J Sys Sc Complex (11) 4: 631 646 COOPERATIVE CONTROL OF MULTI-AGENT MOVING ALONG A SET OF GIVEN CURVES Yangyang CHEN Yupng TIAN DOI: 1.17/s1144-11-9158-1 Receved: 8 June 9 / Revsed: Aprl 1 c The Edoral Offce of JSSC & Sprnger-Verlag Berln Hedelberg 11 Absrac Ths paper deals wh a cooperave conrol problem of a eam of double-negraor agens movng along a se of gven curves wh a nomnaed formaon. A projecon-rackng desgn mehod s proposed for desgnng he pah-followng conrol and he formaon proocol, whch guaranee formaon moon of he mul-agen sysem under a dreced communcaon graph. Necessary and suffcen condons of he conrol gans for solvng he coordnaed problem are obaned when he dreced communcaon graph has a globally reachable node. Smulaon resuls of formaon moon among hree agens demonsrae he effecveness of he proposed approach. Key words Cooperave conrol, formaon moon, graph heory, mul-agen sysem, pah followng. 1 Inroducon In recen years, he conrol of mulple agens movng n formaons has emerged as a opc of wdespread neres due o s broad range of applcaons n mlary mssons, envronmenal surveys, and space mssons. Through effcen coordnaon, mulple smple and nexpensve agens can subsue for a sngle complcaed and expensve sysem o execue he same operaon, examples of whch can be found n hunng operaons [1], ransporaon of a large objec [],omenonjusafew. From a convenonal pon of vew, he problem of formaon conrol of mul-agen sysems falls whn he doman of seerng relave posons or ner-agen dsances o he nomnaed values, regardless of he moon pah of each agen [3 6]. Wh agle sensor neworks appled o collec daa n he sky, on he ground and underwaer, some of he laes research resuls demonsrae ha he resrcon of each agen s pah durng performng formaon moon no only sgnfcanly exers all sensor powers, bu also grealy reduces he fuel and execuve me cos (see, e.g., he adapve samplng and predcon projec n [7] and he operaon of mulple underwaer vehcles for fas acousc coverage of he seabed n [8]). The mposng moon consrans on he ner-agen formaon paern have movaed he need for a sysemac mehod o consruc cooperave conrollers for mul-agen movng along he gven curves. Ths problem has been addressed n a number of recen publcaons. Zhang and Leonard [9 1] presened he mehod of orb funcon o coordnae paerns of un-speed parcles on a closed Yangyang CHEN Yupng TIAN School of Auomaon, Souheas Unversy, Nanjng 196, Chna. Emal: ypan@seu.edu.cn. Ths research s suppored by Naonal Naural Scence Foundaon of Chna under Gran Nos. 697441 and 69346. Ths paper was recommended for publcaon by Edor Jng HAN.
63 YANGYANG CHEN YUPING TIAN loop under he bdreconal chan communcaon opology. In [11], he mehod of orb funcon s exended o solve he coordnaed problem of a famly of double-negraor agens movng along a closed curve based on he assumpon ha he velocy of each parcle s nonzero. By usng he ools of lnearzaon and gan schedulng echnques, Ghabcheloo, e al. [1] nvesgaed he coordnaed pah followng conrol of mulple vehcles, bu only local resuls are yelded and he desred curves are resrced n crcumferences and o sragh lnes. Wh he purpose of overcomng hese lmaons, he rae of progresson of a vrual arge movng along each gven pah s conrved by he backseppng echnque o decouple pah followng and ner-vehcle coordnaon n [13]. Pah followng for each vehcle s accomplshed by dervng a feedback conrol law for he orque of each vehcle. The speed assgnmen, whch depends on he undreced communcaon nformaon of vehcles saes n he formaon, guaranees convergence o he desred formaon wh a consan reference velocy. In [14], a coordnaed pah followng proocol s presened when he