Distributed Adaptive Attitude Synchronization of Multiple Spacecraft

Distributed Adaptiv e A ttitude Sync hronization of Multi ple Spacecraft ∗ Zhongkui Li and Zhisheng Duan Abstract : This pap er addresses the distributed attitude synchronization problem o f multiple spac ecraft with unknown inertia matrices. Two distributed adaptive controllers ar e pr o p o sed for the cases with and without a virtua l leader to which a time-v arying refere nce attitude is a ssigned. The firs t controller achiev es attitude synchronization for a gr oup o f spacecr aft with a leaderles s co mm unication to po logy having a directed spanning tree. The second controller g ua rantees that all spa c ecraft track the reference attitude if the virtual leader ha s a directed pa th to all other spa c e craft. Sim ulation examples are pr esented to illustrate the effectiveness of the results. Keyw ords : attitude synchronization, distr ibuted control, adaptive control, multi-agent system . 1 In tro duc tion In recen t yea rs , consensus and co op erativ e control of multi-v ehicle systems ha ve attracted com- p elling atten tion from v arious scientific comm unities. A large b o dy of theoretical adv ances has b een r ep orted, see [9, 8 , 11, 13, 14, 10 ] and references th er ein. In the aforemen tioned works, the agen t dyn amics are restricted to b e a single, d ou b le integ rators or linear systems. The results prop osed in th ese pap ers b ecome quite limited when dealing with the attitude synchronizatio n problem of multiple spacecraft, wh ic h is more c hallenging than the consensu s of v ehicles w ith in tegrator dynamics, due to the nonlinearity of the attitude dyn amics. A ttitude con trol of a single r igid b o dy has b een extensiv ely stu died, e.g., in [22, 20, 2, 23]. A leader-follo w er strategy is pr op osed in [21] for attitude sync hronization of multiple s p acecraft. Decen tralized con trol la ws using the b eha vioral appr oac h are p resen ted for attitude synchro- nization in [15, 7], where the comm un ication top ology among spacecraft is assumed to b e a bidirectional ring. Adaptiv e consensu s proto cols are prop osed in [3] for multiple manipulators with uncertain dynamics. Co op erativ e attitude cont rol of m ultiple rigid b o dies with a leader- follo w er communicatio n top olog y is considered in [5]. In [4], contract ion analysis theory is used to deriv e attitude syn c hronization strategies with global exp onen tial con verge nce for a group of spacecraft with a bidir ectional comm un ication top ology . Distributed cont rol la ws without v elo cit y measur emen ts are stu d ied for attitude sync h r onization of m ultiple spacecraft in [1 ] by use of qu aternion r epresen tation w hile in [17] by use of Mo d ified Ro driguez Paramete rs (MRPs) for attitude r epresen tation. The resu lts in [1, 17] are applicable to general undirected comm un i- cation top ologies. T he distributed attitude tracking pr oblem is addressed in [16] for sp acecraft whose inf ormation exchange graph can b e simplified to a graph with only one n o de. Motiv ated b y [17, 4, 3], this pap er concerns the distributed adaptive attitude synchronizatio n ∗ Zhongkui Li is with the Sch o o l of Automation, B eijing Institute of T ec hnology , Beijing 1 00081 , P . R. China (E-mail: zhongkli@g mail.com). Zhisheng Duan