2012-IJRNC-Decentralized Sliding-Mode Control For Attitude Synchronization In Spacecraft Formation
INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL
Int.J.Robust.Nonlinear Control(2012)
Published online in Wiley Online Library(https://www.360docs.net/doc/3e18486202.html,).DOI:10.1002/rnc.2812
Decentralized sliding-mode control for attitude synchronization in
spacecraft formation
Baolin Wu1,Danwei Wang1,*,?and Eng Kee Poh2
1Nanyang Technological University,639798,Singapore
2DSO National Laboratories,118230,Singapore
SUMMARY
This paper addresses attitude synchronization and tracking problems in spacecraft formation in the pres-ence of model uncertainties and external disturbances.A decentralized adaptive sliding mode control law is proposed using undirected interspacecraft communication topology and analyzed based on algebraic graph theory.A multispacecraft sliding manifold is derived,on which each spacecraft approaches desired time-varying attitude and angular velocity while maintaining attitude synchronization with the other spacecraft in the formation.A control law is then developed to ensure convergence to the sliding manifold.The sta-bility of the resulting closed-loop system is proved by application of Barbalat’s Lemma.Simulation results demonstrate the effectiveness of the proposed attitude synchronization and tracking methodology.Copyright ?2012John Wiley&Sons,Ltd.
Received11May2011;Revised1December2011;Accepted25January2012
KEY WORDS:attitude synchronization;satellite formation;sliding mode control
1.INTRODUCTION
Attitude synchronization has attracted much attention recently[1–21]because of its importance in realizing spacecraft formation?ying missions.For instance,it is required to maintain relative orientation among spacecraft precisely during formation maneuvers for interferometry application. In interspacecraft laser communication mission,the spacecraft is also required to point to each other with desired relative attitude.Various approaches have been proposed to study the problem of attitude synchronization.
A leader–follower approach is developed for attitude synchronization in[1–4],where the follower spacecraft maintain relative attitude with respect to the leader spacecraft.However,the leader–follower approach results in a centralized control law,and the leader spacecraft is potentially a single point of failure in the formation.Centralized and decentralized implementation of a virtual structure coordination strategy was developed for attitude synchronization in[5,6],respectively.The cross-coupling error concept was proposed for attitude synchronization in[7].In[8–13],a behavioral approach was used for attitude synchronization.In the behavior-based approach,each spacecraft has several basic behaviors,such as station-keeping and formation-keeping.Each behavior gener-ates a control input,and the control action for each spacecraft is a weighted average of control for each behavior.The behavioral approach lends itself naturally to a decentralized implementation. Graph-theoretical approach has been actively studied for cooperative control of a multiagent sys-tem using limited local interaction[22,23],and it has been investigated for attitude synchronization
*Correspondence to:Danwei Wang,School of Electrical and Electronic Engineering,Nanyang Technological University, Nanyang Avenue639798,Singapore.
?E-mail:edwwang@https://www.360docs.net/doc/3e18486202.html,.sg
B.WU,D.W ANG AND E.K.POH
problems in[14–18].A passivity approach was applied for attitude synchronization without using angular velocity measurements in[8,19–21].
This paper presents a decentralized control law for satellite attitude synchronization and tracking in the presence of model uncertainties and external disturbances.Sliding mode control is an ef?cient approach to control spacecraft attitude when there are model uncertainties and external disturbances [24–28].Here,a novel decentralized adaptive sliding mode control law is proposed and analyzed based on algebraic graph theory for attitude synchronization and tracking in spacecraft formation with undirected interspacecraft communication links.A multispacecraft sliding manifold is?rst pro-posed,on which each spacecraft approaches the desired time-varying attitude and angular velocity while achieving attitude synchronization with other spacecraft in the formation.Then,a control law is developed to ensure the reaching and sliding conditions are satis?ed.The stability of the resulting closed-loop systems are proved by virtue of Barbalat’s Lemma.Finally,a continuous control law is introduced to avoid the chattering of control input of the proposed control law.
Sliding mode control has been studied for attitude synchronization in[12]and[18]. Reference[18]addressed distributed?nite-time attitude containment control for multiple rigid bod-ies by use of Modi?ed Rodriguez Parameters for attitude representation.Minimal representation of attitude dynamics,such as Modi?ed Rodriguez Parameters or Euler angles,contain singulari-ties,and are hence not suited for the design of global control algorithms.Reference[12]presented a decentralized sliding mode controller for the attitude synchronization and tracking problem.The synchronization and tracking problem is addressed by use of quaternion for attitude representation in this paper.The proposed decentralized sliding mode control law is based on a novel multispacecraft sliding manifold on which attitude tracking and synchronization can be achieved.
