Formation Control of Nonholonomic Multi-Vehicle Systems based on Virtual Structure

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1 Proceedngs of the 7th World Congress The Internatonal Federaton of Automatc Control Seoul, Korea, July 6-, 8 Formaton Control of Nonholonomc Mult-Vehcle Systems based on Vrtual Structure Chka Yoshoka Toru Namerkawa Dvson of Electrcal Engneerng and Computer Scence, Graduate School of Natural Scence and Technology, Kanazawa Unversty, Kanazawa, JAPAN toru@t.kanazawa-u.ac.jp Abstract: Ths paper deals wth formaton control strateges based on Vrtual Structure (VS for mult- systems. We propose several control laws for networked mult-nonholonomc systems n order to acheve VS consensus, VS Flockng and VS Flockng wth collsonavodance. Frst, VrtualVehcle forthefeedbacklnearzatons consdered, andwe propose VSconsensus and Flockng control laws based on a vrtual structure and consensus algorthms. Then, VS Flockng control law consderng collson avodance s proposed and ts asymptotcal stablty s proven. Fnally, smulaton and expermental results show effectveness of our proposed approaches.. INTRODUCTION Recentlytherehave been alotofprogress fornew theores thatcreates afusonsofgraphtheores andsystem control theores for cooperatve control problems of dstrbuted networkedcontrolsystems; e.g.,ren [5.Amult-agent control problem s one of sgnfcant topcs where each agent works autonomously by usng nformaton of other agents over the communcaton network. In the networked mult-agent systems, consensus means to reach an agreement regardng a certan quantty of nterest that depends on the state of of dynamcal agents. Consensus algorthm usng graph theory s studed as a control problem of mult-agent systems n Olfat-Saber [7,. Formaton control problems are expected at varous felds, e.g. satelltes, arshp, ntellgent transport systems andloadcarrage. The consensus problemscanbe appled to formaton control for multple s that s essentaltobe able tobehave hgh-effcency Tanner [5, 7, Sepulchre [5, Ren [6. A s generally a nonholonomcsystem andthasaveloctyconstrantthat ts wheels cannot move sde-away. Many research results for formaton control of nonholonomc systems have been reported Tanner [5, 7, Ikeda [. Consensus problemswthcollsonavodance for mult-agent systems have been dscussed n Tanner [5, 7, Sepulchre [5. However the control law could not acheve desred formaton because t dose not consder control of relatve poston. In Ren [6, a control law whch can construct any formatons, was proposed for mult-agent systems. However t has been dffcult to applytforgeneralnonholonomc controlsystems. Recently there have been a lot of progress for nonholonomc formaton problems e.g., n Ln [5, Dmarogonas [7, butthealgorthmsproposed ntheprevouspapers were complcated for real-tme control applcatons. On the other hand, a smple control law that makes any formaton usng devaton model (Vrtual model was proposed n leader-follower type, but t had no nformaton exchange among agents n Ikeda [. In ths paper, we construct mult-agent systems based on vrtual structure and propose novel formaton control laws by usng nformaton exchange of other agents. Severalcontrolstrateges fornetworkedmult-nonholonomc systems n order to acheve VS consensus, VS Flockng and VS Flockng wth collson-avodance are proposed. Furthermore, the asymptotcal stabltes of the closed-loopsystem wththenetworkedmult-nonholonomc and the proposed control strateges are proven theoretcally. Fnally, the effect of the proposed control laws are evaluated va control smulatons and experments.. MULTI-VEHICLE SYSTEMS Ourcontrolledplants are networkedmult- systems whch consst N s ( N agents under the followng assumpton. Assumpton. There are an nformaton network between Anyth andjth ( j s connected and can exchange nformaton of states of each. Graph theory s a useful mathematcal tool to represent nformaton network structures. The network structure wth Assumpton s sad to be connected graph f t has bdrectonal communcaton edges, or strongly connected dgraph f t has undrectonal communcaton edges /8/$. 8 IFAC 59.8/876-5-KR-.7