is with with State Key Lab for T urbulence and Complex Sys tems, Depa rtment of Mechanics and Aerospace Engineering, College o f E ngineering, Peking Univ er sity , B e ijing 100 8 71, P . R. China (E-mail: dua nz s @pku.edu.cn) 1 problem of a group of spacecraft with un kno wn iner tia matrices. The attitude dy n amics are represent ed h ere by MRPs. Tw o distributed adaptiv e controll ers are pr op osed for the cases with and w ithout a virtual leader to whic h a time-v arying reference attitude is assigned. Th e first controll er achiev es attitude sync hron ization for a group of spacecraft with a leaderless comm unication top ology h a ving a directed spannin g tree. The second controll er guaran tees that all spacecraft trac k the reference attitude w hic h is av ailable to only a subset of the spacecraft, if the virtu al leader has a dir ected path to all other spacecraft. Differing from the results giv en in [4] which applies only to a bidirectional ring communicat ion top ology , and those in [3] whic h r equires the comm unication graph to b e undir ected, the comm un ication top ology among the spacecraft is relaxed to a general dir ected graph in this pap er. The r esults obtained here generalize Th eorem 4.1 in [17] to the case where th e attitude dynamics are uncertain and th e comm unication top ology is either leaderless or leader-follo w er. It should b e men tioned th at all the resu lts in this pap er are applicable to rob otic man ip ulators with dyn amics represen ted by the Euler-Lagrange equation. The r est of this pap er is organized as f ollo ws. The attitude dynamics and some usefu l results of the graph theory are in tro d uced in Section 2. The distributed adaptiv e attitude synchronizatio n problem f or the cases without and with a reference attitude are studied in Sections 3 and 4, resp ectiv ely . S im ulation examples are presented to illus trate the theoretical resu lts in Section 5. S ection 6 concludes the p ap er. The f ollo win g notation will b e u sed throughout the p ap er. R n × n denotes the set of all n × n real matrices. I N represent s the iden tit y matrix of dimension N . 1 ∈ R p denotes the vect or with all entries equal to one. Matrices, if not exp licitly stated, are assumed to ha ve compat- ible dimensions. k M k represent s the induced 2-norm of matrix M ∈ R n × m . F or a v ector x = [ x 1 , x 2 , x 3 ] T ∈ R 3 , th e cross-pro du ct op erator is denoted by S ( x ) =    0 − x 3 x 2 x 3 0 − x 1 − x 2 x 1 0    . diag( A 1 , · · · , A n ) represent s a b lo c k-diagonal matrix with matrices A i , i = 1 , · · · , n, on its d iag- onal. A ⊗ B d enotes the Kr onec k er pro d uct of matrices A and B . 2 Preliminaries and Problem F orm ula tion This pap er considers the attitude synchronizatio n p r oblem of a n et w ork of N spacecraft. Mo d- ified Ro driguez Paramete rs (MRPs) are used here to rep resen t the attitude of the spacecraft with resp ect to th e inertial frame. T he MRP vect or σ i ∈ R 3 for the i -th spacecraft is defin ed b y σ i = ˆ e i tan( φ i 4 ), where ˆ e i is the E u ler axis and φ i is the Euler angle [18]. T he attitude dynamics of the i -th spacecraft are give n b y [20] J i ˙ ω i = − S ( ω i ) J i ω i + u i , ˙ σ i = G ( σ i ) ω i , i = 1 , 2 , · · · , N , (1) 2 where ω i ∈ R 3 denotes the angular v elo cit y in th e b od y-fixed