communcaon opology s dreced. Smlar deas are used n [15] for he case of mulple fully acuaed underwaer vehcles. However, all he dynamcs of he agens and he predefned curves n foregong references are consdered n he Serre-Frene frame. In he mul-agen sysem, each agen can only know s own and s neghbors (and herefore local) nformaon due o lmed sensor or communcaon capacy. A possble way o ge he local nformaon of each agen s o use he onboard sensors whch measure he relave posons and veloces among mulple agens. In hs crcumsance, he Serre-Frene frame mgh be more suable. However, here are alernave means o provde agen s local nformaon n a fxed Caresan coordnaed sysem, such as sensor neworks or GPS. Through sensor neworks or GPS, he absolue locaon for each agen n a fxed Caresan coordnae sysem s deermned and hen ransmed o s neghbors va communcaon. Based on he absolue locaon nformaon, he desred pah for each agen s also planned n he common recangular coordnaes. Ths movaes he auhors o sudy he coordnaed problem of mul-agen formaon movng along a se of curves wh respec o he Caresan coordnae frame. The man conrbuon of hs paper s he developmen of a new mehod (we call projecon-rackng desgn mehod) for coordnang mulple agens movng along a se of gven curves wh a nomnaed formaon. The gven curves under consderaon are a famly of specal ones, where each curve s vercal coordnae can be explcly expressed by a C smooh funcon of s horzonal ordnae. Dfferng from he problems resrced o he area of coordnaed paerns of un speed parcles on a close loop [9 11], he nal speed of each agen n our consderaon may be varous, and our mehod does no requre ha he velocy of each agen s always nonzero. Nong he common communcaon opology n pracce (e.g., some agens may have ranscevers, whle oher less capable eam member only have recevers), he dreced communcaon opology s consdered n hs paper. Thus, he overall communcaon cos here s much reduced compared wh he b-dreced communcaon used n [9 13]. The desgn procedure n hs paper can be brefly skeched as follows: In he begnnng, we projec vercally each agen on s gven pah. The dynamcs of he projecon of each agen can be easly ganed from he dynamcs of he acual agen whou desgnng an exra conrol law for he vrual arge as done n [13 15]. Then, pah followng s acheved when he error dynamcs beween he acual agen and s projecon s sablzed by he normal proporon of each agen s acceleraon along s pah. A he same me, he horzonal proporon of each agen s acceleraon s also adjused o solve he formaon moon problem hrough achevng he consensus of agens posons and veloces along he gven curves based on he local nformaon. Accordng o he analyss of sysem marx, necessary and suffcen condons of he conrol gans for solvng he cooperave problem are obaned n hs paper, whereas only suffcen condons are shown n [13 15] by usng he Lyapunov approach.
COOPERATIVE CONTROL ALONG GIVEN CURVES 633 The paper s organzed as follows: Secon gves some prelmnary resuls. The man dea of he desgn mehod based on projecon-rackng and he problem saemen are presened n Secon 3. Secon 4 gves our man resuls. Compuer smulaons are ncluded n Secon 5. Concluson s provded n Secon 6. Prelmnares We frs revew some noons n graph heory [16] ha are needed for he furher dscusson. In he mul-agen sysem, agen can only receve he nformaon of s neghbors. I s naural o model he dreced communcaon opology among agens by a dreced graph G = {V, E},whereV s a se of nodes and E V s a se of ordered pars of nodes, called edges. If here s a dreced edge from V o V j, means ha agen can receve he nformaon of agen j and agen j s a neghbor of agen. The se of neghbors