An Approach to Formation Maneuvers of Multiple Nonholonomic Agents using Passivity Techniques

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1 An Approach to Formaton Maneuvers of Multple onholonomc Agents usng Passvty echnques Wu Fan, Geng Zhyong he State Key Laboratory for urbulence an Complex Systems, Department of Mechancs an Aerospace Engneerng, Pekng Unversty, Bejng 87 E-mal: Abstract: hs paper stues the problem of formaton maneuver of multple nonholonomc agents. A ynamc feeback lnearzaton metho s use to transform each agent s ynamc moel nto two chans of 3r orer ntegrator. hen a ecentralze scontnuous formaton control law wth nter-agent ampng njecton s erve. Asymptotcal stablty of the overall system s prove by usng Lyapunov metho. he propose algorthm s apple to a nonholonomc mult-vehcles formaton maneuver smulaton, whch shows the effectveness of our strategy. Key Wors: Multple nonholonomc agents, Formaton, Full-state lnearzaton, Passvty IRODUCIO he major reason for stuyng formaton of multple agents s ue to ts challengng features an many applcatons, e.g., rescue msson, large object movng, troop huntng, an satellte clusterng. Formaton control s closely relate to the consensus problem. Most results on graph theory that relate the cooperatve solutons to the consensus problem are obtane for lnear agents [, ]. In these papers, the structure of the communcaton network between agents was escrbe by Laplacan matrces. Each agent was treate as a vertex an the communcaton lnks between agents were treate as eges. he stablty of the whole system was guarantee by the stablty of each mofe nvual lnear system, where the mofcaton to the lnear system accounts for the structure of the communcaton network []. However, many practcal formaton maneuver applcatons nvolve the agents wth nonlnear ynamcs an nonholonomc constrants. In the well-known paper by Brockett [3], t was shown that the systems wth nonholonomc constrants cannot be stablze wth contnuous statc state feeback. For the case of multple agents, the problem becomes more complex. In [4], a strbute smooth tme-varyng feeback control law was propose wth analyss base on averagng theory for coornatng the moton of multple nonholonomc moble robots to capture/enclose a target. In [5], formaton control of several moble robots was consere wth the a of the ynamc feeback lnearzaton technque, resultng n cooperatve control laws base on multple ouble ntegrator systems. In aton, several general control methos were propose for multple robots. here are behavor-base control [6], vrtual structure [7], an leaer-follower [8] methos. hs work s supporte by atonal ature Scence Founaton uner Grant /9/$5. c 9 IEEE In ths paper, we propose a formaton control law by usng passvty technques for multple agents wth nonholonomc constrants. Frst, we generalze the full-state lnearzaton metho n [9] to transform each agent s ynamcs nto a lnear system consstng of two chans of 3 r orer nput-output ntegrator; Secon, we treat each agent as a MIMO lnear system an use the consensus law n [] to acheve formaton maneuvers; hr, nter-agent ampng s njecte to elmnate the relatve moton oscllatory an formaton steay error. Fnally, the asymptotcal stablty of the overall system s prove by usng Lyapunov metho. he objectve of ths paper s to ntegrate the scontnuous control law for a sngle nonholonomc agent wth the consensus law for multple MIMO lnear systems to obtan an effectve formaton control strategy for the multple nonholonomc agents wth exponental convergence rate an meanwhle epens on less nter-agent communcaton. he rest of the paper s organze as follows. In secton II, the relevant results n graph theory s brefly summarze. In secton III, the agent s ynamcal moel an full-state lnearzaton va ynamc feeback are ntrouce. In secton IV, the man result of formaton control strategy s gven. In secton V, the effectveness of the propose strategy s teste by a smulaton. An fnally, we conclue by summarzng the paper an provng some thoughts on future rectons of research. 57 PRELIMIARIES he concept of Graph s a useful tool to represent the structures of networks. Let G = ( V, A) be a graph of orer wth a set of noes V = { v,, v } an a set of arcs A V V. An arc ( v, vj) of G s graphcally enote by a secton of lne wth arrow rawn from noe v j to noe v. he set of neghbors of noe v s