frame, J i ∈ R 3 × 3 is the inertia matrix, u i ∈ R 3 is the con trol torqu e, and G ( σ i ) = 1 2  1 − σ T i σ i 2 I 3 − S ( σ i ) + σ i σ T i  . F ollo wing [19, 23], (1) can b e rewritten as H ∗ i ( σ i ) ¨ σ i + C ∗ i ( σ i , ˙ σ i ) ˙ σ i = G − T ( σ i ) u i , i = 1 , 2 , · · · , N , (2) where H ∗ i ( σ i ) , G − T ( σ i ) J i G − 1 ( σ i ) , C ∗ i ( σ i , ˙ σ i ) , G − T ( σ i ) J i G − 1 ( σ i ) ˙ G ( σ i ) G − 1 ( σ i ) − G − T ( σ i ) S ( J i G − 1 ( σ i ) ˙ σ i ) G − 1 ( σ i ) . It is assumed that the in er tia matrices J i , i = 1 , 2 , · · · , N , are unkn o wn constan t p ositiv e- definite matrices. Und er this assumption, the attit ud e dynamics (2) h as the follo wing p rop erties: Prop ert y 1 . Matrix H ∗ i ( σ i ) is symmetric and p ositiv e definite. Prop ert y 2 . Matrix ˙ H ∗ i ( σ i ) − 2 C ∗ i ( σ i , ˙ σ i ) is ske w symmetric, i.e., x T ( ˙ H ∗ i ( σ i ) − 2 C ∗ i ( σ i , ˙ σ i )) x = 0 , ∀ x ∈ R 3 . Prop ert y 3 . The attitude d ynamics (2) satisfies the f ollo wing linear parameterization condi- tion: H ∗ i ( σ i ) ¨ y + C ∗ i ( σ i , ˙ σ i ) ˙ y = Y ( σ i , ˙ σ i , ˙ y , ¨ y ) θ i , ∀ y ∈ R 3 , where Y ∈ R 3 × 6 is called the regression matrix and θ i = h J i 11 J i 12 J i 13 J i 22 J i 23 J i 33 i T (3) is the unkn o wn parameter v ector with J i j k b eing the ( j, k )-th ent ry of the iner tia matrix J i in (1). The comm unication top ology among spacecraft is represented by a dir ected graph G consisting of a n o de set V = { 1 , 2 , · · · , N } and an edge set E ⊂ V ×V . The n o de i denotes the i -th spacecraft. An edge ( i, j ) means that spacecraft j can obtain the attitude information of spacecraft i , but not conv ersely . F or an edge ( i, j ) in the directed graph , i is the p aren t no de, j is the c hild no de, and j is neigh b orin g to i . A graph with th e p r op ert y that ( i, j ) ∈ E implies ( j, i ) ∈ E is said to b e und irected. A path on G from no de i 1 to no de i l is a sequence of ord ered edges of the form ( i k , i k +1 ), k = 1 , · · · , l − 1. A directed graph contai ns a d irected spannin g tree if there exists a no de called th e ro ot, wh ic h has n o parent, such that there exists a directed path from this n o de to eve ry other n o de. The adjacency matrix A ∈ R N × N of graph G is defined as a ii = 0, an d a ij > 0 if ( j, i ) ∈ E but 0 otherwise. Th e Laplacian matrix L ∈ R N × N is defin ed as L ii = P j 6 = i a ij , L ij = − a ij for i 6 = j . Given a matrix R = ( r ij ) p × p , the graph of R is the d irected graph with p no des su c h that there is an edge in the graph from no de j to no de i if and only if r ij 6 = 0 [6]. Lemma 1 [13]. Zero is an eigenv alue of L w ith 1 as the corresp onding righ t eigenv ecto r and all the nonzero eigen v alues ha ve p ositiv e real p arts. F urthermore, zero is a simple eigen v alue of L if and only if the graph has a directed spann in g tree. 3 3 Distributed Ad aptiv e At titude Synchronizati on This section consider s the d istr ibuted adaptive attitude synchronizat ion pr oblem of (1) whose comm unication top ology is represented by a leaderless directed graph G . A graph is leader- less, if eac h no de in this graph has at least one p aren t. Before moving forw ard, the attitude sync hr on ization problem is fi r st defin ed. Definition 1 . The distrib uted adaptiv e attitude sync hron