Proceedngs of the Internatonal Conference of Control, Dynamc Systems, and Robotcs Ottawa, Ontaro, Canada, May 15-16 214 Paper No. 46 Dstrbuted Exponental Formaton Control of Multple Wheeled Moble Robots Wene Dong, Felpe De La Torre Department of Electrcal Engneerng, the Unversty of Texas-Pan Amercan 121 Unversty Dr., Ednburg, TX, USA Abstract - Ths paper consders the dstrbuted formaton control of multple nonholonomc wheeled moble robots wth a leader. Dstrbuted trackng control laws are proposed wth the ad of results of cascade systems such that the centrod of the states of a group of moble robot exponentally tracks the leader. Smulaton results show the effectveness of the proposed algorthms. Keywords: Wheeled moble robots, cooperatve control, formaton control, dstrbuted control. 1. Introducton There are lots of applcatons of wheeled moble robots n practce and control strateges have been developed for decades. Wheeled moble robots wth nonslp constrants are typcally nonhonomc systems. Stablzaton of a sngle nonholonomc system s challengng due to the fact that no smooth pure state feedback control law exsts for stablzaton. Moreover, no dynamc contnuous tme-nvarant feedback controller s avalable to render the closed loop system locally asymptotcally stable Pomet 1992). Wth the efforts of researchers, several approaches have been proposed for stablzng nonholonomc systems, whch can be classfed nto the followng three aspects: dscontnuous tmenvarant feedback, the tme-varyng feedback, and the hybrd feedback. For detals, see Kolmanovsky and McClamroch 1995) and the references theren. It s challengng to desgn trackng controllers for wheeled moble robots because of ther nonlnear features. Samson and At-Abderrahm proposed the frst trackng controller for a moble robot n Samson and At-Abderrahm 1991). Then a trackng controller was desgned through lnear approxmaton for nonholonomc systems n Walsh et al 1994). In D'Andrea-Novel et al 1995), the trackng task of wheeled moble robots was fulflled by lnearzng both statc and dynamc feedback. Fless et al solved the trackng problem utlzng results of dfferentally flat nonlnear systems Fless et al 1995). Wth the ad of backsteppng technques, sem-global trackng controllers were proposed for a chaned-form system n Jang and Nmeer 1999). In Jang 2), global state and output trackng controllers were proposed for chaned-form systems wth the ad of Lyapunov technques. Based on the results of cascade systems, lnear trackng controllers were proposed for chaned-form systems n Lefeber et al 2) and Tan and Cao 27). Due to the practcal requrement of specfc tasks, the consensus problem wthout a leader has been extensvely studed n the past decades. In Jadbabae et al 23), matrx theory was appled to propose local control laws for fst-order lnear dscrete-tme systems such that the states of multple systems converge to a constant value. In N and Cheng 21), Laplacan matrx of a communcaton graph was exploted to propose local control laws for the consensus problem of multple frst-order lnear contnuous-tme systems. In Ren and Beard 25), consensus algorthms were proposed wth relaxed assumpton on the communcaton graphs n N and Cheng 21). Consensus problem wth a leader has also been studed systematcally and several control laws have been proposed. In Cao and Ren 21), consensus problems of frst-order and second-order lnear systems were consdered. Dstrbuted controllers were proposed such that the state of each system converge to a desred traectory wthn fnte tme under the condton that the desred traectory s avalable to a porton of the group of systems. In N and Cheng 21), leader-followng consensus of hgh-order lnear systems was consdered over a 46-1