2 enote by = { j :(, j) } v V v v A. In mult-agent systems represente n the nomenclature of graph, the noe v V means the th agent; the arc ( v, v j) s the communcaton connecton that the jth agent sens nformaton to the th agent. A graph Laplacan L = l j s a matrx wth ts components efne by: l = aj, lj = aj ( j) j. a = ( v ) a = ( v ) j j j j Assumng that the graph s strongly connecte, the followng propertes hol []: () here s a unque zero-egenvalue of L. () All egenvalues of L are nonnegatve. (3) he egenvector of the zero-egenvalue of L s a column vector wth all ts components are ones. (4) he egenvalues of L can be presente as: = λ ( L) < λ ( L) λ ( L). 3 HE DYAMICAL MODEL OF A ODE AD FULL SAE LIEARIZAIO In the sequel, we shall conser the formaton maneuver problem for multple agents wth nonholonomc constrants from the vewpont of complex networks. We treat each agent as a noe n the network. he th agent follows the noe s ynamcal equatons below [5]: r x vcosθ r y vsnθ F θ = ω + τ. v m ω J () An the agent subjects to a st orer nonholonomc constrant [4, 9]: rx snθ r y cosθ =. Physcally speakng, the nonholonomc constrant means that the agent cannot move n the lateral recton nstantaneously. It reuces each agent s nstantaneous moblty, e.g., n real lfe, the moble car can t move laterally nstantaneously. In the nertal coornates frame, let ( r, ) x ry represent the nertal poston of the th agent, θ represent the orentaton of the th agent. Let v an ω represent the magntues of the th agent s lnear an angular veloctes. Let τ an F be the apple torque an apple force. Let m an J represent the th agent s mass an moment of nertal. For convenence, we assume unt mass an moment of nertal an let each noe s output be: rx h = r y () Unlke the I/O lnearzaton metho n [5] whch nees a specfc output varable selecton, we generalze the ynamc feeback lnearzaton metho n [9], ntrouce a state varableξ = v : cos h ξ θ = ξsnθ (3) Dfferentatng (3) wth respect to tme gves: cosθ ξsnθ ξ h = snθ ξcosθ θ (4) Dfferentatng agan gves: cosθ ξsnθ ξ h = snθ ξcosθ + θ snθ θ ξsnθ ξcosθ ξ θ cosθ θ ξcosθ ξsnθ θ θ ( 5) Ifξ, the ynamc feeback lnearzaton control s gven by: ξ cosθ ξsnθ = snθ ξcosθ θ u snθ θ ξsnθ ξcosθ ξ θ u cosθ θ ξcosθ ξsnθ θ θ (6) whch gves two chans of 3 r orer ntegrator: u h = u (7) u Where u = u s the new nput for the trple ntegrator system an the resultng ynamc compensator s: ξ = cosθ s + snθ s F = ξ (8) τ = ω = θ + θ ( sn s cos s ) ξ Where: s s = u + snθ ω ξ + ξ cosθ ω = u cosθ ω ξ + ξ snθ ω he orgnal system () has fve states an the ynamc compensator (8) has one atonal state. All these sx states have appeare n the lnear system (7), thus there s no nternal ynamc left. 4 FORMAIO MAEUVERS COROL We aopt the behavor-base approach [5] n ths paper Chnese Control an Decson Conference (CCDC 9)

3 Defne error functon E g whch escrbes the total error between the current poston of agents an the esre formaton: Eg = h Kgh = where h = h h R, (see Fgure ), K g R postve efnte. r y r y y Intal Formaton h= Agent ( r, r ) x y r x θ h = h h r x (, ) h = r r x y Desre Formaton Desre Agent Fg. Multple onholonomc Agents Formaton Smlarly, efne E f as the formaton error: E = ( h h ) K ( h h ) f j f j j where K f R s postve efnte. herefore, E g = f an only f the agents acheve the esre formaton poston; E f = f an only f the agents keep formaton whle n moton. he total error for the formaton maneuvers control s: E = Eg + Ef (9) where K f, K g weght the relatve mportance of formaton keepng versus goal convergence. he formaton control objectve s to rve E asymptotcally. owar ths en, efne the states transformaton: h h h X = h h = h, =,, h h h It s easy to verfy that the formaton control objectve s equvalent to the zero consensus of: lm X = α, X = X X t 6 6 = [,, ], α = We stuy the problem n state space