2 7th IFAC World Congress (IFAC'8 Seoul, Korea, July 6-, 8 VV VV snθ cosθ ω θ Vehcle Vehcle Vehcle Vehcle Vehcle snθ cosθ Fg.. threal Vehcle andcorrespondngvrtualvehcle In ths paper, we use graph Laplacan for network structures expressed mathematcally. Graph Laplacan L = [l j conssts of l = j a j, l j = a j, j f a j = that means jth send some nformaton to th, otherwse a j =.. Vehcle Model The treated n ths paper s a two-wheeled whch s shown n Fg. (lower left. We assume that N s can be expressed va an dentcal model and frcton force can be gnored. The knematc model of th s descrbed as ẋ ẏ θ = [ cos θ [ v snθ, ( ω where (x,y are the postons of center of gravty of th, θ s a headng angle of th and v and ω are the control nputs. It s well known that above models have constrant on ts velocty as ẋ snθ ẏ cos θ =. ( Therefore these s are nonholonomc.. Vrtual Structure (VS We consder Vrtual Structure (VS usng Vrtual Vehcle (VV Ikeda [ foreach as shownnfg. (upper rght. By the postonal relatonshp between and VV n Fg., the knematcs model of th VV s descrbed as [ [ xr x + x d cos θ y d snθ y r θ r = y + x d snθ + y d cos θ θ. ( where (x r, y r are postons of center of gravty of th VV, θ r s headng angle of th VV and x d,y d are dstance between VVs and s. The dervatve of ( are gven by ẋ r [ [ ẏ r B v =, B θ θ ω r where [ cos θ x B = d snθ y d cos θ, ( snθ x d cos θ y d sn θ B θ =[. (5 Fg.. Poston of VSs In ths knematcs model, B s nonsngular matrx f x d. In ths paper, we consder formaton control problems for these VS systems (.. VS CONSENSUS PROBLEMS The goal of formaton control problems s that N s preserve any formaton based on nformaton exchange between them over the network. To mantan any formatons, the VVs of each has to converge to a common poston as shown n Fg... Control Objectves To converge to a common value for VV of each, It s necessary toguarantee consensus forpostonsofcenter of gravty and headng angle of VVs as x r x rj, y r y rj, θ r θ rj (t. (6 Ths consensus s called VS consensus. Lemma. Consder the N N graph Laplacan L wth strongly connected dgraph. If the systems can be descrbed as ẋ = L.m x (7 where x = [x T x T x T N T R Nm are the state of all systems and L.m = L I m, the state x converge as x (x r x T l I m x( = α (t, (8 where x r,x l are rght and left egenvector of zero egenvalue of L wth x T l x r = and x T l =, denotes Kronecker product, α R m s consensus value and = [ T R N Olfat-Saber [. Proof. See Olfat-Saber [ for proof. From Lemma, the all of states converge to a common value α as x = x = = x N = α. (9. Control Law for VS Consensus To acheve VS consensus, we propose the followng control law for the as 55

3 7th IFAC World Congress (IFAC'8 Seoul, Korea, July 6-, 8 Control law. u = B ( k (r r j + ṙ d, ( where u = [v ω T, r = [x r y r T, N s th neghbor set, ṙ d R s constant reference velocty and k > s controller gan. Theorem. Consder a system of the N s wth knematcs ( and Control Law (. If Assumpton and ṙ d are satsfed, then VS consensus acheves asymptotcally. Proof. All of the VS systems ( wthout ts angle θ r can be wrtten as ṙ = N B u, ( = where r = [r T r T rn T T, u = [u T u T u T N T, N = B s matrx that dagonal block elements are B. The Control law ( can be wrtten as u = N = B ( kl. r + ṙ d. ( Let r e = r r d, then we get the followng from ( and (, r e = kl. r e. ( By Lemma, the systems ( acheve consensus as r e α (t. Hence, we can conclude that the postons of VVs converge to a common value as r (α + r d (t. ( The consensus for r s acheved as r r j α + r d. Next, we consder