IN RECENT years, there has been an increasing research interest

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1 IEEE/ASME TRASACTIOS O MECHATROICS, VOL. 4, O. 2, APRIL Robust Fnte-Tme Consensus Trackng Algorthm for Multrobot Systems Suyang Khoo, Lhua Xe, Fellow, IEEE, and Zhhong Man, Member, IEEE Abstract Ths paper studes the fnte-tme consensus trackng control for multrobot systems. We prove that fnte-tme consensus trackng of multagent systems can be acheved on the termnal sldng-mode surface. Also, we show that the proposed error functon can be modfed to acheve relatve state devaton between agents. These results are then appled to the fntetme consensus trackng control of multrobot systems wth nput dsturbances. Smulaton results are presented to valdate the analyss. Index Terms Fnte-tme consensus, multagent systems, multrobot systems, termnal sldng-mode (TSM) control. I. ITRODUCTIO I RECET years, there has been an ncreasng research nterest n the consensus control desgn of multagent systems. Consensus algorthms have applcatons n rendezvous control of multnonholonomc agents 3, formaton control 4 6, and flockng atttude algnment 7. In leader follower multagent system, the leader s usually ndependent of ther followers, but have nfluence on the followers behavors. Hence, by controllng only the leader, the control objectve of the networks can be realzed easly. Such a strategy not only smplfes the desgn and mplementaton, but also saves the control energy and cost 8, 9. The objectve of ths paper s to address the followng ssues ) Under what condtons, a nonsmooth control algorthm can be developed to guarantee the leader follower multagent system to reach consensus n a fnte tme. 2) How to desgn ths fnte-tme control algorthm systematcally. Our nterest n these two ssues s motvated by the prelmnary research work n 0 on fnte-tme consensus desgn for frstorder systems, and the work n on fnte-tme consensus for second-order systems wth undrected communcaton topology. In practce, the network topology mght be drected and the tme-varyng control nput of the actve leader mght not be avalable to all the followers (e.g., multple mssles trackng a Manuscrpt receved July 6, 2008; revsed October 28, 2008 and January 2, Frst publshed March 0, 2009; current verson publshed Aprl 5, Recommended by Guest Edtor M.-Y. Chow. S. Khoo s wth the School of Engneerng, Deakn Unversty, Geelong, VIC. 327, Australa (e-mal: khooyang@yahoo.com). L. Xe s wth the School of Electrcal and Electroncs Engneerng, anyang Technologcal Unversty, Sngapore (e-mal: elhxe@ntu. edu.sg). Z. Man s wth the School of Mechatroncs, Swnburne Unversty, Hawthorn, Melbourne, VIC. 322, Australa (e-mal: m.zhhong@hotmal.com). Color versons of one or more of the fgures n ths paper are avalable onlne at Dgtal Object Identfer 0.09/TMECH fghter arcraft). Therefore, t s mportant to nvestgate how to desgn consensus control algorthm applcable to the case, where only a porton of the followers have drected communcaton wth the leader under the condton that the control nput of the leader s unknown to any follower. Based on the prevous works on termnal sldng-mode (TSM) control 2 5, we present n ths paper a practcal robust fnte-tme consensus trackng (RFTCT) algorthm for multagent systems. In partcular, we show that for leader follower multagent systems domnated by second-order systems, t s possble to acheve global fnte-tme consensus on the TSM surface by swtchng control law. Ths concluson s proved based on the Lyapunov theory for fnte-tme stablty and TSM control desgn methodology. Our proposed RFTCT algorthm s robust to system uncertantes and nput dsturbances. Therefore, the proposed scheme does not requre the nformaton of the tme-varyng control nput of the actve leader. Snce not all followers have drected communcaton wth the leader, our results assume that the agents n the network only need to communcate wth ther neghbors and not the entre communty. In contrast to consensus trackng of the leader s state, we show that the proposed error functon could be modfed to acheve fnte-tme relatve state devaton between agents. The desred devaton between agents could be specfed n real tme and hence dfferent formatons can be formed. The proposed control scheme s then appled to the fnte-tme consensus trackng control of multrobot systems wth m degrees of freedom wth nput dsturbances. The remander of ths paper s organzed as follows. Secton II revews some basc concepts n graph theory, the Lyapunov theory for fnte-tme stablty, and the basc prncple of TSM control. An error functon for consensus control s proposed n Secton III, where the desgn of RFTCT algorthm to guarantee fnte-tme consensus trackng of multagent systems s dscussed n detal. The proposed scheme s then appled to the consensus trackng control for multrobot systems wth m-dof n Secton IV. Secton V gves numercal examples to llustrate our results. Concludng remarks are gven n Secton VI. II. BACKGROUD AD PRELIMIARIES In ths secton, we ntroduce some basc concepts n algebrac graph theory for multagent networks and revew some termnologes related to the noton of fnte-tme stablty and the correspondng Lyapunov stablty theory frst. Then, we brefly study the basc prncple of TSM control, focusng just on the second-order nonlnear system that we shall need /$ IEEE