ization problem is said to b e solv ed, if the cont rol la ws u i , i = 1 , 2 , · · · , N , are designed b y using only lo cal information of neighborin g spacecraft suc h that the attitudes of (1) satisfy lim t →∞ k σ i − σ j k = 0, lim t →∞ k ˙ σ i − ˙ σ j k = 0, i, j = 1 , 2 , · · · , N . A t eac h time instan t, the attitude information of other spacecraft a v ailable to spacecraft i is giv en b y σ d i , P N j =1 a ij σ j P N j =1 a ij , i = 1 , 2 , · · · , N , (4) where A = ( a ij ) N × N is the adjacency matrix of the comm u n ication graph G . T o quanti fy whether the attitude synchronizatio n is ac hiev ed or not, a synchronizatio n error e i ( t ) is defin ed as follo ws: e i , σ i − σ d i , i = 1 , 2 , · · · , N . (5) In ad d ition, furth er defin e a fi ltered syn chronizatio n error s i ( t ) as s i , ˙ e i + Λ i e i , i = 1 , 2 , · · · , N , (6) where Λ i ∈ R 3 × 3 , i = 1 , 2 , · · · , N , are constant p ositiv e-definite matrices. In ligh t of (2), (4), (5), and (6), it can b e obtained that vec tor s i ev olv es according to the follo wing dynamics: H ∗ i ( σ i ) ˙ s i + C ∗ i ( σ i , ˙ σ i ) s i = G − T ( σ i ) u i − H ∗ i ( σ i )( ¨ σ d i − Λ i ˙ e i ) − C ∗ i ( σ i , ˙ σ i )( ˙ σ d i − Λ i e i ) , i = 1 , 2 , · · · , N . (7) By Prop ert y 3, the r igh t sid e of the ab o ve equation can b e written in to a linear combinatio n of the inertia v ector θ i , whic h is defined in (3). T o the end , introd u ce a linear op erator L ( a ) for v ector a = [ a 1 , a 2 , a 3 ] T as L ( a ) =    a 1 a 2 a 3 0 0 0 0 a 1 0 a 2 a 3 0 0 0 a 1 0 a 2 a 3    , and a linear op erator F ( x, v ) for vec tors x = [ x 1 , x 2 , x 2 ] T , v = [ v 1 , v 2 , v 3 ] T as F ( x, v ) =    0 x 1 v 3 − x 1 v 2 x 2 v 3 − x 2 v 2 + x 3 v 3 − x 3 v 2 − x 1 v 3 − x 2 v 3 x 1 v 1 − x 3 v 3 0 x 2 v 1 x 3 v 1 x 1 v 2 − x 1 v 1 + x 2 v 2 x 3 v 2 − x 2 v 1 − x 3 v 1 0    . It can b e v erified that op erators L ( s ) and F ( x, v ) satisfy J i a = L ( a ) θ i , S ( J i x ) v = F ( x, v ) θ i . (8) 4 Therefore, by u s ing (8), (7) can b e written as H ∗ i ( σ i ) ˙ s i + C ∗ i ( σ i , ˙ σ i ) s i = G − T ( σ i ) u i − Y i ( σ i , ˙ σ i , ˙ σ d i − Λ i e i , ¨ σ d i − Λ i ˙ e i ) θ i , i = 1 , 2 , · · · , N , (9) where Y i ( σ i , ˙ σ i , ˙ σ d i − Λ i e i , ¨ σ d i − Λ i ˙ e i ) = G − T ( σ i )  L ( G − 1 ( σ i )( ¨ σ d i − Λ i ˙ e i )) + L ( G − 1 ( σ i )( ˙ σ d i − Λ i e i )) + F ( G − 1 ( σ i ) ˙ σ i , G − 1 ( σ i )( ˙ σ d i − Λ i e i ))  Since the inertia parameter θ i is u nknown, its estimate ˆ θ i is u sed ins tead to construct th e con troller u i to sp acecraft i as follo ws: u i = G T ( σ i )  Y i ( σ i , ˙ σ i , ˙ σ d i − Λ i e i , ¨ σ d i − Λ i ˙ e i ) ˆ θ i − K i s i  , i = 1 , 2 , · · · , N , (10) where K i ∈ R 3 × 3 , i = 1 , 2 , · · · , N , are p ositiv e definite. The p arameter estimate v ector ˆ θ i is generated by the follo wing adaptive up dating la w: ˙ ˆ θ i = − Γ i Y T i ( σ i , ˙ σ i , ˙ σ d i − Λ i e i , ¨ σ d i − Λ i ˙ e i ) s i , i = 1 , 2 , · · · , N , (11) where Γ i ∈ R 6 × 6 , i = 1 , 2 , · · · , N , are p ositiv e-definite d iagonal m atrices. Define the estimation errors ˜ θ i = θ i − ˆ θ i , i = 1 , 2 , · · · , N . Th en, substituting (10), (11) in to (9) giv es H ∗ i ( σ i ) ˙ s i + C ∗ i ( σ i , ˙ σ i ) s i = − Y i ( σ i , ˙ σ i , ˙ σ d i − Λ i e i , ¨ σ d i − Λ i ˙ e i ) ˜ θ i − K