This paper is organized as follows.In Section2,spacecraft attitude dynamics is introduced and mathematical preliminaries of graph theory are reviewed.Subsequently,Section3proposes a novel multispacecraft sliding manifold on which each spacecraft approaches the desired time-varying attitude and angular velocity while achieving attitude synchronization among spacecrafts in the for-mation.Thereafter,Section4presents a decentralized adaptive sliding mode control law to ensure that the error dynamics reach the sliding mode vector even in the presence of satellite inertia matrix uncertainties and external disturbances.Finally,Section5presents simulation results to validate the proposed attitude synchronization and tracking solution.
2.DYNAMICS AND MATHEMATICAL PRELIMINARIES
2.1.Spacecraft attitude dynamics
The rigid spacecraft attitude dynamics is described in this Section.The unit quaternion,Ná,represents the orientation of a body-?xed frame B with respect to an inertial frame I and is de?ned as
NáD ?cos.'=2/
e sin.'=2/
D
?á
á
,(1)
where e is the Euler axis,'is the Euler angle,á2R3,á02R are the vector part and scalar part of the quaternion Ná,respectively,and satisfy the following constraint:
áTáCá20D1.(2) The product of two unit quaternion Ná1and Ná2is de?ned by
Ná1Ná2D "
á0,1á2Cá0,2á1Cá
1
á2
á0,1á0,2 áT
1
á2
#
,(3)
DECENTRALIZED SLIDING-MODE CONTROL FOR ATTITUDE SYNCHRONIZATION
which is also a unit quaternion.The notation a for a vector a D ?a 1
a 2a 3 T is used to denote the skew-symmetric matrix.
a D 240 a 3a 2
a 3
0 a 1 a 2a 10
35.(4)
The conjugate of the unit quaternion N áis de?ned by N á D á0, áT T .
The dynamics of a rigid spacecraft are described as [30]J P !D ! J!C u C d
(5)P áD 12 á C á0I 3 !(6)
P á0D 12áT !,(7)
where !2R 3denotes the angular velocity of the spacecraft with respect to an inertial frame I and expressed in the body-?xed frame B ,J 2R 3 3is the constant,positive de?nite inertia matrix of the spacecraft and expressed in the body-?xed frame B ,u 2R 3denotes the vector of control torque,d 2R 3denotes the vector of external disturbances,I 3denotes a 3 3identity matrix.
In the case of tracking a desired attitude motion,the attitude tracking problem is formulated similarly as in the related work [29].The desired motion of the spacecraft is speci?ed by the atti-tude of the reference frame D .The target attitude of the spacecraft in the reference frame D with respect to the inertial frame I is described by unit quaternion N ád D ád 0,.ád /T T ,which satis?es .ád /T ád C .ád 0/2D 1.Let !d 2R 3be the desired angular velocity of reference frame D with respect to the inertial frame I .Let error quaternion N q D q 0,q T T represent the orientation error of the body-?xed frame B with respect to the reference frame D ,and satis?es q T q C q 20D 1.N q is related to N ád and N áby quaternion multiplication N q D N
ád N á.The angular velocity error Q !of frame B with respect to frame D is then represented by Q !D ! R.N q/!d ,R.N q/2SO.3/is the rotation matrix from the reference frame D to the body-?xed frame B and is given by
R.N q/D q 20 q T q I 3C 2qq T 2q 0q ,(8)
where SO.3/is the Lie group of orthogonal matrices with determinant 1.It follows from Ref.[30]that R T R D 1,k R k D 1,det .R/D 1and P R.N q/D Q ! R.N q/.
The rigid spacecraft attitude tracking error dynamics is described as follows [29]:
J i P Q !i D ! i J i !i C J i Q ! i R.N q i /!d i R.N q i /P !d i áC u i C d i ,i D 1, ,n
(9)P q i D 12 q i C q 0i I Q !i ,i D 1, ,n (10)
P q 0i D 12q T i Q !i ,i D 1, ,n ,(11)
where superscript i denotes the i th spacecraft in a formation.