2 220 IEEE/ASME TRASACTIOS O MECHATROICS, VOL. 4, O. 2, APRIL 2009 A. Concepts n Graph Theory and Multagent Systems Consder a multagent system consstng of one leader and n followers. To solve the coordnaton problems and model the nformaton exchange between agents, graph theory s ntroduced here. Let G {V, E} be a drected graph, where V {0,, 2,...,n} s the set of nodes, node represents the th agent, E s the set of edges, and an edge n G s denoted by an ordered par (, j).(, j) Ef and only f the th agent can send nformaton to the jth agent drectly, but not necessarly vce versa. In contrast to a drected graph, the pars of nodes n an undrected graph are unordered, where the edge (, j) denotes that agent and j can obtan nformaton from each other. Therefore, an undrected graph can be vewed as a specal case of a drected graph. A drected tree s a drected graph, where every node has exactly one parent except for the root, and the root has a drected path to every other node. A drected spannng tree of G s a drected tree that contans all nodes of G 6. A (a j ) R n n s called the weghted adjacency matrx of G wth nonnegatve elements, where a 0 and a j 0 wth a j > 0 f there s an edge between the th agent and the jth agent. Let D dag{d 0,d,...,d n } R n n be a dagonal matrx, where d n j0 a j for 0,,...,n. Then, the Laplacan of the weghted graph can be defned as L D A R n n. () The connecton weght between the th agent and the leader s denoted by b wth b > 0 f there s an edge between the th agent and the leader. The followng theorems present the exstng results on Laplacan matrx and graph theory. Theorem 8: The drected graph G {V, E} has a drected spannng tree f and only f {V, E} has at least one node wth a drected path to all other nodes. Theorem 2 7: The Laplacan matrx L of a drected graph G {V, E} has at least one zero egenvalue and all of the nonzero egenvalues are n the open rght-half plane. In addton, L has exactly one zero egenvalue f and only f G has a drected spannng tree. Furthermore, Rank(L) n f and only f L has a smple zero egenvalue. In ths paper, we assume that the leader s actve, n the sense that ts state keeps changng throughout the entre process 9, 24. In general the behavor of the leader s ndependent of the followers. The dynamcs of the leader s descrbed as follows: ẋ 0 v 0, x 0 R m v 0 u 0, v 0 R m (2) where x 0 s the poston and v 0 s the velocty of the leader. The dynamcs of the th follower agent s descrbed by ẋ v, x R m v u δ, v R m,,...,n (3) where δ represents the dsturbance and u (,...,n), the control nputs. For further analyss, we assume that δ D<. (4) B. Lyapunov Theory for Fnte-tme Stablty Here, we recall some Lyapunov theorem for fnte-tme stablty of nonlnear systems, whch was dscussed prevously n 20 and 2. The classcal Lyapunov stablty theory s only applcable to a dfferental equaton whose soluton from any ntal condton s unque 22. A well-known suffcent condton for the exstence of a unque soluton of a nonlnear dfferental equaton ẋ f(x) s that the functon f(x) s locally Lpschtz contnuous. The soluton of such nonlnear dfferental equaton can have at most asymptotc convergence rate. Snce fnte-tme stablty guarantees that every system state reaches the system orgn n a fnte tme, fnte-tme stablty has a much stronger requrement than asymptotc stablty. The followng theorem presents suffcent condtons for fnte-tme stablty. Theorem 3 23: Consder the non-lpschtz contnuous nonlnear system ẋ f(x) wth f(0) 0. Suppose there are C functon V (x) defned on a neghborhood of the orgn, and real numbers c>0and 0 <α<, such that ) V (x) s postve defnte, 2) V (x)cv α 0. Then, the orgn s locally fnte-tme stable, and the settlng tme, dependng on the ntal state x(0) x 0, satsfes T (x 0 ) V (x 0) α (5) c( α) for all x 0 n some open neghborhood of the orgn. C. Basc Prncple of TSM Control Consder the second-order nonlnear system ż z 2 ż 2 f(z,z 2 )u(t) (6) where z and z 2 are system states, f( ) a nonlnear functon of z and z 2, and u s the control nput. In order to guarantee fntetme convergence of the state varables, the followng frst-order termnal sldng varable s defned: s z z α 2 (7) where α p/q, and p and q are postve odd ntegers, whch satsfy the followng condton: p>q. (8) Usng a sldng-mode controller of the form { u (z), f s>0 u(z) (9) u (z), f s<0 the termnal sldng varable s can be drven to the TSM surface, s 0n fnte tme. On the TSM surface, the system dynamcs are determned by the followng nonlnear dfferental equaton: ż z q/p. (0) If the ntal value of z at t 0s z (0)( 0), then the tme taken for the soluton of the system (0) to reach z (T )0s