of agen s denoed by N. In hs paper, we assume ha N s me-nvaran. A pah of lengh N from V j o V jn n adgraphg s an ordered se of dsnc nodes {V j, V j1,, V jn } such ha ( ) V j 1, V j E, =1,,,N.IfheresapahnGfrom one node V o anoher V j,henv j s sad o be reachable from V. If a node V s reachable from every oher node n he dgraph, hen we say s a globally reachable node. The adjacency marx A =[a j ] of a weghed dgraph s defned as a =anda j > f(v, V j ) E. The Laplacan marx L =[l j ] s defned as l = a j j and l j = a j. The followng lemma plays a key role n he sably analyss hroughou he paper. Lemma 1 [17] Consder a sysem: [ ] [ ][ ] n n I = n γ, γ ζ τ L αi n τ 1 L ζ where γ =[γ 1,γ,,γ n ] T, ζ =[ζ 1,ζ,,ζ n ] T, τ >, α. Assume he dgraph has a globally reachable node. The consensus such ha γ γ j and ζ ζ j as s acheved for all, j =1,,,n f and only f τ 1 > max μ + αq + q ) α q 4p τ (p + q ) p p (p + q ), ( α where μ, =1,,,n, are he egenvalues of L, p =Re(μ ), q =Im(μ ),andre( ) and Im( ) represen he real and magnary pars of a number, respecvely. Furhermore, f α>, lm ζ () =. 3 Projecon-Trackng Desgn Mehod Consder a formaon comprsng n agens movng n he plane. We vew each agen as a double negraor ha obeys ż = υ, υ = u, (1) where z = [x,y ] T R s he poson of agen n he recangular coordnaes, υ = [υ x,υ y ] T R s velocy and u =[u x,u y ] T R s acceleraon npu, =1,,,n. Suppose ha Γ s a gven curve for agen, denfed n he recangular coordnaes by equaon f (x) y =, ()
634 YANGYANG CHEN YUPING TIAN where x R and f s a C smooh funcon of x. In consderaon of a gven curve followed by a movng agen, he man dea nvolved n our dervaon of he pah-followng conrol s o vercally projec agen on s gven pah Γ and hen o drve agen o approach o s projecon β asympocally (see Fgure 1). Due o he fac ha he horzonal coordnae of s projecon β s he same as agen, one can easly fgure ou he poson of s projecon β from he dynamcs of agen such ha z β =[x β,y β ] T =[x,f (x )] T. (3) Γ υ r d υ β y β Tangen vecor along he pah Normal vecor along he pah o x Fgure 1 Pah followng frame Le d () denoe he dfference n he longudnal coordnae beween agen and s correspondng projecon β. The pah followng for each agen s acheved by manpulang d () and s dervave r () =dd ()/d o zero,.e., lm d () =, lm r () =. (4) Each agen s acceleraon along he normal vecor wh respec o s curve wll be desgned o mee hs requremen. When each agen moves along s gven curve, he descrpon of formaon for mulple agens degeneraes no he agreemen of posons and veloces of he agens along her gven curves. Nocng he specal form () of each gven curve, only one value of y assocaes wh agvenx. Once he relave abscssa of each par agens s decded, he relave posons of agens are unquely confrmed. Also, he relave posons of agens correspond wh one relave abscssa for each par. Thus, s a one-o-one correspondence beween he formaon paern and he relave abscssas of agens. In he smples case, curve Γ s obaned by movng a emplae curve (e.g., he desred rajecory of he group cener) along a desred formaon vecor h =[h x,h y ] T as shown n Fgure. The desred formaon z () z j () =h h j s equal o he form x () x j () =h x h xj. Furhermore, s obvous ha hs formaon can be ensured by he consensus of boh x () h x and ẋ (). In realy, formaon paerns are complcaed and pah-dependen. I s hard o use he consensus of x () h x and ẋ () o solve all he complcaed formaon problems. For nsance, consder he coordnaon depced n Fgure 3, where agens 1 and mus follow pah Γ 1 and Γ whle mananng agen 1 ohe-rgh-ahead agen, ha s, along sragh lne ha makes an angle of 15 degree wh he posve drecon of pah Γ 1. Obvously, he relave poson for agens 1 and should sasfy x 1 () =4x (). If we choose ξ 1 = x 1 and ξ =4x, one can easly observe ha he