form: s x X = AX + Bu h = CX A I, B () = = I, C = [ ] I where I n s the nth orer entty matrx an s the noton A B of kronecker prouct. It s obvous that: s C controllable an observable. he overall system can be presente as: X = ( I A) X + ( I B) u () h = ( I C) X Frst, we utlze the consensus algorthm n []: u = K ( x x ) or u = ( L K) x j j Usng ths state feeback control, () s expresse n a close form as: X = (( I A) ( L BK) ) X () Lemma [] Assume that Λ e +Λ e >, the communcaton network structure of () s strongly connecte an () s also controllable an observable. hen the states of () acheve consensus value: / α = exp( At)( l I ) X() Only f there s a postve efnte matrx P an gan matrx K such that: ( ) A P + PA λ ΛeΛ e PBB P < λ ( ΛΛ e e) K = BP (Dgraph case) λ ( Λ e +Λe) K = λ ( LBP ) (Unrecte case) where Λ= = SLS, S s any regular matrx Λe whch frst row s /, l s the left-egenvector corresponng to zero-egenvalue of L satsfyng / l =. As ponte n [] that when usng ths metho n graph case, the states non-zero consensus value brngs formaton steay error an oscllatory n agents relatve moton. hs problem can be solve by ntroucng a vrtual zero-state agent to rve the non-zero consensus to zero. Here, we propose an nter-agent ampng njecton technque base on passvty metho [, 3]. Snce the map from u h s passve, so the feeback h u nee to be strctly passve. We construct a strctly postve real lnear 9 Chnese Control an Decson Conference (CCDC 9) 59

4 system Cs () from h to u : = A + z = C p p From KYP lemma [4], there exst postve efnte matrces P anq such that: AP p + PAp = QP, = C p (3) So the map h us: Ap I sc() s = PA p P hen the augmente state space representaton of each agent s: X A BKX B = + u C A p X h = [ C ] Usng Lemma agan, we gve the consensus control law wth nter-agent ampng njecton: = Ap + h u = K h D h Ah K h h h Ac = k I, D= k I, K = k I k >, =,,3 g 3 ( ) g c f j j P h PA (4) p 3 Where s + ks + ks+ k3 = satsfes Hurwtz crteron an P satsfes (3). Remark : ote the presence of the agent s velocty an acceleraton n the control law. Snce ths nformaton s requre for ynamc feeback lnearzaton, we assume that t s avalable to the controller. Remark : he state of ynamc controller represents, n a sense, the estmate of relatve veloctes between neghbors. It serves as nter-agent njecte ampng to elmnate the relatve moton oscllaton an formaton steay error. heorem : he error functon (9) converges to zero asymptotcally, f the mult-agents system of () s subject to control strategy (8) an (4). Proof: (4) can be wrtten n a collectve form: = ( I Ap) + h u = ( I Kg) + ( L K f ) + ( I h I D h I A h I P I A ( ) ( c) ( ) ( p) Conser the Lyapunov functon canate: (5) V = h ( I Kg + L K f ) h + h h + h h ( + ( I 6) he tme ervatve of V s gven by V = h ( I Kg + L K f ) h + h + h u (7) + ( I P) + ( I where: = ( I Ap) + h,usng the control law (5) an the fact that: ( I ( I A ) + ( I A ) ( I p p ( I Q) = we obtan: V = h Ah c h Dh ( I Q) (8) whch s negatve sem-efnte. Let: Ω= hhh,,,, V = {( ) } an Ω be the largest nvarant set n Ω. On Ω, we haveu =, therefore, (5) mples that the followng two equaltes hol: ( I Ap) + h = ( I Kg) + ( L K f ) + ( I h + ( I ( I Ap) = Combng these two equatons gves: ( I Kg) ( L K f ) + h = (9) From whch asymptotc stablty follows by applcaton of LaSalle s nvarance prncple. 5 SIMULAIO We conser a nonholonomc four-agent formaton maneuver problem where each agent s a two-wheele vehcle governe by the noe s ynamcal moel (). he nter-agent communcaton graph s fxe as shown n Fgure : 4 Fg. Inter-agent communcaton graph he ntal posture( r,, ) x ry θ of the agents are: (-,,), (-,,), (-,-,), (-,-,). We plan the moton frst to format a tght formaton: (,.5,), (.5,.5,), (.5,-.5,), (,-.5,); then move along the x-axs at the Chnese Control an Decson Conference (CCDC 9)