headng angles θ r of VVs. Substtutng Control law ( nto θ r n ( and consderng ṙ d = [v d cos θ d v d snθ d T, we get that θ r = v d x d sn(θ r θ d. (5 Hence, We have that θ r θ d (t. Therefore VS consensus s acheved asymptomatcally. Furthermore, the any formaton shape s guaranteed. B s nonsngular matrx, there s not sngular value n Control Law. Then, the s can make any formatons when VVs converge to a common value. By selectng the dstance for VVs (x d, y d approprately as shown n Fg., the s acheve any formaton shapes. The Control law can be extended and the s can acheve anyformatonsevenfdstances forvvs are same as x d = x d = = x dn, y d = y d = = y dn, We propose the new control law for the th as (6 Control law. ( ( u = B k (r r r (r j r rj + ṙ d (7 where r r s reference relatve poston to r. Theorem. Consder a system of the N s wth knematcs ( and Control Law (7. If assumpton and ṙ d are satsfed, then VS consensus acheve asymptotcally. Proof. Thscanbe provennasame waywththeorem.. Control Law wth Velocty Trackng for VS Consensus The Control laws and nclude feedforward terms whch are reference sgnals ṙ d. In case of physcal s, the moton of s are not exactly same between them. Therefore, the error of veloctes (ṙ d ṙ do not converge to. Consequently we propose new control law wth velocty control for th as Control law. v r = v k vr(v r v (8 ( ( u=b k (r r r (r j r rj + v r where v s constant reference velocty and k vr > s controller gan. Theorem. Consder a system of the N s wth knematcs ( and Control law (8. If Assumpton and v are satsfed, then VS consensus acheve asymptotcally. Proof. Substtutng Control law (8 nto the th knematcs (, we get that v r = v k v (v r v ˆr = kl.ˆr + v r. (9 Usng v re = v r v, r e = ˆr t v dτ, [ [ [ ṙe kl. I = N re. ( v re k vr I N v re By Lemma, the systems ( acheve consensus and velocty errors r e converge to as r e α v re ( Therefore any formaton shape s guaranteed.. Control Objectves. VS FLOCKING PROBLEMS Flockngs defnedthatveloctyandnter- dstances converge to common value. It could be as ṙ ṙ j ( VS consensus problem consders only relatve postons between s. Here, we dscuss VS Flockng problems 55

4 7th IFAC World Congress (IFAC'8 Seoul, Korea, July 6-, 8 that s consdered both relatve postons and relatve veloctes between VVs. The veloctes s defned as v r = [v x v y T. Then t s expressed as v r = a, ṙ = v r, ( where a s control nput.. Control Law for VS Flockng The followng control law s proposed Control law. v r = k ( (ˆr ˆr j + k v (v r v rj u = B v r, ( where k v, k > are controller gans. Theorem. Consder a system of the N s wth knematcs ( and Control law (8. If Assumpton and > + /(k vλ, then VS Flockng acheve asymptotcally, where λ are egenvalues of weghted graph Laplacan L w ncludng k and v v j. Proof 5. The control nput v r for mult- systems can be wrtten as v r = L w.ˆr k v L w. v r. (5 By B, the poston coordnate of VS system ( can be also descrbed as (. Therefore, f flockng problem acheve n second order system (, VS systems wth ( acheve VS flockng problem. By ( and (5, we have followng result [ [ [ ˆr IN ˆr = I v r L w k v L (6 w v r }{{} Σ Σ has zero egenvalues. Selectng k v to satsfy as > + /(kvλ, (7 where λ s th egenvalue of L w, All of egenvalues wthout zero have negatve real parts Ren [6. Fnally, we consder tme response of (6 and transform Σ to Σ = SJS where J s Jordan form composed of any vector as S = [ω ω ω N, S = [ν ν ν N T. ω, ν are rght and left egenvector of Σ to λ(σ =. ω, ν are vectors that Σω = ω,ν T Σ = ν T. The state of mult- at t s expressed as, [ ˆr vr = lm t S exp(jts I [ ˆr( v r ( (ω ν T + ω ν T t + ω ν T. [ ˆr( v r ( The each vector s wrtten as [ ω =, ω = [, ν = [ [ p, ν =, p. (8 where = [ T R N, p s egenvector of λ( L w = and p T =. Then, we get [ [ ˆr (p T.