3 KHOO et al.: ROBUST FIITE-TIME COSESUS TRACKIG ALGORITHM FOR MULTIROBOT SYSTEMS 22 gven by 0 dz T z (0) z q/p z (0) q/p ( q/p). () Ths means that, on the TSM surface s 0, the system state z converges to zero n fnte tme and also z 2 converges to zero n fnte tme dentcally. III. RFTCT ALGORITHM FOR MULTIAGET SYSTEMS The system consdered here conssts of n agents, where an agent ndexed by 0 acts as the leader and the other agents ndexed by,...,n, are referred to as the followers. The topology relatonshps among the leader and followers s descrbed by a drected graph G {V, E} wth V {0,,...,n} and the adjacent matrx a A 0 a... a n R(n) (n). (2) a n0 a n... a nn Denote Ḡ { V, Ē} as the subgraph of G, whch s formed by the n followers, where a a 2... a n Ā R (n) (n). (3) a n a n2... a nn Let D dag{ d, d 2,..., d n } R n n be a dagonal matrx wth d n j a j for, 2,...,n. Then, t s clear that the Laplacan of the graph Ḡ can be defned as L D Ā. (4) In ths paper, for smplcty, we assume that {, f (j, ) E a j (5) 0, otherwse. Meanwhle, the connecton weght between agent and the leader s denoted by B where B dag{b,b 2,...,b n } (6) such that {, f agent s connected to the leader b (7) 0, otherwse. For further analyss, we have the followng assumptons. Assumpton : The tme-varyng control nput u 0 s unknown to any follower but ts upper bound ū 0 s avalable to ts neghbors. Assumpton 2: The poston of the leader x 0 and ts velocty v 0 are avalable to ts neghbors only. Remark : In ths paper, the tme-varyng control nput of the leader s not requred. Assumpton s practcal n the sense that not all the real-tme control nputs of the leader s known by the followers, and the upper bound of the control nput can be found easly based on the physcal lmtaton of the plant. In ths paper, we shall consder the case, where the velocty of the leader s avalable to ts neghbors only. Let E e,...e n T and E 2 e 2,...e 2 n T, and by defnng the error functons as e e 2 a j (x x j )b (x x 0 ) (8) j a j (v v j )b (v v 0 ) (9) j the error dynamcs of the nterconnecton graph can be expressed as E E 2 E 2 ( L B)U ( L B)δ Bu 0 (20) where U u,...,u n T and δ δ,...,δ n T. Usng Theorem 2, we are able to prove the followng theorem that state under what condton, consensus can be reached. Theorem 4: Consder the leader follower system (2) and (3). If the drected graph G has a drected spannng tree and E 0 and E 2 0, then Furthermore, f (8) s defned as e x x n T x 0 (2) v v n T v 0. (22) a j (x x j j ) j and E 0, then x.. x n b (x x 0 0 ) (23) x 0.. n. (24) Proof: Wth e n j a j(x x j )b (x x 0 ) n mnd, after E 0t s easy to see that x ( L B). Bx 0. (25) x n Snce L 0,wehave x ( L B).. x n ( L B)x 0 Lettng ( L B)x 0. (26) b M n (n). L B (27) b n

4 222 IEEE/ASME TRASACTIOS O MECHATROICS, VOL. 4, O. 2, APRIL 2009 t s clear that L D A 0 (n) M n (n). (28) ote that f L has a drected spannng tree, by Theorem 2, t follows that Rank(L) n. Ths n turn mples that Rank(M n (n) )nbecause all of the entres n the frst row of L are zero. otng that M n (n) has n columns and each of ts row sums s zero, t follows that the frst column of M n (n) depends on ts last n columns, where b... L B. (29) b n As a result, t follows that Rank( L B) Rank(M n (n) ) n. Ths shows the nvertblty of L B and hence, from (26), we have x x n T x 0. (30) Smlarly, we can prove (22) and (24). Remark 2: It s seen n (24) that we could specfy the desred separaton between follower agents and leader, ths error functon s useful for relatve state devatons wth a tme-varyng consensus reference state and formaton control. Before proceedng further, the notaton of the fractonal power of vector s ntroduced. For a varable vector M R n,the fractonal power of the vector s defned as M α m α,m α 2,...,m α n T (3) dag ( M α ) Ṁ α ṁ α, ṁ α 2,...,ṁ α n T (32) m α... m α n. (33) To obtan the suffcent condton of the exstence of the fntetme controller for the leader follower system, we have the followng theorem. Theorem 5: If the drected graph G has a drected spannng tree, then there exst a termnal sldng varable vector and a nonsngular TSM control law for the leader follower system (2) and (3) such that on the TSM