The attitude tracking problem of the i th spacecraft is solved if lim t !1q i .t/!0and lim t !1Q !i .t/!0.It can be seen that the attitude tracking problem is equivalent to an asymptotic stabilization problem for Q !i and q i .
B.WU,D.W ANG AND E.K.POH
The following assumptions are made about attitude dynamics systems:
(A1)Let J i D N J i C J i,where N J i, J i are the nominal part and uncertain part of the inertia matrix of the i th spacecraft,respectively.The uncertainty of inertia matrix J i is assumed to satisfy k J i k6 i;
(A2)The external disturbances d i satisfy k d i k6 i;
(A3)The desired angular velocity of spacecraft with respect to the inertial frame I,denoted by
!d i ,and its time derivative P!d
i
are assumed to be bounded.
(A4)The control torque u i satisfy k u i k6 i,where i, i,and i,i D1, ,n are unknown non-negative but constant,and k k denotes the standard Euclidean vector norm and induced matrix norm.
2.2.Algebraic graph theory
A directed graph G n consists of a?nite set of vertices,denoted C,and a set of arcs<,where a D.?,ˇ/2 The adjacency matrix of G n,denoted A.G/is a square matrix of size j C j with entries ′ a i,j>0if ?j,?i 2< a i,j D0otherwise ?i,?j2C ,(12) where,the non-negative a i,j is subsequently used as the control weight parameter for attitude synchronization between the i th and j th spacecraft.Note that a i,i D0from(12). The in-degree matrix of G n is the diagonal matrix D with diagonal entries d i,i D n X j D1,j¤i a ij,i D1, ,n.(13) Following[31],the weighted Laplacian L2R n n of the graph G n is de?ned as L D D A.G/.(14) The following lemmas will be employed in the subsequent sections to derive the proof of stability. Lemma1([31]:) For a directed digraph G n with N vertices,all eigenvalues of the weighted Laplacian L have non-negative real part(use Gershgorin’s theorem).Moreover,if G n is undirected,then all the eigenvalues of L are real. Lemma2([32]:) Suppose that M2R m m,and N2R n n.We have the following results:(i)If M and N are symmetric,so is M?N,where?denotes the Kronecker product;and(ii)let 1, , m be the eigenvalues of M and 1, , n be those of N.Then the eigenvalues of M?N are i j, i D1, ,m,j D1, ,n. 3.MULTISPACECRAFT SLIDING MANIFOLD In this section,the multispacecraft sliding mode vector is developed to achieve the attitude synchronization and tracking in spacecraft formation. The following assumption is made about the interspacecraft communication links: (A5)The interspacecraft communication links are assumed to be undirected. DECENTRALIZED SLIDING-MODE CONTROL FOR ATTITUDE SYNCHRONIZATION The multispacecraft sliding mode vector is de?ned as S D?s1, ,s n T,(15) where s i D b i.Q!i C C q i/C n X j D1 a ij Q!i Q!j C C q i q j ,i D1, ,n,j¤i(16) with C being a positive de?nite constant matrix,scalar b i>0is the control weight parameter for attitude tracking of the i th spacecraft(station-keeping behavior),scalar a ij>0de?ned in(12)is the control weight parameter for interspacecraft attitude synchronization between the i th and j th spacecraft(formation-keeping behavior).The proposed multispacecraft sliding manifold enables the designer to prioritize attitude tracking and attitude synchronization by judiciously choosing b i and a ij.The attitude synchronization weights a ij are restricted to be a ij D a j i.Note that this restriction is valid with assumption(A5). Remark1 In the multispacecraft sliding mode vector de?ned in(15)and(16),in addition to attitude error and angular velocity error of each individual spacecraft,the relative attitude errors and relative angular velocity errors among spacecrafts are also included.Hence,under the sliding mode control pro-posed in the next section,each spacecraft approaches its desired attitude and angular velocity while achieving attitude synchronization among spacecrafts in the formation.This gives rise to the concept of attitude synchronization and tracking. Using the Kronecker product,the multispacecraft sliding vector in(15)can be rewritten as S D?.L C B/?I3 Q C Q C Q ,(17) where D?Q!1, ,Q!n T,Q D?q1, ,q n T,B D diag?b1, ,b n ,Q C D diag?C, ,C 2 R3n 3n,L is the weighted Laplacian matrix corresponding to the interspacecraft bidirectional communication topology,control gain a ij in(16)for interspacecraft attitude synchronization are the weights in L. The multispacecraft sliding mode surface is then de?ned as S D0or ?.L C B/?I3 Q C Q C Q D0.