ˆr( + (p T. v(t vr (p T (9. v( Therefore VS Flockng s acheved asymptotcally.. ControlLaw wth CollsonAvodanceforVS Flockng From Theorem, the formaton shape was guaranteed n VS Flockng problem. However, ncase ofphyscal s, the collson avodance s also mportant problem. It s wellknownthatartfcalpotentalapproachs effectve to avod collsontanner [5. The artfcal potental gves repulsve force to other s f a come close to other s. Here, we use followng artfcal potental functon Tanner [5 U = U j, U j = d r j + log r j, ( where r j = r r j and d s controller gan. We have to select d that satsfes d > ( x d + y d + R v where R v s the largest radus of the s. Then we propose followng control law wth collson avodance as Control law 5. where v r = u co + u ca u = B v r ( u co = v k vr (v r v ( ( (ˆr ˆr j + k v (v r v rj k u ca = r U k (v r v rj ( where k vr, k v, k > are controller gans. ( s the control law to acheve consensus and ( s the control law to acheve collson avodance. Theorem 5. Consder a system of the N s wth knematcs ( and Control law 5 (. If Assumpton and assumpton of the bdrectonal communcaton for the network, and k vr + k v λ f max L w. > are satsfed, then VS Flockng acheves asymptotcally. Where λ s the smallest egenvalue of L w wthout zero egenvalue and f max s the maxmum potental force of and v. Proof 6. Let v e = v r v, then the control nput v e for mult- systems s wrtten as v e = k vr v e L w.ˆr k v L w. v e r U L w. v e ( where r U s matrx that the dagonal block element are r U. Now, we defne the functon V for the system as V (x = (vt e v e + ˆr T L w.ˆr. (5 where x = [v e, ˆr T. Because of network structure of mult systems wth bdrectonal communcaton can be represented undrected graph. Then we have that L w. = L T w.. The dervatve of ths functon along trajectores of the V are gven by V= ˆr T L w. ˆr + v T e v e 55

5 7th IFAC World Congress (IFAC'8 Seoul, Korea, July 6-, 8 dgraph 5 lne graph 5 Fg.. Graph structure Fg.. Formaton Fg. 5. Trajectory of fve s (VS consensus (k vr + k v λ f max L w. v e, (6 where λ s smallest egenvalue of L w wthout zero egenvalue and f max s maxmum potental force. Choosng 5 VR Fg. 6. Trajectory of the fve s (VS Flockng Σ( - Σ( Σ( - Σ( - Σ( 5- Σ( - Σ( - Σ( Σ( - Σ( 5- Σ( - Σ( tme[s Fg. 7. Error of VSs velocty (VS Flockng 5 k vr + k v λ f max L w. >, (7 the V s negatve sem-defnte. Furthermore, V = s satsfed by only v e =. Applyng LaSalle s nvarant prncple, we can see that v e converge to asymptotcally. Therefore, the consensus s acheved as v r v. Furthermore, we can see that 8 6 flockng+collson avodance flockng v r = L w.ˆr = (8 Therefore, ˆr ˆr j. Thus, VS Flockng wth collson avodance s acheved asymptotcally. 5. SIMULATIONS Consder a group of 5 s that has network structure as shown n Fg. (upper. Fg. shows the desred formaton and dstances of VS. 5. VS Consensus Problems We verfy the Control law (. The parameter for VS and control law are selected as k =.5. The reference veloctes are ṙ d = [.cos(π/.sn(π/ T. Fg.5 shows the trajectory of the s. From ths result, the s acheve desred formaton and the poston of VVs converge to a common value. 5. VS Flockng Problems The Control law ( s examned. The parameters for VS and control law are selected as k =. and k v =. The reference veloctes are ṙ d = [. cos(π/.sn(π/ T Fg. 8. Trajectory Fg.6shows thetrajectoryofthes andfg.7shows the velocty errors between VVs. From these results, the s acheve formatonandthepostonandveloctyof VVs converge to a common value. 5. VS Flockng Problems wth Collson Avodance We verfy the proposed Control law 5 (. A group of 5 s that has the network structure of lne graph s consdered as shown n Fg.(lower. The parameter for VS are selected as x d =.5, y d =,.e. the dstances of VVs s a common value. The parameter for control law are selected as k vr =, k v =, k =.. The parameter for collson avodance functon s selected as d =. by reason of the largest radus of the physcal s s R v =.8. The reference veloctes are v = [. π T. The desred formaton structure s shown n Fg.. Fg.8shows smulatonresults ncase wthcollsonavodance and wthout collson avodance as u ca =. Ths shows that s acheve formaton wth collson avodance. 55

