Decentralized Formation Tracking of Multi-vehicleSystems with Nonlinear Dynamics

Lei Fang and Panos J. Antsaklis

Abstract- The problem of formation tracking can be statedas multiple vehicles are required to follow spatial trajectorieswhile keeping a desired inter-vehicle formation pattern in Jtime. This paper considers vehicles with nonlinear dynamics d trajectoryand nonholonomic constraints and very general trajectories timethat can be generated by some reference vehicles. We specifyformations using the vectors of relative positions of neighboringvehicles and use consensus-based controllers in the context leader

of decentralized formation tracking control. The key ideais to combine consensus-based controllers with the cascadedapproach to tracking control, resulting in a group of linearlycoupled dynamical systems. We examine the stability properties Fig. 1. Six vehicles perform a formation tracking task.of the closed loop system using cascaded systems theoryand nonlinear synchronization theory. Simulation results arepresented to illustrate the proposed method.

In many scenarios, vehicles have limited communicationI. INTRODUCTION ability. Since global information is often not available to each

Control problem involving mobile vehicles/robots have vehicle, distributed controllers using only the local infor-attracted considerable attention in the control community mation are desirable. One approach to distributed formationduring the past decade. One of the basic motion tasks control is to represent formations using the vectors of relativeassigned to a mobile vehicle may be formulated as following positions of neighboring vehicles and the use of consensus-a given trajectory [12], [24]. The trajectory tracking prob- based controllers with input bias [4], [10].lem was globally solved in [19] by using a time-varying In this paper, we study the formation tracking problemcontinuous feedback law, and in [3], [11], [15] through the for a group of vehicles/robots using the consensus-baseduse of dynamic feedback linearization. The backstepping controllers combined with the cascade approach [16]. Thetechnique for trajectory tracking of nonholonomic systems idea is to specify a reference path for a given, nonphysi-in chained form was developed in [7], [9]. In the special cal point. Then a multiple vehicle formation, defined withcase when the vehicle model has a cascaded structure, the respect to the real vehicles as well as to the nonphysicalhigher dimensional problem can be decomposed into several virtual leader, should be maintained at the same time aslower dimensional problems that are easier to solve [16]. the virtual leader tracks its reference trajectory. The vehiclesAn extension to the traditional trajectory tracking problem exchange information according to a communication digraph,

is that of coordinated tracking or formation tracking (see G. Similar to the tracking controller in [16], the controllerFig. 1). The problem is often formulated as to find a for each vehicle can be decomposed to two "sub-controllers,"coordinated control scheme for multiple robots that make one for positioning and one for orientation. Different fromthem maintain some given, possibly time-varying, formation the traditional single vehicle tracking case, each vehiclewhile executing a given task as a group. The possible tasks uses information from its neighbors in the communicationcould range from exploration of unknown environments digraph to determine the reference velocities and stay atwhere an increase in numbers could potentially reduce the their designation in the formation. Based on nonlinear syn-exploration time, navigation in hostile environments where chronization results [26], we prove that consensus-basedmultiple robots make the system redundant and thus robust, formation tracking can be achieved as long as the formationto coordinated path following; see recent survey papers [2], graph had a spanning tree and the controller parameters are[20]. large enough (They can be lower-bounded by a quantity

In formation control of multi-vehicle systems, different determined by the formation graph.)control topologies can be adopted depending on applications. Related work includes [1], [5], [6], [18], [21]. In [1], theThere may exist one or more leaders in the group with other vehicle dynamics were assumed to be linear and formationvehicles following one or more leaders in a specified way. control design was based on algebraic graph theory. In [1 8],

L. Fang and P. J. Antsaklis are with Dept. of Electrical En- ouptfebc ierzto oto a obndwtgineering, Univ. of Notre Dame, Notre Dame, IN 46556. E-mail: a second-order (linear) consensus controller to coordinate{lfang, antsaklis.l}@nd. edu. the movement of multiple mobile robots. The problem of

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Lei Fang, Panos Antsaklis, “Decentralized Formation Tracking of Multi-vehicle Systems with Nonlinear Dynamics,” 14th Mediterranean Conference on Control and Automation (MED’06), Università Politecnica delle Marche, Ancona, Italy, June 28-30, 2006.

