I , where ρij denotes the relative attitude configuration. It can be

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found that Theorem 1 is also capable of this case. Note that ρij ¼ ρi  ρj , where ρi represents the attitude configuration of the ith spacecraft. Define q~ i ¼ qi  ρi , iA I . Then, following the same lines of proof in Theorem 1, the effectiveness of Theorem 1 can be directly verified. Remark 2. It is worth mentioning that the proposed strategy can be adopted to deal with the attitude synchronization problem with bounded but time-varying external disturbances. Suppose m that jτm di joΩ , m ¼ 1; 2; 3. Then, redefine 0 1 X X eij ðt k ÞAθ^ i ¼  γ C^ i ðqi ; q_ i Þ eij ðt k Þ: Y @qi ; q_ i ; 0;  jANi

jANi

Then, the distributed control input (8) can be redesigned as 0 1 X eij ðt k ÞAθ^ i ; τi ¼  K i si þ k i sgnðsi Þ þ Y @qi ; q_ i ; 0;  jANi m m where ki 40 is a constant and sgnðÞ denotes the sign function. By choosing km i 4Ω þ Δ , m Δ 40, m ¼ 1; 2; 3, the attitude synchronization of spacecraft formation flying can be achieved in the presence of time-varying external disturbances. Similar results can be found in [38].

4. Numerical example In this section, a numerical example is provided to show the effectiveness of the theoretical results. Consider a group of four spacecrafts with attitude kinematics and dynamics described by Eq. (1). The inertia matrices of the spacecrafts are set as J 1 ¼ diagf16; 12; 8g; J 3 ¼ diagf15; 10; 9g;

J 2 ¼ diagf15; 13; 11g; J 4 ¼ diagf16; 8; 14g;

and the external disturbances are set to be d1 ¼ ½2; 1:8; 2T ; d 2 ¼ ½1:8; 1:8; 1:5T ; d3 ¼ ½1:2; 1:5; 1:5T ; d4 ¼ ½1:2; 1:2; 1:3T : The Laplacian of 2 1 6 1 6 L¼6 4 0 0

the communication topology is given as 3 0 0 1 1 0 0 7 7 7: 1 1 0 5 0

1

1

All the parameters are set to be 30–80% of accuracy of their real values. The initial values of qi are randomly chosen within ½ 3; 3 and the initial values of ωi are set as zero. The parameters γ, α3, μ are given as 0.2, 0.1, 0.1. The control gains K i and Λi are given as 25I 3 and 5I 3 , respectively. By solving Πo0, it can be obtained that the maximum of h is 1.2654 s. By choosing h ¼ 0:5, it can be seen from Figs. 1–3 that the attitude synchronization is achieved in the sense of Eqs. (4) and (5). In order to show the advantage of the proposed method from the energy point of view, it is assumed that each information exchange between the two spacecrafts consumes 1 unit energy and the continuous-time communication interval is considered as 0.01 s. For simplicity, the

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q

i3

Spacecraft 1 Spacecraft 2 Spacecraft 3 Spacecraft 4

4 3 2 1 0 -1 -2 -3 -4 3

2

1

0 q

i2

-1

-2

-3

-2

-3

-1

2

1

0 q

4

3

i1

Fig. 1. The attitudes of spacecrafts. 0.8 0.6 0.4

i

0.2 0 -0.2 -0.4 -0.6 -0.8

0

10

20

30

40

50

60

Time (s)

Fig. 2. The angular velocities of spacecrafts.

communication topology is set to be a ring configuration. Fig. 4 shows the comparison of communication energy consumption between the proposed sampled-data communication and continuous-time communication strategies. It can be found that the communication energy consumption can be significantly reduced by the proposed method, such that the working time of the spacecrafts can be effectively increased. In addition, it should be pointed out that fast sampling rates will increase the communication network load and need more energy supply, such that this trade-off should be considered in the formation flying design. 5. Conclusion In this paper, the attitude synchronization problem of spacecraft formation flying under directed topology was addressed. Based on the introduction of sampled-data information flow between

C. Ma et al. / Journal of the Franklin Institute 352 (2015) 2796–2809

2807

6 4

eij

2 0 -2 -4 -6 -8

0

10

20

30

40

50

60

Time (s)

Fig. 3. The sum of relative attitude errors of spacecrafts.

Energy consumption (Unit)

Continuous-time communication Sampled-data communication

12000 10000 8000 6000 4000 2000 0 12 10 Nu mb er o 8 f sp a ce cra

6 fts (N)

4

0

2

4

6 ( Time

8

10

s)

Fig. 4. The communication energy consumption of the proposed sampled-data communication and continuous-time communication strategies.

neighboring spacecrafts, adaptive controllers were designed in a totally distributed fashion, which allow for unknown parameters and external disturbances. It was shown that the attitude synchronization of spacecraft formation flying can be guaranteed under the established conditions. Furthermore, compared with the common continuous-time commutation strategy, the formation performance can be effectively improved and the heavy communication network burden can be significantly reduced. Finally, the effectiveness and potential advantages of the proposed strategy were demonstrated through a simulation example. Future work includes extending the results in this paper to the cases where the sampling period is time-varying or exhibits random behaviors, and the cases with switching topologies.

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