swtchng communcaton topology. Dstrbuted controllers were proposed wth the ad of Rccatnequalty-based approach. In Lu and Ja 21), consensus control of mult-agent systems was consdered. Output feedback controllers were proposed wth the ad of theory. In Scardov and Sepulchre 29), consensus of hgh-order lnear systems was consdered for tme-varyng and drected communcaton topologes. Dstrbuted controllers were proposed wth the ad of the observer desgn approach. In L et al 211) and L et al 21), consensus of multple lnear systems was consdered n an unfed vewpont and a notaton of concensus regon was ntroduced. In Meng et al 212), the leaderfollowng consensus problem for a group of agents wth dentcal lnear systems subect to control nput saturaton was consdered. Lnear feedback laws were proposed for fxed and swtchng communcaton topology. In Cao and Ren 212a,212b), consensus of frst-order and second-order nonlnear systems was consdered. Fnte-tme control laws were proposed wth the ad of a comparson lemma. In Dong 212), cooperatve control of multple moble robots was consdered. Dstrbuted control laws were proposed wth the ad of a consensus approach. In ths paper, we study dstrbuted formaton control of multple nonholonomc wheeled moble robots wth a leader whose state s not avalable to each system such that the group of robots converges to a desred geometrc pattern whose centrod follows the leader. New dstrbuted control laws are proposed based on the results of cascade systems and the propertes of persstently exctng sgnals. Compared to the results n lterature, n ths paper cooperatve trackng control s solved for multple nonlnear systems. Compared to the results n Dong 212), a new approach s proposed for cooperatve trackng control problem of multple wheeled moble robots and the proposed control laws can make the trackng errors unformly exponentally converge to zero, whch s much more applcable n practce. The remanng parts of ths paper are organzed as follows. In Secton 2, the consdered problem s formulated and some prelmnary results are presented. In Secton 3, dstrbuted trackng controllers are proposed. In Secton 4, controllers are proposed for swtchng communcaton graphs. In Secton 5, smulaton results are presented. The last secton concludes ths paper. 2. Problem Statement It s consdered a group of m wheeled moble robots whch move on a horzontal plane. The moton of robot s descrbed by x v cos, y v sn, 1) where x, y ) s the poston of robot n a coordnate system, s the orentaton of robot, v s the speed of robot, and s the angular speed of robot. The control nputs are v and. For m systems, each system knows ts own state and ts neghbors' states by communcaton and/or sensors. For smplcty, t s assumed that the communcatons between the systems are bdrectonal. If we consder each system as a node, the communcaton between the systems can be descrbed by an undrected) graph G { V, E}, where V {1,2,, m} s a node set, and E s an edge set wth unordered par, ) whch descrbes the communcaton between node and node. If the state of node s avalable to node, node s called a neghbor of node. The set of all neghbors of node s denoted by N. A graph s called connected f for any two dfferent nodes there exsts a set of edges whch connect the two nodes. A formaton of m robots s defned by a geometrc pattern P. The pattern P can be descrbed by orthogonal coordnates p, p ) 1 m). Wthout loss of generalty, we assume that x y m m p and p,.e., the center of the geometrc pattern P s at the orgn of a local 1 x 1 y 46-2
orthogonal coordnate system. It s gven a reference traectory q t) x t), y t), t)) whch satsfes x v cos, y v sn, 2) where v and are known tme-varyng functons. The state q s assumed to be avalable to a porton of the m wheeled moble robots. Let q [ x, y, ], the control problem consdered n ths artcle s defned as follows. Control Problem: Desgn control laws v and for system usng ts own state q, ts neghbor's state q l, the relatve poston wth ts neghbor p lx, ply) for l N, and the desred traectory q f t s avalable to the system such that x x px p x lm t y y py p y 3) lm ) 4) t m m xl yl lm x,lm y t l1 m t l1 m for 1 m. In order to solve the defned problem, the followng assumpton s made on the desred traectory. d t T Assumpton 1 The 2 ) are bounded and 2 ) d > dt for some > and T >. t 3. Cooperatve Controller Desgn Defne the varables z, z x p )cos y p )sn, z x p )sn y p ) cos 6) 1 2 x y 3 x y 5) for m, where k > and p p. The transformed state space model s 3 x y z 1 z2 v z3 z3 z2. 8) 9) 7) We have the followng results. Lemma 1 If lm t z1 z1), lm t z2 z2), and lm t z3 z3) for 1 m, then 3)-5) hold. System 7)-9) can be consdered as a cascade system of 7) and 8)-9). We frst desgn control law for system 7) such that lm t z1 z1). For a group of m robots 1 m ), the communcaton between the robots s descrbed by a graph G { A, E}. Gven an m m constant matrx 46-3