i s i , i = 1 , 2 , · · · , N . (12) Theorem 1 . If the leaderless communicatio n graph G has a directed spann ing tree, then distributed con trollers (10) and adaptiv e up dating la ws (11) solv es the attitude sync hr onization problem for (1). Pro of . Let e ( t ) = [ e T 1 , · · · , e T N ] T and σ ( t ) = [ σ T 1 , · · · , σ T N ] T . Th en, (5) can b e w ritten as e = M L σ, (13) where L is the Lap lacian matrix asso ciated with graph G , and M = diag( P N j =1 a 1 j , · · · , P N j =1 a N j ). Since graph G is leaderless and has a directed spanning tree, one obtains 1) there exists at least one nonzero entry for eac h row of the adjacency matrix A , thus matrix M is p ositiv e d efinite; 2) the Laplacian matrix L h as a simple eigen v alue 0 with 1 as the corresp onding eigen v ector, and the other eigen v alues hav e p ositive real parts. Then, it follo ws from (13) that e = 0 if and only if σ 1 = σ 2 = · · · = σ N . That is, the attitude synchronizati on problem is s olved if and only if e ( t ) → 0, ˙ e ( t ) → 0, as t → ∞ . Consider the follo wing L y apunov function candidate V ( t ) = 1 2 N X i =1 s T i H ∗ i ( σ i ) s i + N X i =1 ˜ θ T i Γ − 1 i ˜ θ i ! . (14) 5 By (12), (11), and Prop erty 2, the time deriv ativ e of V ( t ) is ˙ V ( t ) = N X i =1 s T i  ˙ H ∗ i ( σ i ) s i + H ∗ i ( σ i ) ˙ s i  + N X i =1 ˙ ˜ θ T i Γ − 1 i ˜ θ i = N X i =1 s T i  ˙ H ∗ i ( σ i ) s i − C ∗ i ( σ i , ˙ σ i ) − Y i ( σ i , ˙ σ i , ˙ σ d i − Λ i e i , ¨ σ d i − Λ i ˙ e i ) ˜ θ i − K i s i  − N X i =1 ˙ ˆ θ T i Γ − 1 i ˜ θ i = − N X i =1  s T i Y i ( σ i , ˙ σ i , ˙ σ d i − Λ i e i , ¨ σ d i − Λ i ˙ e i ) ˜ θ i + ˙ ˆ θ T i Γ − 1 i ˜ θ i + s T i K i s i  = − N X i =1 s T i K i s i ≤ 0 . (15) Let S = { ( σ i , s i , ˜ θ i ) | ˙ V = 0 } . No te that ˙ V = 0 implies that s i = 0, i = 1 , 2 , · · · , N . By LaSalle’s in v ariance principle [19], it follo ws from that s i → 0, i = 1 , 2 , · · · , N , as t → ∞ , which b y (6) in turn sh o ws that e ( t ) → 0, ˙ e ( t ) → 0, as t → ∞ , i.e, attitude synchronizati on is ac hiev ed.  Remark 1 . It should b e n oted that the controlle r (10) to spacecraft i dep ends only on its o wn attitude v ectors σ i , ˙ σ i , ¨ σ i , and the attitudes of its neigh b orin g sp acecraft, therefore is distribu ted. Theorem 1 giv es a sufficien t condition for ac hieving attitude s ync hronization. Ho w ev er, it is generally quite hard to expressly d eriv e the final syn c hronized attitude v alue, whic h dep ends on the initial attitudes of the N spacecraft, matrices Λ i , K i , i = 1 , 2 , · · · , N , and the comm unication top ology G . 4 Distributed Ad aptiv e At titude T rac king Differen t fr om the leaderless communication graph discussed as in the ab o ve section, the sp ace- craft’ attitudes ma y b e desired to follo w a given time-v arying reference attitude σ r in certain circumstance. It is sup p osed that σ r is a v aila ble to only a s u bgroup of the spacecraft, other- wise co op eration b etw een neigh b oring spacecraft by exchanging attitude in formation b ecome not so necessary . Assume that σ r , ˙ σ r , and ˙ σ r are all b ounded. F or this case, the attitude sync hr on ization p roblem is called attitude tr acking p roblem in [4, 17], which is form ulated as follo ws. Definition 2 . The d istributed adaptiv e attitude trac king p roblem is said to b e solv