(18) The following theorem de?nes the condition under which the attitude synchronization and tracking will be satis?ed. Theorem1 Consider spacecraft formation attitude tracking systems described by(9)–(11),if the interspacecraft communication links are bidirectional,then on the multispacecraft sliding mode surface S D0,the spacecraft within the formation will converge to the desired attitude trajectory. Proof Because the interspacecraft communication links are assumed to be bidirectional,all eigenvalues of the weighted Laplacian matrix L are non-negative through Lemma1.Moreover,B is a diagonal matrix with all positive elements in the diagonal.Thus,all eigenvalues of L C B are positive.With Lemma2,all eigenvalues of.L C B/?I3are positive.Furthermore,L is symmetrical from the restriction a ij D a j i.Thus,.L C B/?I3is also symmetrical from Lemma2.Consequently,it follows that.L C B/?I3is positive de?nite and nonsingular. Because.L C B/?I3is nonsingular,on the sliding mode surface S D0,(18)yields Q C Q C Q D0,(19) B.WU,D.W ANG AND E.K.POH which is equivalent to Q!i C C q i D0,i D1, ,n.(20) As proved in[33],the above equation implies that lim t!1k Q!i k D lim t!1 k q i k D0,i D1, ,n.(21) It can be concluded that on the multispacecraft sliding mode surface S D0,the attitude error and angular velocity error of each spacecraft will converge to zero as t!1. 4.DECENTRALIZED ADAPTIVE SLIDING-MODE CONTROL DESIGN In this section,a decentralized adaptive sliding mode control law is?rst introduced.Then,a continuous control law is introduced to avoid the chattering of control input of the proposed control law. 4.1.Control law design On the basis of the proposed multispacecraft sliding manifold in Section3,this Section formulates a sliding mode control law to ensure that the error dynamics reach the sliding mode vector. To develop the control law,the following equations are derived from(9)–(11): P Q! i C C P q i D h i C i C N J 1 i u i,i D1, ,n(22) with h i.t/ D N J 1 i ! i N J i!i C Q! i R.N q i/!d i R.N q i/P!d i á C 1 2 C q i C q i,0I Q!i(23) i.t/ D N J 1 i h d i J i P Q!i ! i J i!i C J i Q! i R.N q i/!d i R.N q i/P!d i ái ,(24) i.t/represents the inertia matrix uncertainties and external disturbances.Under the assumptions (A1)–(A4),it can be veri?ed that i.t/is bounded by the following function: k i.t/k16?i,i D1, ,n,(25) where?i are positive constant numbers,and k k1denotes the induced matrix1-norm. Sliding mode control is deployed in this section for attitude synchronization and tracking in the presence of model uncertainties and external disturbances.However,in conventional sliding mode control laws[24–28],an important assumption is that inertia uncertainties and external disturbances are bounded and that their bounds are available to the designer.These bounds provide important information to guarantee the stability of the closed-loop system.However,bounds on the uncer-tainties of spacecraft are not easily obtained because of the complexity of the structure of the uncertainties.In particular,the magnitude of external disturbances cannot be simply estimated because of existence of various environmental disturbances such as gravity-gradient torque,aerody-namic torque,earth magnetic torque,and solar radiation torque.In many practical situations,even if the bounds can be estimated,they may be conservative.The implementation of the control law, based on these conservative bounds,may result in impractically large control authority and con-trol chattering.Control chattering is highly undesirable in spacecraft mission,because it implies extremely high control activity and may excite neglected high-frequency dynamics.Therefore,a simple methodology is required to determine the bounds on uncertainties.The adaptive approach introduced below offer a simple and effective solution to this problem. Let O?i denote the estimate of?i.Then,the estimation errors are de?ned as Q?i D O?i ?i. Now,consider the simple adaptation laws for the upper bound of the norm k i.t/k1such that P Q?i D? i k s i k1,(26) where?i,i D1,:::,n are positive scalars. DECENTRALIZED SLIDING-MODE CONTROL FOR