A [ a ] wth a a >, the Laplacan matrx L [ L ] of the graph G wth weght matrx A s defned by L l a,, ln, a l, f N f N f and and. For the Laplacan matrx, the followng result s useful n ths paper. Lemma 2 Dong 212)) If the communcaton graph G s connected, then L dag )) s a postve [ 1, 2,, m], defnte symmetrc matrx, where constant vector 1 m) and at least one of the elements of s nonzero. Wth the ad of Lemma 2, we have the followng lemma. Lemma 3 For the m 1) systems n eqn. 7) m ), f the communcaton graph G s connected and the system s avalable to at least one of the m systems, the control laws 1 a z1 z1 ) b z1 z1) 1) N 1 a 1 1 ) b 1 1) 1 sgn a 1 1 ) b 1 1) N N 11) exp for 1 m guarantee that lmt z1 z1) and lmt ), where 1, b >, s suffcently large, the parameter 1 f system s avalable to system and f 1 system s not avalable to system. The proof of Lemma 3 s the same as the proof of Lemma 4 n Dong 212) and s omtted here for space lmtaton. Wth the ad of Lemma 3 and the results of cascade system, we have the followng results. Theorem 1 For the m 1) systems n eqn. 8) m ), under Assumpton 1, f the communcaton graph G s connected, then the dstrbuted control laws 1)-11) and v k1z2 k2 z3 z3 2 12) 2 a 2 2 ) b 2 2) sgn a 2 2 ) b 2 2) N N 13) for 1 m ensure that z, z, z ) unformly exponentally converges to z, z, ) and exp 1 2 z3 1, 2 ) exponentally converges to ), where s a suffcently large postve constant, k >, k >, and v z k z k. 1 2 1 2 3 1, 2 2 3 1 2 2 z3 In Dong 212), dstrbuted control laws for multple wheeled moble robots were proposed such that the state of each system asymptotcally converges to a desred state. In ths paper, new dstrbuted 46-4
trackng control laws are proposed wth the ad of the results of cascade systems. Moreover, the proposed control laws ensure that the state of each system globally unformly exponentally converges to a desred state. 4. Smulatons To show the effectveness of the proposed results, smulaton has been done for three robots. The desred geometrc pattern P s shown n Fgure 1. The pattern P can be descrbed by orthogonal coordnates p, p ) 1,1.7), p, p ) 1, 1.7), and p, p ) 2,). Assume the 1x 1y, y, 2x 2 y 3x 3y reference traectory s x ) 1sn.5t ), 1cos.5 t),.5t ), by 2) v 5 and.5. So, Assumpton 1 s satsfed. Assume the communcaton graph s shown n Fgure 2. The cooperatve controllers can be obtaned by Theorem 1. We chose the control parameters a 2, k 2, b 2, 2, and 2. Fgure 3 3 shows the centrod of x 1 3 ).e., x /3 1 ) and x. Fgure 4 shows the centrod of y 3 1 3 ).e., y /3 1 ) and y. Fgure 5 shows ) 1 3 ). Fgure 6 shows the path of the centrod of the three robots and ts desred path. From the smulaton 4)-5) are satsfed. Eqn. 3) s also verfed and the response of them s omtted here. 3 1 1 2 Fg. 1. Desred formaton. Fg. 2. Informaton exchange graph G. 46-5
Fg. 3. Response of the centrod of x sold) for 1 3 and x dashed). Fg. 4. Response of the centrod of y sold) for 1 3 and y dashed). Fg. 6. Response of ) for 1 3. 46-6