COOPERATIVE CONTROL ALONG GIVEN CURVES 635 formaon can be mananed when ξ 1 ξ =and ξ 1 ξ =. For he purpose of fulfllng he more general formaon paerns, here we use he consensus of performance ndex ξ () and s dervave η () =dξ ()/d o mee hs requremen. h j j Γ j j k h h j h h j k Γ h h k Γ k h y Templae curve h k k o x Fgure Trangle formaon y Γ o x 15 o 1 x 1 Γ 1 x Fgure 3 A general formaon paern Assumpon 1 The performance ndex ξ s a C smooh funcon of abscssa x such ha ξ x ε1 > for all x. I s sad ha he mul-agen sysem manans he desred formaon when he consensus of performance ndex and s dervave s acheved,.e., lm (ξ () ξ j ()) =, lm (η () η j ()) =. (5) Based on each agen s own and s neghbors nformaon, he acceleraon of each agen along x-axs wll be desgned o ensure he consensus. Problem 1 (formaon moon along curves whou any specfed speed) Desgn a cooperave conrol law usng he local nformaon u = g (z,z j,υ,υ j,ξ,ξ j,η,η j ), where agen j N, such ha requremens (4) and (5) are sasfed. In some praccal suaons, s requred ha he formaon be movng along he curves wh some specfed speed [13 15]. Accordng o hese scenaros, he acceleraon of each agen along x-axs mus be changed o yeld he consensus of he performance ndex bu s changng raes o he reference velocy η (),.e., lm (ξ () ξ j ()) =, lm (η () η ()) =, (6)
636 YANGYANG CHEN YUPING TIAN Desgn a cooper- where η () s a unformly bounded sgnal. Problem (formaon moon along curves wh he reference velocy) ave conrol law usng he local nformaon u = g (z,z j,υ,υ j,ξ,ξ j,η,η j,η, η ), where agen j N, such ha requremens (4) and (6) are sasfed. Remark 1 If each predefned curve can be descrbed wh equaon x κ (y) =where y R and κ s a C smooh funcon of y, one can also desgn a cooperave algorhm n he smlar procedure herenafer by buldng up he projecon β of agen on s gven curve whose longudnal coordnae s equal o agen. Remark In [13 15], he desred formaon s mananed by achevng he consensus of generalzed arc-lengh ξ (s ), where ξ (s ) s a monooncally ncreasng funcon of arclengh s,all ξ / s s posve and unformly bounded for all s and ξ / s s unformly bounded. Alhough hs descrpon of formaon s also feasble n hs paper, o compue s whch s an negral of nonlnear funcons may ake more sysem resources n he praccal mplemenaon. By choosng a smpler form of ξ (x ), he numercal negral can be avoded, and hen he compuaon load s reduced. Moreover, from () he arc-lengh can be expressed 1+( f / τ) dτ, wherex as s = x () x arc-lengh ξ (s ) can be regarded as a specal form of ξ (x ). 4 Conrol Desgn s a fxed pon on he curve. Thus, he general In hs secon, we analyze he cooperave conrol of mul-agen movng along a se of gven curves. The pah-followng conrol law s presened n Subsecon 4.1. Subsecon 4. gves he frame of consrucng he formaon conrollers. The whole cooperave algorhms and he man resuls are presened n Subsecon 4.3. Fnally, he boundedness of he conrol npu for each agen s dscussed n Subsecon 4.4. 4.1 Pah Followng Ths subsecon descrbes he precedence of desgnng he pah-followng conroller for each agen. From (3) one can ge he poson dfferen d () beween agen and s projecon β as d = y β y = f (x ) y, (7) whch s also called he poson error of pah followng for each agen. Dfferenang boh sdes of (7), he velocy error r () of pah followng for agen s gven by r = f υ x υ y. (8) x Then, he dervaon of r wh respec o me s ṙ = f u x u y + f x x υx. (9) Nocng ha he normal vecor feld of Γ s [ ] T f n (x )=, 1, (1) x