surface, consensus can be reached n a fnte tme. Proof: By desgnng the termnal sldng varable as s e ( e 2 ) α, for,...,n (34) the termnal sldng varable vector can be wrtten as S E E2 α. (35) One can smply choose the control nput as wth u 0 U u 0,...,u 0 n T u,...,u n T (36) j,j ( e 2 a j b α ) 2 α (37) u j,j a j b (2nD b ū 0 κ ) sgn(s ) j,j a j ( u j ) (38) κ > 0 (39) and ths results n U (E2 ) D B 2 α α ĀU (dag(2nd κ ) Bū 0 )sgn(s). (40) otng that I D B Ā D B D B Ā D B L B. Equaton (40) s rewrtten as U L B (E2 ) 2 α α (dag(2nd κ ) Bū 0 )sgn(s). (4) Consder the Lyapunov functon V 2 ST S. Usng (35), (20), and (4), a smple computaton gves V S T { E 2 α dag(e2 α )( L B)(U δ) Bu 0 } { ( S T E 2 α dag(e2 α ) (E 2) 2 α dag(2nd α ) } κ ) Bū 0 sgn(s)( L B)δ Bu 0 S T {α dag(e2 α ) (dag(2nd κ ) Bū 0 )sgn(s)( D B Ā)δ Bu 0 } α s k ( e 2 ) α k (2nD) α s k ( e 2 ) α k (κ ) k k k α s k ( e 2 ) α ( ) k (bk ū 0 )α s k e 2 α k j,j k j,j k k ( ) a kj b k (δ k ) α s k e 2 α k k ( ) a kj δ j α s k e 2 α k (bk u 0 ). k

5 KHOO et al.: ROBUST FIITE-TIME COSESUS TRACKIG ALGORITHM FOR MULTIROBOT SYSTEMS 223 Usng Assumpton, (4), and the fact that ( e 2 k ) α 0, we get V α s k ( e 2 ) α k (2nD) α s k ( e 2 ) α k (κ ) k k k α s k ( e 2 ) α k (bk ū 0 )α s k ( e 2 ) α k (nd) α α k s k ( e 2 ) α k (nd)α s k ( e 2 ) α k (bk ū 0 ) k k s k ( e 2 ) α k (κ ). (42) k Lettng η(e 2 )mn{ακ e 2 α,ακ e 2 2 α,...,ακ e 2 n α }. For e 2 k 0, k, 2,...,n, η(e 2) > 0, wehave V η(e 2 ) η(e 2 ) s k k ( k s k 2 ) η(e2 )V 2. (43) Therefore, for case E 2 0, the condton for fnte-tme Lyapunov stablty s satsfed and the termnal sldng varable vector S can reach the TSM surface S 0n a fnte tme 2, 3. ow t remans to show that E 2 0s not an attractor for S 0. Substtutng the control nput (4) nto the error dynamcs of the nterconnected graph (20), we have E 2 E 2 2 α α (dag(2nd κ ) Bū 0 )sgn(s) ( L B)δ Bu 0. (44) Then for E 2 0, t s clear that E 2 (dag(2nd κ ) Bū 0 )sgn(s) ( L B)δ Bu 0. (45) For S 0, ė 2 κ or ė 2 κ, showng that E 2 0s not an attractor. Then, t can be easly concluded from (43) that the TSM surface S 0can be reached n a fnte tme. Interested reader may refer to 3 for further explanatons. We clam that on ths new TSM surface, consensus trackng of multagent system can be reached n fnte tme. To prove ths clam, consder the Lyapunov functon V E 2 E T E.Onthe TSM surface, and t follows that V E E 2 E α (46) E T E α 2 α 2 α (VE ) α 2 α. (47) By Theorem 3, the error functons E and E 2 wll converge to zero n fnte tme. It follows from Theorem 4 that consensus s reached n fnte tme. Remark 3: It has been seen n (35) that, ths TSM surface s the conventonal TSM proposed n 3. In order to ncorporate the nformaton of the nterconnecton graph nto the TSM surface, the sldng varable can be desgned as ( s a j e e ) ( ) ( ) j b e e 2 α, j for,...,n. (48) It s noted that the convergence property of the error dynamcs on ths TSM surface s substantally dfferent from the one n (35). For example, nstead of convergng toward the leader drectly, the followers wll frst converge to one of the leader s neghbors, and then, together wth ths neghbor, all the followers converge to the leader. Remark 4: It s noted from (8) and (9) that, because the nformaton of the nterconnecton graph s used to defne the error functons, convergence of these error functons mply consensus of the leader follower systems. Hence, based on the second-order error dynamcs equaton (20), many control methods can be used to ensure consensus of the leader follower or tme-varyng reference state systems. Remark 5: It s easy to extend the error functons n (8) and (9) for hgh-order systems n pure-feedback form. Hence, by usng these error functons, the consensus control problem for hgh-order system s transformed to the conventonal control problem for hgh-order multnput systems. Remark 6: For asymptotc consensus trackng of multagent systems, under the condton that the velocty of the leader s avalable to ts neghbors, the sldng mode surface can be defned as s a j (x x j )b (x x 0 ) j a j (v v j )b (v v 0 ), j for,...,n. (49) IV. RFTCT ALGORITHM FOR MULTIROBOT SYSTEMS In ths secton, we consder a group of n fully actuated moble robots whose dynamcs of the robot wth m-dof can be descrbed as 25, 26 M(q) q C(q, q) q D(q, q) q g(q) u r ρ (50) where q R m s a generalzed coordnate, M(q) R m m s a symmetrc postve defnte nerta matrx, C(q, q) R m m s a matrx of Corols and centrpetal terms, D(q, q) R m m represents the dampng force, g(q) R m denotes a gravtatonal force vector, ρ R m represents the nput dsturbances and system uncertantes, and u r R m denotes the control nputs. Wthout loss of generalty, n the followng analyss, let m for notatonal smplcty.