ATTITUDE SYNCHRONIZATION Because ?i are assumed to be constant,P O ?i D P Q ?i .Then the adaptive parameters can be obtained by integrating O ?i .t/ D O ?0i C ?i Z t t 0 k s i k 1d t , (27)where O ?0i are the initial values of O ?i .By choosing appropriate O ?0i and ?i ,the rate of parameter adaptation can be adjusted. The control input for the i th spacecraft is proposed as u i D N J i ?h i C K i s i C O ?i sgn.s i / ,i D 1, ,n ,(28) where N J i is the nominal inertia matrix of the i th spacecraft,K i is the positive de?nite gain matrix,h i is de?ned in (23),s i is the component of multispacecraft sliding mode vector S ,which is de?ned in (16),and sgn .s i / D sgn .s i ,1/sgn .s i ,2/sgn .s i ,3/ T ,s i ,j ,j D 1,2,3is the j th component of s i ,sgn . /denotes the sign function,that is, sgn .x/D 8<:1, x >00,x D 0 1,x <0 .The following theorem gives the condition for the existence of the multispacecraft sliding mode surface (18)for spacecraft formation attitude tracking systems described by (9)–(11). Theorem 2 Consider spacecraft formation attitude tracking systems described by (9)–(11)with the decentralized adaptive sliding mode control law (28)and the parameter adaptation law (26).If the assumptions (A1)–(A5)are valid,then the multispacecraft sliding mode surface S D 0de?ned in (15),(16)will be reached asymptotically. Proof The candidate Lyapunov function is chosen as V D V 1C V 2 (29) with V 1D 12 S T ?.L C B/?I 3 1S V 2D 12n X i D 1? 1i Q ?2i .As shown in the previous Section,.L C B/?I 3is positive de?niteˇc ˇn so ?.L C B/?I 3 1is positive de?nite.Thus,V is positive de?nite and radially unbounded. Taking the ?rst derivative of V 1and using(17)yields P V 1D S T ?.L C B/?I 3 1P S D S T ?.L C B/?I 3 1?.L C B/?I 3 2 64P Q !1C C P q 1...P Q !n C C P q n 375D n X i D 1s T i P Q !i C C P q i á. (30) B.WU,D.W ANG AND E.K.POH Substituting(22)into the above equation leads to P V 1D n X i D1 s T i h i C i C N J 1 i u i .(31) Substituting the control input(28)into the above equation and applying(25)yields P V 1D n X i D1 s T i K i s i s T i.O?i sgn.s i//C s T i i D n X i D1 s T i K i s i O?i k s i k1C s T i i 6 n X i D1 s T i K i s i O?i k s i k1C k s i k1k i k1 6 n X i D1 s T i K i s i O?i k s k1C?i k s i k1 D n X i D1 s T i K i s i Q?i k s i k1 . (32) Taking the?rst derivative of V2and adopting(26)yields P V 2D n X i D1 ? 1i,0Q?i P Q?i D n X i D1 Q?i k s i k1 .(33) Adding the above equation to(32)leads to P V6 n X i D1 s T i K i s i.(34) Therefore,it follows that s i2L1,and Q?i2L1.Consequently,we have u i2L1with(28) and assumption(A3).It follows that P Q!i,P q i hence P s i are all bounded from(9)–(11).Integrating P V gives the results that s i2L2.Hence,using the corollary of Barbalat’s Lemma,it follows that lim t!1s i.t/D0,i D1, ,n.Thus,lim t!1 S.t/D0. Remark2 There is no extra restriction on parameters b i>0,a D ij a j i>0in(16)unlike[9,12]and[15].There- fore,the proposed attitude synchronization and tracking scheme will enable the designer to prioritize between station-keeping behavior and formation-keeping behavior.For example,if one wants to pri-oritize formation-keeping behavior,a ij should be increased while b i should be decreased or remain invariant. Remark3 In the proposed control law,the desired attitude of each spacecraft with respect to inertial frame I is not restricted to be the same.Thus,the given relative attitude among spacecrafts can be maintained. Note that Nád i,j D Nád j á Nád i de?nes the desired relative attitude of the i th spacecraft with respect to the j th spacecraft,where Nád i and Nád j denote the desired attitude of the i th and j th spacecraft with respect to inertial frame I,respectively. DECENTRALIZED SLIDING-MODE CONTROL FOR ATTITUDE SYNCHRONIZATION Remark 4 Reference [12]presents a sliding mode controller for attitude synchronization and stabilization.Unlike [12],the approach adopted in this paper is based on graph theory and a judiciously selected multispacecraft sliding mode vector.Also,the designed controller is amenable for satellite attitude synchronization and tracking application. 