COOPERATIVE CONTROL ALONG GIVEN CURVES 637 s easy o see ha f x u x u y s he normal componen of each agen s acceleraon along he gven curve. We desgn hs varable as f u x u y = g1 c x + ge 1, (11) where g1 c denoes he nonlnear compensaon par of pah followng and ge 1 s he pah-followng error conrol par. Two conrol erms are gven by g1 c = f x υx, (1) g1 e = k 1 d k r, (13) where k 1 and k are posve consans. I follows ha he correspondng pah-followng error dynamcs becomes [ ] [ ] d d = B ṙ, (14) r where [ B = ] 1. k 1 k One can solve he equaon λ + k λ+k 1 = o fnd he egenvalues of B n he followng form: λ = k ± k 4k 1. (15) I s sraghforward o see ha all he egenvalues of B have negave real pars f and only f k 1 and k are posve consans. Therefore, requremen (4) s sasfed, namely, agen fnally moves along s arge curve Γ. 4. Formaon Moon The soluon o pah followng for each agen s obaned n he prevous subsecon. In he followng, we ry o consder ha agens move along a se of gven curves n such a way as o manan he desred formaon paern compable wh hese pahs. Snce he performance ndex ξ s a C smooh funcon of x, he dervave of ξ () canbe wrennfollowngform η = ξ υ x. (16) x Takng he me dervave of η (), one gans η = ξ u x + ξ x x υx. (17) Consder Problem 1. If we choose he componen of acceleraon npu along he x-axs as ξ u x = g c x + ge, (18) where g c expresses he nonlnear compensaon par of formaon moon such ha g c = ξ x υx, (19)
638 YANGYANG CHEN YUPING TIAN and g e s he error conrol par of formaon moon gven by g e = k 3 a j (ξ ξ j ) k 4 a j (η η j ), () j N j N where k 3 >, hen he whole formaon moon dynamcs becomes [ ] [ ][ ] ξ n n I = n ξ, (1) η k 3 L k 4 L η where ξ = [ξ 1,ξ,,ξ n ] T and η = [η 1,η,,η n ] T. Assume he dgraph has a globally reachable node. Accordng o Lemma 1, demand (5) s me f and only f k3 q k 4 > max μ μ, () p ha s o say, ha mulple agens acheve he nomnaed formaon. For he purpose of achevng formaon moon along curves wh he reference velocy, we renewedly choose he componen of each agen s acceleraon npu along he x-axs as follows: ξ u x = g c + g es, (3) x where he he error conrol par of formaon moon s changed o he followng form: g es = η k 5 (η η ) k 3 a j (ξ ξ j ) k 4 a j (η η j ), (4) j N j N and k 5 s a posve consan. Le ξ = ξ ξ,whereξ = η (τ)dτ and η = η η,he whole formaon moon dynamcs can be rewren as [ ξ η ] [ n n I = n k 3 L k 5 I n k 4 L ][ ξ η ], (5) where ξ =[ ξ 1, ξ,, ξ n ] T and η =[ η 1, η,, η n ] T. Accordng o Lemma 1, when he dgraph has a globally reachable node, he consensus such ha lm ( ξ () ξ j ()) = and lm η () = s acheved f and only f ( k 5 k 4 > max + k 5q + q ) k 5 q 4p k 3 (p + q ) μ p p (p + q ), (6) whch mples ha requremen (6) s sasfed. 4.3 Man Resuls From he above dscusson, he man resuls are esablshed as follows. Theorem 1 Consder a famly of smooh curves gven by Equaon (). Suppose all he performance ndces sasfy Assumpon 1 and he dgraph has a globally reachable node and k 3 >. Problem 1 s solved va he cooperave conrol ( ) [ ] 1 ξ ux x = (g c + g e ) ( ) u 1, (7) y (g1 c + ge f 1 )+ ξ x x (g c + g e )
COOPERATIVE CONTROL ALONG GIVEN CURVES 639 f and only f k 1,k are posve consans and k 4 sasfes he nequaon (). Proof If he normal componen of each agen s acceleraon along he gven curve s desgned as (11), where k 1,k are posve consans, he pah-followng errors are manpulaed o zero because all he egenvalues of he sysem marx B of he pah-followng error dynamcs have negave real pars. Smulaneously, f he componen of acceleraon npu along he x-axs s consruced as (18), where k 3 > and he communcaon opology has a globally reachable node, mulple agens acheve he nomnaed formaon f and only f k 4 sasfes