6 224 IEEE/ASME TRASACTIOS O MECHATROICS, VOL. 4, O. 2, APRIL 2009 Suppose that the dynamcs of the leader robot s descrbed as follows: M 0 (q 0 ) q 0 C 0 (q 0, q 0 ) q 0 D 0 (q 0, q 0 ) q 0 g 0 (q 0 )u r 0 (5) and the dynamcs of the th follower s descrbed as M (q ) q C (q, q ) q D (q, q ) q g (q )u r ρ. (52) To proceed further, we need the followng assumptons: Assumpton 3: The tme-varyng control nput, u r 0 of the leader robot s unknown to any follower robot but ts upper bound, ū r 0 s avalable to ts neghbors. Assumpton 4: The followng bounds for the follower robots n expresson (52) are assumed to be known: where D r > 0, M mn > 0. It s obvous that, by defnng q... Q U r q n u r.. u r n ρ < D r <, (53) M mn mn{m,...,m n } (54) ρ q... Q ρ.. ρ n q n M 0 C 0 M... C... 0 M n 0 C n D 0 g D... G. 0 D n g n wth E r e r e r n T E r2 e r2 e r2 n T e r e r2 a j (q q j )b (q q 0 ) j a j ( q q j )b ( q q 0 ) j the error dynamcs of the multrobot system can be expressed as E E 2 E 2 L B M (C D ) Q G U r ρ BM 0 (C0 D 0 ) q 0 g 0 u r 0. (55) The objectve of ths secton s to desgn a robust consensus controller for the multrobot system based on Assumptons 3 and 4 so that, for any follower robot wth bounded uncertantes, the error dynamcs, (55) can be brought to the TSM surface n fnte tme, and on the TSM surface, the error dynamcs can then converge to zero n fnte tme, whch mples fnte-tme consensus trackng of multrobot system. Theorem 6: Consder the multrobot systems n expressons (5) and (52). If the drected graph of ths multrobot system G has a drected spannng tree, and the TSM surface s defned as wth s r e r where the TSM control s desgned as u r (C D ) q g { (e r2 ) 2 α α S r s r,...,s r n T (56) ( e r2 ) α, for,...,n (57) j,j j,j b M 0 (C 0 D 0 ) q 0 g 0 (a j b) M a j M j (C j D j ) q j g j u r j ( 2nM mn D r b M mnūr 0 κ ) sgn(s ) }. (58) Then, consensus can be reached n fnte tme, whch means the follower robot can track the leader robot n fnte tme. Proof: Consder the followng Lyapunov functon: V r 2 ST r S r. (59) Dfferentatng V r wth respect to tme, and substtutng (58) nto t yelds V r Sr T {E r2 α dag ( Er2 α ) ( L B)M ( (C D ) Q G U r ρ) B M0 (C 0 D 0 ) q 0 g 0 u r 0}. (60) otng that I ( D B) M ĀM M ( D B) ( D B Ā)M M ( D B) ( L B)M, the control nput U r (C D ) Q G ( D B) M { (E 2 ) 2 α ĀM α (C D ) Q G U r BM 0 (C 0 D 0 ) q 0 g 0 dag(m mn (2nD r b ū r 0)κ )sgn(s) } (6)

7 KHOO et al.: ROBUST FIITE-TIME COSESUS TRACKIG ALGORITHM FOR MULTIROBOT SYSTEMS 225 can be rewrtten as U r M ( L B) ( D B)M { (C D ) Q G ( D B) (E 2 ) 2 α M ĀM α (C D ) Q G BM 0 (C 0 D 0 ) q 0 g 0 dag(2nm mn D r b M mnūr 0 κ )sgn(s) From (60), t follows that { V r Sr T E r2 α dag(er2 α ) ( L B)M }. (62) ( (C D ) Q G ρ) BM 0 ( (C 0 D 0 ) q 0 g 0 u r 0)( D B)M ((C D ) Q G ) E 2 α 2α ( (C D ) Q G ) BM 0 ( (C 0 D 0 ) q 0 g 0 ) dag(2nm } b M mnūr 0 κ )sgn(s r ) Sr T {α dag(er2 α )( L B)M ĀM mn D r ρ BM 0 (u r 0) dag(2nmmn D r b M mnūr 0 k )sgn(s r )} α s r k (e 2r k ) α (2nMmn D r b k M mnūr 0 k k ) α (( s r k (e 2r k ) α k ) b k (M k ρ k ) b k M k u r 0 α j,j k j,j k a kj (M j ρ j ) a kj s r k (e 2 k ) α (κ ) (63) k that s V r 2 2 η(e2 )V 2 r. (64) Therefore, based on the explanaton n Secton III, the TSM surface S r E r Er2 α 0can be reached n fnte tme. On the TSM surface, E r E r2 0s reached n fnte tme. Ths completes the proof. ) Fg.. Drected graph used n (36) and (58). V. UMERICAL EXAMPLES Ths secton presents some smulaton results to llustrate the performance of the proposed RFTCT algorthms. A. Consensus Trackng Control of Multagent Systems Here, we consder one leader ndexed by 0 and four followers ndexed by, 2, 3, and 4, respectvely. Suppose that the leader dynamcs are ẋ 0 v 0 v 0 u 0 (65) and the dynamcs of th follower are descrbed as follows: ẋ v v u 0.0 sn (x ),, 2, 3, 4. (66) Let the ntal condton of the four followers ndexed by, 2, 3, and 4 be x (0), x 2 (0).2, x 3 (0) 2, and x 4 (0).2, respectvely. Suppose the drected graph n Fg. s used to model the nformaton exchange among agents, where the nformaton of leader s avalable only to followers 3 and 4. ote that follower 4 has no drected path to all other followers, but there exsts a drected path from the leader to all followers. The adjacent matrx of the graph can be wrtten as A (67) The Laplacan of the follower system can be wrtten as L (68) and the dagonal matrces for the nterconnecton relatonshp between the leader and the followers s B dag(0 0 ). (69)