Fg. 6. The path of the centrod of the three robots dashed lne), the desred path sold lne) of the centrod of robots, and formaton of the three robots at several moments red trangles). 4. Concluson Ths paper has dscussed the formaton control of multple wheeled moble robots under the condton that a desred traectory s avalable to only a porton of the systems. Dstrbuted control laws were proposed wth the ad of Lyapunov technques and results from graph theory. Smulaton results show the effectveness of the proposed control laws. In ths paper, the nformaton exchange graph s assumed to be bdrectonal. The future work s to extend our results to more general nformaton exchange graphs. Acknowledgements Ths work was supported by DoD ARO under Grant No. W911NF-12-1-94. References Cao Y., Ren W. 21). Dstrbuted coordnated trackng va a varable structure approach - part I: Consensus trackng. Proc. Of Amercan Control Conference, pp. 4744-4749. Cao Y., Ren W. 212a). Fnte-tme consensus for second-order systems wth unknown nherent nonlnear dynamcs under an undrected swtchng graph. Proc. of Amercan Control Conference. pp. 26-3. Cao Y., Ren W. 212b). Fnte-tme consensus for sngle-ntegrator knematcs wth unknown nherent nonlnear dynamcs under a drected nteracton graph. Proc. of Amercan Control Conference. pp. 163-168. D'Andrea-Novel B, Bastn G., Campon G. 1995). Control of nonholonomc wheeled moble robots by state feedback lnearzaton. Int. J. of Robotc Research, 14, pp. 543-559. Dong W. 212). Trackng control of multple wheeled moble robots wth lmted nformaton of a desred traectory. IEEE Trans. on Robotcs, 28, pp. 262-268. Fless M., Levne J., Martn P., Rouchon P. 1995). Flatness and defect of nonlnear systems: ntroductory theory and examples. Int. J. Control, 61, pp. 1327-1361. Jadbabae A., Ln J., Morse A.S., 23). Coordnaton of groups of moble autonomous agents usng nearest neghbor rules. IEEE Trans. On Automatc Control, 48, pp. 988-11. Jang Z.-P. 2). Lyapunov desgn of global state and output feedback trackers for nonholonomc control systems. Int. J. of Control, 73, pp. 744-761. Jang Z.-P., Nmeer N., 1999). A recursve technque for trackng control of nonholonomc systems n chaned form. IEEE Trans. on Auto. Contr., 44, pp. 265-279. Kolmanovsky I., McClamroch N.H., 1995). Development n nonholonomc control problems. IEEE Control System Magazne, pp. 2-36. 46-7
Lefeber E., Robertson A., Nmeer H., 2) Lnear controllers for exponental trackng of systems n chaned form. Int. J. of Robust and Nonlnear control, 1, pp. 243-263. L Z., Duan Z., Chen G., 211). Dynamc consensus of lnear multagent systems. IET Control Theory and Applcatons, 5, pp. 19-28. L Z., Duan Z., Chen G., Huang L., 21). Consensus of multagent systems and synchronzaton of complex networks: a unfed vewpont. IEEE Trans. on Crcuts and Systems - I: Regular Papers, 57, pp. 213-224. Lu Y., Ja Y., 21). H_nfty consensus control of mult-agent systems wth swtchng topology: a dynamc output feedback protocol. Int. J. of Control, 83, pp. 527-537. Meng Z., Zhao Z., Ln Z., 212). On global consensus of lnear mult-agent systems subect to nput saturaton. Proc. of Amercan Control Conference, pp. 4516-4521. N W., Cheng D., 21). Leader-followng consensus of mult-agent systems under fxed and swtchng topologes. Systems and Control Letters, 59, pp. 29-217. Olfat-Saber R., Murray R.M., 24). Consensus problems n networks of agents wth swtchng topology and tme-delays. IEEE Trans. on Auto. Contr., 49, pp. 11-115. Pomet J.-B., 1992). Explct desgn of tme-varyng stablzng control laws for a class of controllable systems wthout drft. Systems and Control Letters, 18, pp. 147-158. Ren W., Beard R. W. 25). Consensus seekng n mult-agent systems under dynamcally changng nteracton topologes. IEEE Trans. on Automatc Control, 5, pp. 655-661. Samson C., At-Abderrahm K., 1991). Feedback control of a nonholonomc wheeled moble cart n cartesan space. Proc. of IEEE Conf. on Robotcs and Automaton, pp. 1136-1141. Scardov L., Sepulchre R., 29). Synhronzaton n networks of dentcal lnear systems. Automatca, 45, pp. 2557-2562. Tan Y.P., Cao K.C., 27). Tme-varyng lner controllers for exponental trackng of non-holonomc systems n chaned form. Int. J. of Robust and Nonlnear control, 17, pp. 631-647. Walsh G., Tlbury D., Sastry S.S, Murray R.M., Laumond J.P., 1994). Stablzaton of traectores for systems wth nonholonomc constrants. IEEE Trans. on Auto. Contr., 39, pp. 216-222. 46-8