4.2.Continuous control law The proposed control law in the previous subsection is discontinuous across the surface S.t/,and may lead to control chattering.This situation can be remedied by smoothing out the control dis-continuity in a thin boundary layer neighboring the switching surface.This can be accomplished by replacing the sign function in the control law (28)with a saturation function de?ned as sat .x/D 8? if x= >1x= if 1 ,(35)where is the boundary layer thickness.As tends to be zero,the performance of control law with this boundary layer can be made arbitrarily close to that of the original control law (28).The prac-tical advantages of a control law with this boundary layer may be signi?cant,although it will lead to small terminal tracking error.However,it should be pointed out that in this case the estimated gains Q ?i may become unbounded in the boundary layer because the restriction to the sliding surface cannot always be achieved. To address this problem,the adaptive sliding mode control law given in (28)is modi?ed as u i D N J i h i C K i s 0i C O ?i sat .s i / ,i D 1, ,n ,(36) where sat .s i / D sat .s i ,1/sat .s i ,2/sat .s i ,3/ T ,s 0i D s 0i ,1,s 0i ,2,s 0i ,3 T with s 0i ,j D s i ,j i sat s i ,j ,i D 1, ,n ,j D 1,2,3is a measure of the algebraic distance of the current state to the boundary layer [34]. The adaptation laws (26)are modi?ed as P Q ?i D ?i s 0i 1.(37) The following Lyapunov function is de?ned to prove the convergence to the boundary layer V 0 D 12 S 0 T ?.L C B/?I 3 1S 0C 12n X i D 1? 1i Q ?2i ,(38) where S 0D s 01, ,s 0n .Noting that P S 0D P S outside the boundary layer,while S 0D 0inside the boundary layer leads to P V 06 n X i D 1 s 0i T K i s 0 i (39) so that P V 060outside the boundary layer.Finally,(38)implies that P V 0D 0inside the boundary layer,which shows that P V 060is valid everywhere and thus guarantees further that states eventually converge to the boundary layer. It can be shown that the system is globally uniformly ultimately bounded inside the boundary layer [34]. Remark 5 As evident from (37)that adaptation ceased as soon as the boundary layer is reached.This prevents undesirable long term drift found in many adaptive schemes,and provides a consistent rule on when to stop adaptation [34]. B.WU,D.W ANG AND E.K. POH Sat 1 Sat 2 Sat 4 Sat 3 Figure 1.Bidirectional communication topology. 5.ILLUSTRATIVE EXAMPLE Simulation results are presented in this section to illustrate the performance and stability character-istics of the proposed decentralized sliding mode control law.A scenario with four spacecrafts is considered in the simulation.The interspacecraft bidirectional communication topology is shown in Figure 1.The corresponding weighted Laplacian matrices for this topology is L D 2641 100 12 100 12 100 12 375 The actual inertia matrices of the spacecrafts are assumed to be as follows (the unit of which is kg m 2/J 1D 242020.92170.50.90.51535,J 2D 242210.91190.50.90.51535,J 3D 241811.51150.51.50.51735,J 4D 2418111200.510.515 35. To validate the robustness against model uncertainties and external disturbances of the proposed control laws,it is assumed that the nominal inertia matrices of the spacecrafts are N J 1D N J 2D N J 3D N J 4D diag ?20,20,20 T ákg m 2and different sinusoidal-wave disturbances as in (40)are added to each spacecraft randomly.The external disturbances used in the simulation are far worse than those typically observed in practice. d 1.t/D ?0.10sin .0.4t/,0.05cos .0.5t/,0.08cos .0.7t/ T Nm d 2.t/D ?0.06cos .0.4t/,0.10sin .0.5t/,0.05sin .0.7t/ T Nm d 3.t/D ?0.08sin .0.4t C pi =4/,0.06cos .0.5t C pi =4/,0.07cos .0.7t C pi =4/ T Nm d 4.t/D ?0.06cos .0.4t C pi =4/,0.08sin .0.5t C pi =4/,0.10sin .0.7t C pi =4/ T Nm .(40) The initial angular velocity errors of all spacecrafts are chosen to be zeros,and the initial attitude tracking errors are chosen as N q 1.0/D 0.89860.4 0.10.15 T ,N q 2.0/D 0.8888 0.20.10.4 T ,N q 3.0/D 0.80620.1 0.50.3 T ,N q 4.0/D 0.8426 0.4 0.20.3 T . The initial desired attitudes are N ád i .0/D 1000 T ,i D 1,2,3,4.The time-varying desired angular velocities of the spacecrafts are identical as follows: !d i .t/D 0.1cos .t=40/ 0.1sin .t=50/ 0.1cos .t=60/ T i D 1,2,3,4. DECENTRALIZED SLIDING-MODE CONTROL FOR ATTITUDE SYNCHRONIZATION The controller parameters are chosen as C D I3,K i D0.01I3,b i D1,'i D0.01,K i D0.01I3,and the parameters of the adaptation law(37)are chosen