8 226 IEEE/ASME TRASACTIOS O MECHATROICS, VOL. 4, O. 2, APRIL 2009 Fg. 2. onholonomc dfferentally drven wheeled moble robot. Fg. 4. Formaton control of four followers wth nput dsturbances. Fg. 3. Poston trackng of four followers wth nput dsturbances. The smulaton results are obtaned wth the tme-varyng control nput to the leader beng desgned as u 0 sn (x 0) exp t (70) where x 0 π 2 and v 0 0. Fg. 3 shows the results of the proposed TSM control n (36). It s seen that the followers can track the leader n a fnte tme under the nosy condton. To llustrate the formaton control strategy, we use the error functons for formaton control (23), where, 2 2, 3 3, and 4 4. As shown n Fg. 4, x x 0, x 2 x 0 2, x 3 x 0 3, and x 4 x 0 4 n a fnte tme. B. Consensus Trackng Control Of Multrobot Systems In ths subsecton, we valdate the proposed TSM control on a multrobot system. We consder the AmgoBots gven n 8. Based on Fg. 2, the knematc equatons for the th robot are ṙ x v cos (θ ), ṙ y v sn (θ ), θ ω. (7) The hand poston s gven by hx rx h y r y L cos (θ ) sn (θ ). (72) Fg. 5. Poston trackng of four robots wth nput dsturbances. Dfferentatng (72) twce wth respect to tme gves ḧx cos (θ ) L sn (θ ) v ḧ y sn (θ ) L cos (θ ) ω sn (θ ) v ω L cos(θ ) ω 2. (73) cos (θ ) v ω L sn (θ ) ω 2 Lettng v cos (θ ) sn(θ ) ux (74) ω L sn (θ ) L cos (θ ) u y g sn (θ ) v ω L cos (θ ) ω 2 (75) g 2 cos (θ ) v ω L sn (θ ) ω 2 gves ḧx ux g (76) ḧ y u y g 2 whch takes n the form of (50), mplyng that the consensus algorthms n Theorem 6 can be drectly appled. Suppose the drected graph n Fg. s used to model the nformaton exchange among robots. Let the ntal condton of the four robots ndexed by, 2, 3, and 4 be h x h y T 0 0. T, h x2 h y 2 T T, h x3 h y 3 T T, and h x4 h y 4 T T, respectvely. Fg. 5 shows the

9 KHOO et al.: ROBUST FIITE-TIME COSESUS TRACKIG ALGORITHM FOR MULTIROBOT SYSTEMS 227 Fg. 6. Formaton control of four robots wth nput dsturbances. effectveness of the proposed consensus algorthm n (58). To llustrate the formaton control strategy, we set T, T, T, and T. It s seen n Fg. 6, h x h y T h x0 h y 0 T T, h x2 h y 2 T h x0 h y 0 T T, h x3 h y 3 T h x0 h y 0 T T, and h x4 h y 4 T h x0 h y 0 T T n fnte tme. VI. COCLUSIO Ths paper has presented an RFTCT control scheme for leader follower multagent systems wth applcatons to multrobot systems. A new error functon s proposed for the system to ensure consensus. By usng ths error functon, the consensus control problem s transformed to the conventonal control problem. It s proven that fnte-tme consensus can be reached on the TSM surface. Smulaton results have valdated the analyss. The proposed RFTCT algorthm can be easly appled to practcal control of multrobot systems. REFERECES D. V. Dmarogonas and K. J. Kyrakopoulos, On the rendezvous problem for multple nonholonomc agents on the rendezvous problem for multple nonholonomc agents, IEEE Trans. Autom. Control, vol. 52, no. 5, pp , May J. Huang, S. M. Farrtor, A. Qad, and S. Goddard, Localzaton and follow-the-leader control of a heterogeneous group of moble robots, IEEE/ASME Trans. Mechatroncs, vol., no. 2, pp , Apr R. C. Luo and T. M. Chen, Development of a mult-behavor based moble robot for remote supervsory control through the Internet, IEEE/ASME Trans. Mechatroncs, vol. 5, no. 4, pp , Dec P. A. Blman and F. T. Gancarlo, Average consensus problems n networks of agents wth delayed communcatons, Automatca, vol. 44, no. 8, pp , R. Olfat-Saber and R. M. Murray, Consensus problems