ed, if the lo cal con trol la ws u i , i = 1 , 2 , · · · , N , are designed for (1) su c h that lim t →∞ k σ i − σ r k = 0, lim t →∞ k ˙ σ i − ˙ σ r k = 0, i = 1 , 2 , · · · , N . T ak e the reference attitude σ r as the attitude of a virtu al leader, lab eled as spacecraft N + 1. Since the vir tual leader d o es not obtain an y in f ormation fr om the N spacecraft, the communi- cation top ology among these N + 1 spacecraft (the N sp acecraft and the virtual leader) is in the leader-follo w er f orm . 6 Assume that the comm un ication top ology among the N spacecraft is still denoted by G . The attitude inf ormation of other spacecraft a v aila b le to spacecraft i is giv en by σ d f i , P N j =1 a ij σ j + a i ( N +1) σ r P N j =1 a ij + a i ( N +1) , i = 1 , 2 , · · · , N , (16) where A = ( a ij ) N × N is the adjacen t matrix of G , a i ( N +1) > 0, i = 1 , · · · , N , if spacecraft i has access to the virtual leader and a i ( N +1) = 0 otherw ise. Similar to the ab o ve s ection, a sync hr on ization error e f i ( t ) is defin ed as follo ws: ˙ e f i = σ f i − σ d f i , i = 1 , 2 , · · · , N . ( 17) Corresp ond ingly , the fi ltered synchronizatio n errors s f i is defin ed as s f i , ˙ e f i + b Λ i e f i , i = 1 , 2 , · · · , N , (18) where matrices b Λ i ∈ R 3 × 3 , i = 1 , 2 , · · · , N , are p ositiv e definite. By (2), (16), (17), and Prop ert y 3, vect or s f i satisfies the f ollo wing dynamics: H ∗ i ( σ i ) ˙ s f i + C ∗ i ( σ i , ˙ σ i ) s f i = G − T ( σ i ) u i − Y i ( σ i , ˙ σ i , ˙ σ d f i − b Λ i e f i , ¨ σ d f i − b Λ i ˙ e f i ) θ i , i = 1 , 2 , · · · , N . (19) The d istributed con trollers u i , i = 1 , 2 , · · · , N , to the N spacecraft are pr op osed as u i = G T  Y i ( σ i , ˙ σ i , ˙ σ d f i − b Λ i e f i , ¨ σ d f i − b Λ i ˙ e f i ) ˆ θ f i − b K i s f i  , i = 1 , 2 , · · · , N , (20) where b K i ∈ R 3 × 3 , i = 1 , 2 , · · · , N , are p ositiv e definite and ˆ θ f i is the estimate of θ i , generated b y the follo wing adaptiv e up dating la w: ˙ ˆ θ f i = − b Γ i Y T i ( σ i , ˙ σ i , ˙ σ d f i − b Λ i e f i , ¨ σ d f i − b Λ i ˙ e f i ) s f i , i = 1 , 2 , · · · , N , (21) where b Γ i ∈ R 6 × 6 , i = 1 , 2 , · · · , N , are p ositiv e-definite d iagonal m atrices. Theorem 2 . Denote by b G N +1 the directed grap h of matrix A N +1 = " A b 0 0 # , wh ere b = [ a 1( N +1) , · · · , a N ( N +1) ] T . If graph b G N +1 has a spannin g tree with n o de N + 1 as the ro ot, then distributed con trollers (20) and adaptiv e up dating la ws (21) solv e the attitude trac king problem for the N spacecraft in (1). Pro of . Let σ N +1 , σ r , ˆ σ ( t ) = [ σ T 1 , · · · , σ T N , σ T N +1 ] T , and e f ( t ) = [ e T f 1 , · · · , e T f N , 0] T . Th en, (17) can b e written as e f = c M b L ˆ σ , (22) where c M = diag( P N j =1 a 1 j , · · · , P N j =1 a N j , 1), and b L = ( b L ij ) ( N +1) × ( N +1) is th e Laplacian matrix of graph b G , defin ed as b L ii = P N +1 j =1 a ij , b L ij = − a ij ( j 6 = i ), ∀ i ∈ { 1 , 2 , · · · , N } , and b L ( N +1) j = 0, ∀ j ∈ { 1 , 2 , · · · , N + 1 } . Under the assu mption of the theorem, matrix c M is p ositiv e definite and 0 is a simple eigen v alue of matrix b L with 1 as the corresp ondin g eigen v ector, implying that e f = 0 if and only if σ 1 = σ 2 = · · · = σ N = σ r . That is, the distribu ted attitude trac king problem is solv ed if and only if e f ( t ) → 0, ˙ e f ( t ) → 0, as t → ∞ . 