6 7th IFAC World Congress (IFAC'8 Seoul, Korea, July 6-, 8 6. EXPERIMENTS We verfy the effcacy of the proposed control laws va control experments for VS consensus problem and VS Flockng problem. The experments were carred out on s as shown n Fg.9. We use the dspace as realtme calculatng machne and a samplng rate s choosen as. [s because ofthetme delayofthewreless network. 6. VS Consensus Problem Frst, the proposed Control law (8 for VS consensus s verfed. The parameters for VS and control law are selected as x d = x d =.5, y d = y d =, r r = [.5 T,r r = [.5 T, k vr =., k =. The ntal condtons are R ( = [.7.8 T, R ( = [.7.8 T. The reference velocty s v d = [.7 T. Fg. shows the trajectory of the postons of the s n the feld. We can see that the VVs acheve consensus. 6. VS Flockng Problem We verfy proposed Control law 5 ( for VS Flockng problems. The parameters for VS and control law are selected as x d = x d =., y d = y d =, r r = [.5 T,r r = [.5 T, k vr =.5, k =.5, k v =.. The ntal condtons are R ( = [.. T, R ( = [.. T. The reference velocty s v d = [.7 T. Fg.shows thetrajectores ofthepostonsofthes n the feld and ths shows the VVs acheve flockng. Vehcle Fg. 9. Expermental setup Camera Computer Modem Image Processng Board DS. experment.5. Vehcle smulaton Vehcle Fg.. Trajectory of two s (VS consensus Vehcle Vehcle experment smulaton Fg.. Trajectory of two s (VS Flockng 7. CONCLUSIONS Inthspaper, we proposed theformatoncontrolstrateges for networked mult- systems usng vrtual structure. Our proposed control laws could acheve desred formatons for nonholonomc systems. Severalcontrolstrateges fornetworkedmult-nonholonomc systems n order to acheve VS consensus, VS Flockng and VS Flockng wth collson-avodance were proposed. The asymptotcal stabltes of the closed-loop system wth the networked mult-nonholonomc and the proposed control strateges were proven theoretcally. Fnally, the effect of the proposed control laws were evaluated va control smulatons and experments whch demonstrated the effectveness of our approaches. REFERENCES We Ren, Randal W. Beard and Ella M. Atkns, A Survey ofconsensus ProblemsnMult-agentCoordnaton, n Proc. of Amercan Control Conference, pp , June, 5. Reza Olfat-Saber, J.Alex Fax, Rchard Murray, Consensus andcooperatonnnetworkedmult-agentsystems, n Proc. IEEE, Vol. 95, No., 7. Reza Olfat-Saber andrchardmurray, Consensus Problems n Networks of Agents Wth Swtchng Topology and Tme-Delays, IEEE Trans. Automatc Control, Vol. 9, No. 9, pp. 5-5,. Herbert G. Tanner, Al Jadbabae and George J. Pappas, Flockng n Teams of Nonholonomc Agents, Cooperatve Control, LNCIS 9, pp. 9-9, 5. HerbertG. Tanner, AlJadbabae, andgeorge J. Pappas, Flockng n Fxed and Swtchng Networks, IEEE Trans. on Automatc Control, Vol.5, No., 7. Rodolphe Sepulchre, Derek Paly and Naom Leonard, Collectve Moton and Oscllator Synchronzaton, Cooperatve Control, LNCIS, Vol. 9, pp. 89-5, 5. We Ren, Consensus Based FormatonControlStrateges for Mult- Systems, nproc. of Amercan Control Conference, pp. 7-, 6. Takash Ikeda, Jurachart Jongusuk, Takayuk Ikeda and Tsutomu Mta, Formaton Control of Multple Nonholonomc Moble Robots, IEEJ Trans. IA, Vol., No. 8, pp. 8-89,. (n Japanese Zhyun Ln, Bruce Francs and Manfred Maggore, Necessary and Suffcent Graphcal Condtons for Formaton Control of Uncycles, IEEE Trans. on Automatc Control, Vol. 5, No., pp. -7, 5. Dmos V. Dmarogonas and Kostas J. Kyrakopoulos, On the Rendezvous Problem for Multple Nonholonomc Agents, IEEE Trans. on Automatc Control, Vol. 5, No. 5, pp. 96-9, 7. 55