n networks of agents wth swtchng topology and tme-delays, IEEE Trans. Autom. Control, vol. 49, no. 9, pp , Sep E. Yang and D. Gu, onlnear formaton-keepng and moorng control of multple autonomous underwater vehcles, IEEE/ASME Trans. Mechatroncs, vol. 2, no. 2, pp , Apr F. Cucker and S. Smale, Emergent behavor n flocks, IEEE Trans. Autom. Control, vol. 52, no. 5, pp , May C. Q. Ma, T. L, and J. F. Zhang, Leader followng consensus control for mult-agent systems under measurement noses, n Proc. 7th World Congr. Int. Fed. Autom. Control, 2008, pp W. Ren, R. W. Beard, and T. W. Mclan, Coordnaton varables and consensus buldng n multple vehcle systems, n Proc. Block Island Workshop Cooperatve Control., Sprnger-Verlag Seres: Lecture otes n Control and Informaton Scence, 2007, vol. 390, pp J. Cortés, Fnte-tme convergent gradent flows wth applcatons to network consensus, Automatca, vol. 42, pp , X. Wang and Y. Hong, Fnte-tme consensus for mult-agent networks wth second order agent dynamcs, n Proc. 7th World Congr. Int. Fed. Autom. Control, 2008, pp Z. Man, A. P. Paplnsk, and H. R. Wu, A robust MIMO termnal sldng mode control scheme for rgd robotc manpulators, IEEE Trans. Autom. Control, vol. 39, no. 2, pp , Dec Y. Feng, X. Yu, and Z. Man, on-sngular termnal sldng mode control of rgd manpulators, Automatca, vol. 38, no. 2, pp , S. Khoo, Z. Man, and S. Zhao, Termnal sldng mode control for MIMO T-S fuzzy systems, n Proc. 6th Int. Conf. Inform. Commun. Sgnal Process., 2007, pp S. Khoo, Z. Man, and S. Zhao, Adaptve fast fnte tme control of a class of nonlnear uncertan systems, n Proc. 3rd IEEE Int. Conf. Ind. Electron. Appl., 2008, pp W. Ren, Mult-vehcle consensus wth a tme-varyng reference state, Syst. Control Lett., vol. 56, no. 7 8, pp , W. Ren and R. W. Beard, Consensus seekng n multagent systems under dynamcally changng nteracton topologes consensus seekng n multagent systems under dynamcally changng nteracton topologes, IEEE Trans. Autom. Control, vol. 50, no. 5, pp , May W. Ren and R. W. Beard, Dstrbuted Consensus n Mult-vehcle Cooperatve Control. ew York: Sprnger-Verlag, Y. Hong, G. Chen, and L. Bushnell, Dstrbuted observers desgn for leader followng control of mult-agent networks, Automatca, vol. 44, no. 3, pp , W. Ln and C. Qan, Adaptve control of nonlnearly parameterzed systems: A nonsmooth feedback framework, IEEE Trans. Autom. Control, vol. 47, no. 5, pp , May S. P. Bhat and D. S. Bernsten, Geometrc homogenety wth applcatons to fnte-tme stablty, Math. Control Sgnals Syst. (MCSS), vol. 7, no. 2, pp. 0 27, X. Huang, W. Ln, and B. Yang, Global fnte-tme stablzaton of a class of uncertan nonlnear systems, Automatca,vol.4,no.5,pp , S. P. Bhat and D. S. Bernsten, Fnte-tme stablty of contnuous autonomous systems, SIAM J. Control Optm., vol.38,no.3,pp , Y. Hong, J. Hu, and L. Gao, Trackng control for mult-agent consensus wth an actve leader and varable topology, Automatca, vol. 42, no. 7, pp , C. C. Cheah, S. P. Hou, and J. J. E. Slotne, Regon followng formaton control for mult-robot systems, n Proc. IEEE Int. Conf. Robot. Autom., 2008, pp J. J. E. Slotne and W. L, Appled onlnear Control. Englewood Clffs, J: Prentce-Hall, 99. Suyang Khoo receved the B.Eng. degree n electroncs and communcatons engneerng from Tasmana Unversty, Hobart, Australa, and the Ph.D. degree n computer engneerng from anyang Technologcal Unversty, Sngapore, n 2005 and 2007, respectvely. In 2008, he was a Lecturer n the Department of Electrcal and Computer Systems Engneerng, Monash Unversty, Sunway, Selangor, Malaysa. Snce 2009, he has been wth the School of Engneerng, Deakn Unversty, Geelong, Australa, where he s currently a Lecturer n Electroncs Engneerng. Hs current research nterests nclude varable-structure control, robust fast fnte-tme control, cooperatve control, robotcs, adaptve sgnal processng, tme-varyng systems, nonlnear systems, neural networks, fuzzy systems, and automatc generaton control for multarea power networks.