7 Consider the follo wing L y apunov function candidate b V ( t ) = 1 2 N X i =1 s T f i H ∗ i ( σ i ) s f i + N X i =1 ( θ i − ˆ θ f i ) T b Γ − 1 i ( θ i − ˆ θ f i ) ! . (23) The time deriv ativ e of b V ( t ) can b e obtained as ˙ b V ( t ) = N X i =1 s T f i  ˙ H ∗ i ( σ i ) s f i + H ∗ i ( σ i ) ˙ s f i  − N X i =1 ˙ ˆ θ T f i b Γ − 1 i ( θ i − ˆ θ f i ) = − N X i =1 s T f i b K i s f i . (24) Since b V ( t ) ≥ 0, ˙ b V ( t ) ≤ 0, b V ( t ) remains b oun ded, w hic h b y (23) implies that s f i , ˜ θ f i , i = 1 , 2 , · · · , N , are b oun ded. By u s ing standard signal c hasing arguments, it is easy to sho w that ˙ s f i , i = 1 , 2 , · · · , N , are b oun ded. Therefore, ¨ b V ( t ) = − 2 P N i =1 ˙ s T f i b K i s f i is b oun ded. I n light of Barbalat’s lemma [19], ˙ b V ( t ) → 0, as t → ∞ . Thus, s i → 0, i = 1 , 2 , · · · , N , as t → ∞ , whic h by (17) sho ws that e f ( t ) → 0, ˙ e f ( t ) → 0, as t → ∞ , i.e, the distributed attitude trac king p roblem is solv ed.  Remark 2. Theorems 1 and 2 pr esent sufficient conditions f or the adaptive attitude syn- c hronization of multiple spacecraft ha ving general dir ected communicati ons in the p r esence of unknown inertia m atrices, for b oth the cases with an d withou t a reference attitude. By con trast, the results giv en in [4] are applicable only to a bidirectional ring comm unication top ology , and the result in [3] requires the communication graph to b e u ndirected. Theorems 1 and 2 gen- eralize Theorem 4.1 in [17] to the case wh ere the attitude dynamics (1) are uncertain and the comm unication top ology is either leaderless or leader-follo w er. It should b e n oted that differen t from Theorem 1, Barbalat’s lemma is utilized to derive Theorem 2, d ue to the f act th at the time-v aryin g reference attitude σ r ma y render the closed-lo op s y s tem nonautonomous. 5 Numerical Examples In this section, the effectiv eness of the prop osed con trol la ws is illus trated through numerical sim ulations. 1 6 2 5 3 4 Figure 1: Th e comm un ication top ology . 8 Consider a group of six sp acecraft, wh ose inertia matrices are sh o wn in T able 1 [15]. The initial attitude states σ i (0), ω i (0), i = 1 , 2 , · · · , 6, are c hosen r andomly . The communicati on top ology is giv en b y Fig. 1, s o the corresp ondin g adjacency matrix is A =            0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0            . F or (10) and (20), tak e matrices Λ i = b Λ i = I 3 , K i = b K i = 3 I 3 , Γ i = b Γ i = 3 I 6 , i = 1 , 2 , · · · , 6. Th e parameter estimates ˆ θ i and ˆ θ f i are in itialized to b e zero, i.e., ˆ θ i (0) = 0, ˆ θ f i (0) = 0, i = 1 , 2 , · · · , 6, in (11), (21). F or sim p licit y , let the referen ce attitude σ r = [0 . 1 , 0 . 3 , 0 . 5] T . Su p p ose that σ r is a v aila ble only to spacecraft 1. In this case, a 17 = 1, a i 7 = 0, i = 2 , · · · , 6. J 1 [1 , 0 . 1 , 0 . 1; 0 . 1 , 0 . 1 , 0 . 1; 0 . 1 , 0 . 1 , 0 . 9] kgm 2 J 2 [1 . 5 , 0 . 2 , 0 . 3; 0 . 2 , 0 . 9 , 0 . 4; 0 . 3 , 0 . 4 , 2 . 0] kgm 2 J 3 [0 . 8 , 0 . 1 , 0 . 2; 0 . 1 , 0 . 7 , 0 . 3; 0 . 2 , 0 . 3 , 1 . 1] kgm 2 J 4 [1 . 2 , 0 . 3 , 0 . 7; 0 . 3 , 0 . 9 , 0 . 2; 0 . 7 , 0 . 2 , 1 . 4] kgm 2 J 5 [0 . 9 , 0 . 15 , 0 . 3; 0 . 15 , 1 . 2 , 0 . 4; 0 . 3 , 0 . 4 , 1 . 2] kgm 2 J 6 [1 . 