as?i D0.1,and O?0 i D0,where i D1,2,3,4. The attitude error,angular velocity error and control torque of the?rst spacecraft are shown in Figures2–4,respectively.For ease of interpretation,attitude errors are expressed by Euler angles converted from unit quaternion.It is observed that attitude tracking of the?rst spacecraft is achieved in spite of the presence of external disturbances and model uncertainties.The closed-loop system Figure2.Attitude tracking error of the?rst spacecraft. Figure3.Angular velocity error of the?rst spacecraft. Figure4.Control torque of the?rst spacecraft. B.WU,D.W ANG AND E.K.POH Figure5.Adaptive parameter O?1of the?rst spacecraft. Figure6.Relative attitude errors of the second spacecraft with respect to the?rst spacecraft. Figure7.Relative attitude errors of the?rst spacecraft with respect to the fourth spacecraft. responses of the other three spacecrafts are similar to those of the?rst spacecraft,and are not plot-ted here because of space constraint.The adaptive parameter O?1is bounded as shown in Figure5, thus the proposed adaptation laws(37)is veri?ed.The relative attitudes of the second spacecraft with respect to the?rst spacecraft,and the?rst spacecraft with respect to the fourth spacecraft are plotted in Figures6and7,respectively.Relative attitude errors between other pairs of spacecrafts are similar to those of the above two. DECENTRALIZED SLIDING-MODE CONTROL FOR ATTITUDE SYNCHRONIZATION Figure8.Attitude tracking error of the?rst spacecraft without coupling between neighbors. Figure9.Relative attitude error from the?rst to the second spacecraft without coupling between neighbors. Figure10.Relative attitude error from the fourth to the?rst spacecraft without coupling between neighbors. For comparison,Figures8–11show attitude error,relative attitude errors and control torque when there is no coupling between neighbors[i.e.,a ij D0,i,j D1,2,3,4in(16)].It is observed from Figures7and10that relative attitude errors among spacecrafts,especially steady-state relative atti-tude errors,in the case with coupling between neighbors are reduced as compared with the case without coupling.Furthermore,even the relative attitude errors between spacecrafts without direct interspacecraft communication are reduced,such as relative attitude errors between the?rst and B.WU,D.W ANG AND E.K.POH Figure11.Control torque of the?rst spacecraft without coupling between neighbors. fourth spacecraft as seen from Figures8and10.Although steady-state relative attitude errors are greatly reduced in the case with coupling,the control torques in the steady-state process for the two cases are almost identical,as can be seen from Figures4and11.Thus,simulation results val-idate the effectiveness of the proposed decentralized adaptive sliding-mode control law for attitude synchronization and tracking. 6.CONCLUSIONS The requirement of attitude synchronization and tracking in spacecraft formation is a challenging and important objective for many practical satellite missions.In this paper,a decentralized adap-tive sliding-mode control law is proposed to achieve this objective for a satellite formation using undirected intersatellite communication link.The design methodology uses unit quaternion parame-terization and a carefully selected multispacecraft sliding mode vector,which includes attitude error, angular velocity error of each individual spacecraft,relative attitude errors,and relative angular velocity errors among spacecrafts.Numerical simulations are performed to validate the effectiveness of the proposed control law in the presence of severe satellite model uncertainties and disturbances. Simulation results of the control performance demonstrate that each individual spacecraft converges to the desired attitude and angular velocity.Furthermore,relative attitude errors among spacecrafts, especially steady-state relative attitudes errors,are reduced by including the relative attitude errors and relative angular velocity errors in the proposed control law. ACKNOWLEDGEMENTS Funded in part under Project Agreement No.POD0513235with Defense Science&Technology Agency, and Project Agreement No.DSOCL10004with DSO National Laboratories,Singapore. 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