nequaly () based on Lemma 1. Accordng o (11) and (18), he acceleraon npu for each agen sasfes [ 1 ξ x ][ f x 1 1 ][ ux ] [ g c = 1 + g1 e u y g c + ge ]. (8) Obvously, Equaon (8) has a unque soluon (7) f ξ x. I has proved ha Problem 1 s solved va he conroller (7). Smlarly, Problem s solved by he followng heorem. Theorem Consder a famly of smooh curves gven by Equaon (). Suppose all he performance ndces sasfy Assumpon 1. Assume he dgraph has a globally reachable node and k 3,k 5 are posve consans. Problem s solved va he cooperave conrol ] = u y [ ux ( ) 1 ξ x (g c + g es ( ) ) 1 (g1 c + ge f 1 )+ ξ x x (g c + g es) f and only f k 1,k are posve consans and k 4 sasfes he nequaly (6). (9) 4.4 Dscusson on Boundedness of Conrol In he praccal applcaons, he conrol npu for each agen mus be fne. Hence, he dscusson on he boundedness of conrol s called for. Owng o he smlar procedure n analyss of he boundedness of conrol beween (7) and (9), we jus gve he whole analyss of he boundedness of conroller (7) herenafer. Rewre he acceleraon npu (7) n he followng form: where u x = u e x + uc x, u y = u e y + uc y, (3) ( ) 1 u e ξ x = g x, e (31) u e y = ge 1 + f ( ) 1 ξ g e x x (3) ( ) 1 u c x = ξ ξ x x υx, (33) [ u c f y = x f ( ) ] 1 ξ ξ x x x υx. (34)
64 YANGYANG CHEN YUPING TIAN When he dynamcs of formaon moon s sable, η () s bounded f he nal saes z () and υ () are fne. From (16) and Assumpon 1, one gans η () c 1 < υ x c 1 c ( ) 1 ξ mn, (35) ε 1 x ha s, υ x s bounded. Thanks o he sably of he dynamcs of pah-followng error and formaon moon, he error conrol pars g1 e and ge canno end o nfny f he nal saes z () and υ () are fne. From (31) and Assumpon 1, one can obvously see ha u e x s always bounded. Thus, he boundedness of conroller (7) s deermned by u e y, uc x and uc y. Snce only he form of performance ndex ξ (x ) can be chosen n he conroller (7), we dvde he analyss no / sx cases based on x and he characerscs of he gven curve such ha f / x and f x o show whch knd of ξ (x ) can ensure bounded npu for each agen. 1) Each agen moves on he se Ω = {x R x c < }. As such, f and ξ x x, f x, ξ x are bounded because f (x) andξ (x )arewoc smooh funcons. Due o he fac ha υ x s bounded, from (33) and (34), s obvous ha u c x and uc y are bounded. Also, he error conrol pars g1 e and ge are bounded f he nal saes z () and υ () are fne. Thus u e y remans bounded on Ω. Therefore, we conclude ha he conrol (7) s bounded on Ω,ha s, he conrol s always bounded when he abscssa of each agen s fne. ) We dscuss he boundedness of conroller when f x c 3 and f c x 4 as x, where c 3 and c 4 are wo bounded numbers. Owng o he he fneness of υ x, g1 e and ge,from (3) (34), one can obvous see ha u e y s bounded also uc x and uc y don end o nfne f and only f ( ) 1 ξ ξ x x c 5 (36) as x where c 5 s a bounded number. Under Assumpon 1, we can choose a suable ξ (x ) sasfyng (36) o ensure bounded conrol n hs case. 3) Consder ha f x c 3 and f as x x. Obvously, u c x s bounded f and only ( ) 1 f (36) s enable. Under hs condon, f ξ ξ x x s bounded bu u c x y ends o nfne due o lm f x =. Thus, no performance ndex can guaranee he bounded conrol n x hs suaon. 4) Consder he case ha f x and f c x 4 as x. Due o he fne of g1 e and g e, from (3) ue y remans bounded f and only f ( ) 1 f ξ c 6 (37) x x as x,wherec 6 s a bounded number. From (33) and (34), u c x and uc y reman bounded f and only f ( ) 1 f ξ ξ x x x c 7 (38) as x,wherec 7 s a bounded number. Suppor all he performance ndces sasfy Assumpon 1. The conroller remans bounded f and only f we choose he performance ndex ξ (x ) sasfyng (37) and (38) n hs suaon.