1 , 0 . 35 , 0 . 45; 0 . 35 , 1 . 0 , 0 . 5; 0 . 45 , 0 . 5 , 1 . 3] kgm 2 T able 1: Sp acecraft sp ecificati ons. Figs. 2(a), 2(b), and 2(c) depict, resp ectiv ely , the attitudes, angular v elo cities, and control torques of the six spacecraft with (10) and (11), from which it can b e obs erv ed that attitude sync hr on ization is in d eed ac hieved. T he parameter estimates ˆ θ i , i = 1 , 2 , · · · , 6, are shown in Fig. 3. Figs. 4(a), 4(b), 4(c), an d 5 depict, resp ectiv ely , the attitudes, angular velocities, con trol torques, and the p arameter estimates ˆ θ f i , i = 1 , 2 , · · · , 6, of the six spacecraft with (20) and (21). 6 Conclusions This pap er has ad d ressed the distr ib uted adaptiv e attitude sync hr onization problem of a grou p of spacecraft with unknown inertia m atrices. Tw o distrib uted adaptiv e con trollers h a v e b een prop osed for the cases with and without a virtual leader to w hic h a time-v arying reference atti- tude is assigned. Th e first control ler ac hieve s attitude synchronizatio n for a group of spacecraft with a leaderless communicatio n top ology ha ving a directed spanning tree. The second con troller guaran tees that all spacecraft trac k the r eferen ce attitude if the virtual leader has a d irected path to all other spacecraft. Th is pap er has extended some existing results in the literature. An in teresting topic for fu ture research is th e distributed adaptiv e attitude s ync hronization of m ultiple spacecraft without v elo cit y measuremen ts. 9 0 10 20 30 40 50 −2 0 2 t (s) σ j (1) 0 10 20 30 40 50 −0.5 0 0.5 t (s) σ j (2) 0 10 20 30 40 50 −2 0 2 t (s) σ j (3) (a) Attitudes 0 10 20 30 40 50 −0.5 0 0.5 t (s) ω j (1) 0 10 20 30 40 50 −1 0 1 t (s) ω j (2) (rad/s) 0 10 20 30 40 50 −0.5 0 0.5 t (s) ω j (3) (b) Angular velocities 0 10 20 30 40 50 −0.5 0 0.5 t (s) τ j (1) 0 10 20 30 40 50 −1 0 1 t (s) τ j (2) 0 10 20 30 40 50 −0.5 0 0.5 t (s) τ j (3) (c) Control torq ues Figure 2: A ttitudes, angular ve lo cities, and con trol torques of (1) w ith con troller (10) and adaptiv e la w (11). 0 10 20 30 40 50 −1 0 1 t θ j (1) 0 10 20 30 40 50 −0.5 0 0.5 t (s) θ j (2) 0 10 20 30 40 50 −0.5 0 0.5 t (s) θ j (3) (a) ˆ θ i , i = 1 , 2 , 3 0 10 20 30 40 50 0 0.5 1 t (s) θ j (4) 0 10 20 30 40 50 −0.5 0 0.5 t (s) θ j (5) 0 10 20 30 40 50 0 0.5 1 t (s) θ j (6) (b) ˆ θ i , i = 4 , 5 , 6 Figure 3: Paramet er estimates of (11). 10 0 50 100 150 200 −0.5 0 0.5 t (s) σ j (1) 0 50 100 150 200 −1 0 1 t (s) σ j (2) 0 50 100 150 200 −1 0 1 t (s) σ j (3) (a) Attitudes 0 50 100 150 200 −0.5 0 0.5 t (s) ω j (1) 0 50 100 150 200 −0.5 0 0.5 t (s) ω j (2) (rad/s) 0 50 100 150 200 −1 0 1 t (s) ω j (3) (b) Angular velocities 0 50 100 150 200 −0.5 0 0.5 t (s) τ j (1) 0 50 100 150 200 −0.5 0 0.5 t (s) τ j (2) 0 50 100 150 200 −0.5 0 0.5 t (s) τ j (3) (c) Control torq ues Figure 4: A ttitudes, angular v elo cit y , and con trol torques of (1) with con troller (20) and adaptiv e la w (21). 0 50 100 150 200 −1 0 1 t θ j (1) 0 50 100 150 200 −1 0 1 t (s) θ j (2) 0 50 100 150 200 −1 0 1 t (s) θ j (3) (a) ˆ θ f i , i = 1 , 2 , 3 0 50 100 150 200 −1 0 1 t (s) θ j (4) 0 50 100 150 200 −1 0 1 t (s) θ j (5) 0 50 100 150 200 −1 0 1 t (s) θ j (6) (b) ˆ θ f i , i = 4 , 5 , 6 Figure 5: Paramet er estimates of (21). 11 References [1] A. 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