10 228 IEEE/ASME TRASACTIOS O MECHATROICS, VOL. 4, O. 2, APRIL 2009 Lhua Xe (F 07) receved the B.E. and M.E. degrees n electrcal engneerng from anjng Unversty of Scence and Technology, Jangsu, Chna, n 983 and 986, respectvely, and the Ph.D. degree n electrcal engneerng from the Unversty of ewcastle, Sydney, Australa, n 992. Snce 992, he has been wth the School of Electrcal and Electronc Engneerng, anyang Technologcal Unversty, Sngapore, where he s currently a Professor and the Drector of the Centre for Intellgent Machnes. He held teachng appontments n the Department of Automatc Control, anjng Unversty of Scence and Technology, from 986 to 989. He has authored or coauthored over 60 journal papers and two patents and the books H-nfnty Control and Flterng of Two-dmensonal Systems (wth C. Du, Sprnger, 2002); Optmal and Robust Estmaton (wth F. L. Lews and D. Popa, Automaton and Control Engneerng, 2007), and Control and Estmaton of Systems wth Input/Output Delays (wth H. Zhang, Sprnger, 2007). Hs current research nterests nclude robust control and estmaton, networked control systems, tme-delay systems, control of dsk drve systems, and sensor networks. Dr. Xe s a Fellow of the Insttuton of Engneers, Sngapore. He was the General Charman of the 9th Internatonal Conference on Control, Automaton, Robotcs and Vson. He s an Assocate Edtor of Automatca, theieee TRAS- ACTIOS O COTROL SYSTEMS TECHOLOGY,theTransactons of the Insttute of Measurement and Control, and an Assocate Edtor at Large of the Journal of Control Theory and Applcatons, and s also a member of the Edtoral Board of the IET Proceedngs on Control Theory and Applcatons. Heservedasan Assocate Edtor of the IEEE TRASACTIOS O AUTOMATIC COTROL from 2005 to 2007, the IEEE TRASACTIOS O CIRCUITS AD SYSTEMS II from 2006 to 2007, the Internatonal Journal of Control, Automaton, and Systems from 2004 to 2006, and the Conference Edtoral Board, IEEE Control Systems Socety, from 2000 to Zhhong Man (M 02) receved the B.E. degree from Shangha Jaotong Unversty, Shangha, Chna, the M.S. degree from the Chnese Academy of Scences, Bejng, Chna, and the Ph.D. degree from the Unversty of Melbourne, Melbourne, Australa, n 982, 996, and 993, respectvely, all n electrcal and electronc engneerng. From 994 to 996, he was a Lecturer n the Department of Computer and Communcaton Engneerng, Edth Cowan Unversty, Perth, Australa. From 996 to 200, he was a Lecturer and then Senor Lecturer n the School of Engneerng, Unversty of Tasmana, Australa. In 200, he was a Vstng Senor Fellow n the School of Computer Engneerng, anyang Technologcal Unversty (TU), Sngapore. From 2002 to 2007, he was an Assocate Professor of Computer Engneerng at TU. From 2007 to 2008, he was a Professor of Electrcal and Computer System Engneerng n the School of Engneerng, Monash Unversty, Malaysa. Snce 2009, he has been a member of the Faculty of Engneerng, Swnburne Unversty of Technology, Melbourne, Australa, where he s a Professor of Robotcs and Mechatroncs. Hs current research nterests nclude neural networks, fuzzy systems, tme-varyng systems, nonlnear systems, sgnal processng, robotcs, ntellgent control, transmsson control protocol (TCP), congeston control, and network control. He has authored or coauthored more than 40 journal and conference papers n these areas. Prof. Man served as the Char of the Techncal Commttee of the Thrd IEEE Conference on Industral Electroncs and Applcatons (ICIEA) 2008 and has been on the program commttees and nternatonal advsory commttees for many IEEE conferences snce 994. In addton, he receved the TU Best Teacher Award n 2004 and the Most Popular Lecturer Award from the School of Computer Engneerng at TU, from 2002 to 2006.