COOPERATIVE CONTROL ALONG GIVEN CURVES 641 ( ) p 5) Assume f x and f f x c x 8 as x where <p 1andc 8 s a bounded number and nonzero. From (3), u e y remans bounded f and only f (37) s enable. Also, u c x remans bounded f and only f (36) s enable accordng o (33). From (34), we know ha u c y remans bounded f and only f f x f ( ξ x x ) 1 ξ x c 9 (39) as x,wherec 9 s a bounded number. Once (39) s enable, he condon (36) s naural sasfed. To show hs resul, we frs assume he nverse s rue. As such, because ( ) p ( ) 1 p ( ) 1 f f x c x 8, f x and ξ ξ x as x x,wehave lm x ( ) 1 ( ) p ( ) 1 p ( ) 1 f f f ξ ξ x x x x x =. (4) However, from (39) one gans lm x ( ) 1 ( ) p ( ) 1 p ( ) 1 f f f ξ ξ x x x x x =1, (41) whch s n conradcon wh he earler equaly (4). Therefore, for purpose of keepng he conroller boundedness, he seleced ξ (x ) should sasfy Assumpon 1, (37) and (39) n hs case. ( ) p 6) Le us consder he case ha f x and f f x x From (33) u c x remans bounded f and only f (36) s enable. Also, uc y and only f (39) s enable. ( ) 1 f f x x Due o ( f x ) p f x asx. Snce (36) s enable, s easy o see ha c 8 as x where p>1. remans bounded f c 8 as x where p>1, we have ( ) 1 ( ) 1 f f ξ ξ x x x x. (4) However, from (39) we have he equaly (41) whch s n conradcon wh he earler equaly (4). Therefore, no performance ndex can guaranee he bounded conrol n hs case. From he above dscusson, he condons of he boundedness of conrollers are esablshed as follows. Theorem 3 Suppose all he performance ndces sasfy Assumpon 1. Theconrollers(7) and (9) are always bounded when he abscssa of each agen s fne. Also, when f c 3 and f x c 4 as x, he conrol npu for each agen remans bounded f and only f he chosen ( ) p performance ndex ξ (x ) sasfes (36). Even n he case of f x and f f x cons x as x,where <p 1, he conrol npu can reman bounded f and only f he performance ndex ξ (x ) s suably chosen o mee he condons (37) and (39). Furhermore, he condon (39) degeneraes no (38) when f x and f c x 4 as x. x
64 YANGYANG CHEN YUPING TIAN 5 Smulaon Resuls In hs secon, we apply he cooperave conrol laws o coordnae he movemen of hree agens along a se of gven curves. The communcaon opology among hree agens s shown n Fgure 4. Here we choose a j =1,when(V, V j ) E. Accordng o he pah-followng conroller (11), k 1 and k mus wo posve consans. Le k 3 =1. k 4 mus be more han.415 accordng o (). From (6), k 4 mus be more han.337 f k 3 =1andk 5 = 5. Thus, we choose hese conrol parameers as follows: k 1 =15, k =15, k 3 =1, k 4 =7, k 5 =5. 1 Fgure 4 Communcaon opology In he smulaon case, agens 1 and 3 follow he parabola pahs gven by he equaons Γ 1 : x y =andγ 3 : x + y =, whle agen s requred o follow he x-axsashehree manan an n-lne formaon along he y-axs, ha s, agen 1 s uprgh above agen and agen s uprgh above agen 3. The nal condons of hree agens are z 1 () = [1, 7] T, z () = [, 1] T, z 3 () = [1.5, ] T, υ 1 () = [.3,.4] T, υ () = [.,.3] T, υ 3 () = [.8,.3] T. In order o acheve he desred formaon and ensure ( bounded ) npu for each agen, he performance ndex ξ (x ) s seleced as ξ = x 1+x. Assumpon 1 s sasfed because ξ x =1+3x 1. I s smpler o selec he performance ndex such as ξ = x, bu he conrol npu may end o nfne when x because he condon (37) sn sasfed. Frsly, we use he cooperave conrol (7) o acheve he formaon movng along hese gven curves. The movemen of agens s shown n Fgure 