5 spee of.5m / s whle mantanng the tght formaton. he ynamc controller s ntal contons are chosen as: v() = vmax =.3m / s, ξ() = v(). ξ () = Fmax =.m / s he controller s parameters are chosen as: k =, k = 7, k3 =. o avo sngularty of the ae ynamcs,.e., ξ =, n (3). [9] proposes a suffcent conton on the control gans so as to guarantee the correct relatve rates of exponental convergence for the state varables, n such a way that v oes not go to zero n fnte tme an ω s always boune. But here, we aopt a straghtforwar soluton that conssts n resettng the stateξ of the compensator whenever ts value falls below a gven threshol. An mmeate consequence of v,ω s exponental convergence to zero s that the orentatonθ also converges to zero exponentally. he smulaton results are as n Fgure. 3 an Fgure. 4: Y [m] Consensus error Hnf norm rajectores of four vehcles v=.5m/s X [m] Fg. 3 rajectores of four vehcles Consensus Error vehcle vehcle vehcle 3 vehcle me [s] Fg. 4 Consensus error In realty, the vehcle s veloctes have been mpose wth bouns constrants: v vmax =.3m/s, ω ωmax =.5ra / s In vew of these saturatons, we aopt the metho n [9] to perform a velocty scalng so as to preserve the curvature raus corresponng to the nomnal veloctes v, ω. he actual velocty nputs are shown n Fgure. 5: Veloctes Vehcles' lnear veloctes v an angular veloctes w v w v w v3 w3 v4 w me [s] Fg. 5 Vehcles lnear an angular veloctes 6 COCLUSIO In ths paper, a formaton control law usng passvty technques s propose for the formaton maneuver problem of multple agents wth nonholonomc constrants. We frst construct a ynamc compensator to full-state lnearze each agent s ynamcal moel. Secon, we obtan a ecentralze scontnuous consensus law wth exponental convergence rate base on lnearzaton. hr, we nject the consensus law wth nter-agent ampng to elmnate the relatve moton oscllatory an steay formaton error. Fnally, we prove the stablty of the overall system va Lyapunov metho. he propose algorthm s apple to a mult-vehcles formaton maneuver smulaton to emonstrate the effectveness of the strategy. he meanngful future researches may nclue robust formaton control esgn wth respect to moel uncertantes an communcaton elays. REFERECES [] R. Olfat-Saber an R. M. Murray, Consensus problems n networks of agents wth swtchng topology an tme-elays, IEEE rans. Automat. Control, Sep. 4, vol. 49, no. 9, pp [] W. Ren an R. W. Bear, Consensus seekng n mult-agent systems uner ynamcally changng nteracton topologes, IEEE rans. Automat. Control, vol. 5, no. 5, pp , May 5. [3] R.W. Brockett. Asymptotc stablty an feeback stablzaton. In Drerent Geometrc Control heory, volume 7, pages 8-9. Sprnger Verlag, 983. [4] H. Yamaguch, A strbute moton coornaton strategy for multple nonholonomc moble robots n cooperatve huntng operatons, Robotcs an Autonomous Systems, 3, vol. 43, pp [5] J. Lawton, R. W. Bear, an B. Young, A ecentralze approach to formaton maneuvers, IEEE rans. on Robotcs an Automaton, 3, vol. 9, no. 6, pp [6]. Balch an R. C. Arkn, Behavor-base formaton control for mult-robot teams, IEEE rans. on Robotcs an Automaton, 998, vol. 4, no. 6, pp [7] W. Ren an R. W. Bear, Formaton feeback control for multple spacecraft va vrtual structures, IEE Proceengs - Control heory an Applcatons, 4, vol. 5, no. 3, pp Chnese Control an Decson Conference (CCDC 9) 5

6 [8] M. Mesbah an F. Y. Haaegh, Formaton flyng control of multple spacecraft va graphs, matrx nequaltes, an swtchng, AIAA J. of Guance, Control, an Dynamcs,, vol. 4, pp [9] G. Orolo, A. De Luca an M. Venttell, WMR control va ynamc feeback lnearzaton: esgn, mplementaton an expermental valaton, IEEE ransactons on Control Systems echnology, ovember, vol., no.6. [] oru amerkawa an Chka Yoshoka, Consensus Control of Observer-base Mult-Agent System wth Communcaton Delay, SICE Annual Conference, August 8, pp. -. [] Olfat Saber, J Alex Fax, Rchar Murray, Consensus Cooperaton n networke Mult-Agent Systems, n Proc. IEEE, Jun, 7, Vol.95,o.pp [] F. Lzarrale an J.Wen, Atttue control wthout angular velocty measurement: A passvty approach, IEEE rans. Automat. Contr., 996, vol. 4, pp , Mar. [3] P. sotras, Further passvty results for atttue control problem, IEEE rans. Automat. Contr., ov. 998, vol. 43, pp [4] H K. Khall. onlnear Systems [M]. 3 r e. Upper Sale Rver. J: Prentce-Hall,, Chnese Control an Decson Conference (CCDC 9)