5(a). From hs fgure, we can see ha hree agens fnally move along he se of gven curves drawn by he dashed lnes and form he desred formaon denoed by he double lnes when he sysem reaches seady sae. Fgure 5(b) and 5(c) demonsrae ha ξ and s dervave η reach consensus. The pah-followng errors d and r end o zero are ploed n Fgure 5(d) and 5(e), respecvely. Fgure 5(f) llusraes he formaon error x x j. Fgure 5(g) and 5(h) show ha he conrol npu u x and u y doesn become larger and larger when x ncreases. Accordng o Fgure 5, we can conclude ha Problem 1 s solved va he cooperave conrol (7). Secondly, we use he cooperave conrol (9) o acheve he formaon movng along he predefned curves wh he reference velocy η () =.5 +.5sn(). The movemen of he agens s shown n Fgure 6(a). From hs fgure, we can see ha hree agens fnally move along he se of gven curves drawn by he dashed lnes and form he desred formaon denoed by he double lnes when he sysem reaches seady sae. Fgure 6(b) demonsraes ha ξ reaches consensus and Fgure 6(c) shows s dervave η converges o he reference one. Fgure 6(d) and 6(e) show ha he pah-followng errors d and r end o zero. Fgure 6(f) llusraes he formaon error x x j. Fgure 6(d) and 6(e) show ha he conrol npu u x and u y sand n a bounded range when x ncreases. From Fgure 6, s obvous ha Problem s solved va he cooperave conrol (9). 3
COOPERATIVE CONTROL ALONG GIVEN CURVES 643 4 r 1 3 z 1 z z 3 8 ξ 1 ξ 3 ξ ξ 1 ξ 3 ξ 6 1 4 y ξ ξ j 1 3 4 4 1 1.5.5 3 3.5 4 4.5 5 5.5 6 x 6 5 1 15 5 3 35 4 45 5 (a) (b) 4 3 η 1 η 3 η η 1 η 3 η 15 d 1 d d 3 1 1 η η j 1 d 5 3 4 5 5 5 1 15 5 3 35 4 45 5 1 5 1 15 5 3 35 4 45 5 (c) (d) 1 1 5 r 1 r r 3.8 x 1 x 3 x x 1 x 3 x.6.4 5 x x j. 1. 15.4 5 1 15 5 3 35 4 45 5.6 5 1 15 5 3 35 4 45 5 (e) (f)
644 YANGYANG CHEN YUPING TIAN 1 5 8 u 1x u x u 3x u 1y u 1y u y u y u 3y u 3y 6 15 4 1 u x u y 5 5 4 5 1 15 5 3 35 4 45 5 1 5 1 15 5 3 35 4 45 5 (g) (h) Fgure 5 (a) Movemens of agens; (b) Plo of ξ ξ j; (c) Plo of η η j; (d) Plo of d ; (e) Plo of r ; (f) Plo of x x j; (g) Plo of u x; (h) Plo of u y 15 1 1 z 1 z z 3 8 ξ 1 ξ 3 ξ ξ 1 ξ 3 ξ 5 6 4 y ξ ξ j 5 1 15 4 1 1.5.5 3 3.5 x 6 1 3 4 5 6 7 (a) (b) 7 6 η 1 η η 3 η 15 d 1 d d 3 5 4 1 η 3 d 5 1 5 1 1 3 4 5 6 7 1 1 3 4 5 6 7 (c) (d)
COOPERATIVE CONTROL ALONG GIVEN CURVES 645 1 1 5 r 1 r r 3.8 x 1 x 3 x x 1 x 3 x.6.4 r 5 x x j. 1 15..4 1 3 4 5 6 7 (e).6 1 3 4 5 6 7 (f) 1 8 u 1x u x u 3x 5 u 1y u y u 3y 6 4 15 1 u x u y 5 4 6 5 8 1 3 4 5 6 7 1 1 3 4 5 6 7 (g) (h) Fgure 6 (a) Movemens of agens; (b) Plo of ξ ξ j; (c) Plo of η ; (d) Plo of d ; (e) Plo of r ; (f) Plo of x x j; (g) Plo of u x; (h) Plo of u y 6 Conclusons In hs paper, we sudy he coordnaon problem for a sysem of mulple agens, whch are expeced o follow a se of gven curves wh a nomnaed formaon. A projecon-rackng desgn mehod s presened o solve he pah followng problem for mul-agen. Then, wo formaon conrol laws usng he performance ndes and her dervaves are proposed o coordnae he moon of he agens accordng o he dreced communcaon opology. When he dgraph has a globally reachable node, he sably of formaon moon s ensured by choosng approprae conrol gans. The boundedness of he conrol npus s also analyzed n deal. References [1] H. Yamaguch, A dsrbued moon coordnaon sraegy for mulple nonholonomc moble robos n cooperave hunng operaons, Robo. Auon. Sys., 3, 43(4): 57 8.
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