Xk iIt can be verified that in these coordinates the errordynamics become

/Ye F Xe OJye-V+Vrcos OeYr ----------------------------------)I Ye oxe vr sin Oe (4)

tv TOLe L (or ocoy k 0;8xeThe aim of (single robot) trajectory tracking is to find

appropriate velocity control laws v and c of the form

v V(t,XeYe, Oe) (50) = 0)(tXe,Ye, Oe)

x,. such that the closed-loop trajectories of (4) & (5) are stablein some sense (e.g., uniform globally asymptotically stable).As discussed in Sect. I, there are numerous solutions to

this problem in the continuous time domain. Here, we revisitthe cascaded approach proposed in [ 16]. Let us first introduce

vehicles moving in a formation along constant or periodic the notion of globally K-exponential stability.trajectories was formulated as a nonlinear output regulation Definition 1. A continuous function a: [0, a) -. [0, ) is(servomechanism) problem in [5]. The solutions adopted in said to belong to class K if it is strictly increasing and a(O)[6], [21] for coordinated path following control of multiple t.marine vessels or wheeled robots built on Lyapunov tech- [0,-) is said to belong to class KL if for each fixed s thenqes were mapping /B (r, s) belongs to class K with respect to r, and forwere de cupled. each fixed r the mapping ,B(r,s) is decreasing with respect

The contributions ofthis work are: 1) The consensus-based tosad/(rs)-0ass- *formation tracking controller for nonlinear vehicles is novel Definito(3,C)onsider t systemand its stability properties are examined using cascaded sys-

D

tems theory and nonlinear synchronization theory; 2) Global x = g(t,x), g(t, 0) = Vlt > 0 (6)results allow us to consider a large class of trajectories with where g(t,x) is piecewise continuous in t and locally Lip-arbitrary (rigid) formation patterns and initial conditions. schitz in x.

II. PRELIMINARIES We call the system (6) globally K-exponentially stable ifthere exist 4 > 0 and a class K function k(.) such that

A. Tracking Control ofMobile Vehicles

A kinematic model of a wheeled mobile robot with two Theorem 1 ([16]): Consider the system (4) in closed-loopdegrees of freedom is given by the following equations with the controller

V = Vr + c2xe, (7)Y=vcos0, yvsinO, 6=o, (1) =)=Dr+ClOe,

where cl > 0 c2 > 0. If Wr(t), (.r(t), and Vr(t) are boundedwhere the forward velocity v and the angular velocity c are ahere ei 3 andk such thatconsidered as inputs, (x, y) is the center of the rear axis ofthe vehicle, and 0 is the angle between heading direction | (T)2dT > k, Vt > to (8)and x-axis (see Fig. 2). t

For time varying reference trajectory tracking, the refer- then the closed-loop system (4) & (7), written compactly asence trajectory must be selected to satisfy the nonholonomicconstraint. The reference trajectory is hence generated using pe h(Xe, Ye: Oe) VrW= h(pe) VrWr (9)a virtual reference robot [8] which moves according to the is globally K-exponentially stable. Dmodel In the above, the subscriptions for h(.) Vr,W mean that the

error dynamics are defined relative to reference velocities vrXr =Vr Cos 0r, Yrvr sin 0r, 6r =, (2 and oJr. The tracking condition (8) implies that the reference

T * trajectories should not converge to a point (or straight line).where [Xr yr Or] iS the reference posture obtained from the g ( g )virtual vehicle. Following [8] we define the error coordinates This also relates to the well-known persistence-of-excitation(cf. Fig. 2) condition in adaptive control theory.

Note that control laws in (7) are linear with respect to[ Xe 1 cosO sinO [Xr 1 F xXe and Oe. This is critical in designing consensus-based

Pe = Ye si cos OI I Y-Y (3) controller for multiple vehicle formation tracking as we shall[ Oe, , si 0 I jL Or-0 J see below.

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Lei Fang, Panos Antsaklis, “Decentralized Formation Tracking of Multi-vehicle Systems with Nonlinear Dynamics,” 14th Mediterranean Conference on Control and Automation (MED’06), Università Politecnica delle Marche, Ancona, Italy, June 28-30, 2006.

B. Formation Graphs VIWe consider formations that can be represented by acyclic r2

directed graphs. In these graphs, the agents involved are iden- r ltified by vertices, and the leader-following relationships by trajt(directed) edges. The orientation of each edge distinguishesthe leader from the follower. Follower controllers implement VNstatic state feedback-control laws that depend on the state ofthe particular follower and the states of its leaders. Fig. 3. Formation tracking using baseline FTC. The reference vehicle sends

Definition 4 ([23]): A formation control graph G to vehicle i the formation specification di as well as the reference velocities(V,E,D) is a directed acyclic graph consisting of the fol- Vr and Wr.lowing.

* A finite set V = {vl,...,VN} of N vertices and a map where x = (xl,...,xN)T, u= (ui,. ..,uv)' and C(t) is a zeroassigning to each vertex a control system xi = f (t,xi, u1) sums matrix for each t. C0 D is the Kronecker product ofwhere xi e RIn and ui C R.m matrices C and D.

. An edge set encoding leader-follower relationships be- Theorem 2 ([26]): Let Y(t) be an n by n time-varyingtween agents. The ordered pair (v1, vj) = eij belongs to matrix and V be an n by n symmetric positive definite matrixE if uj depends on the state of agent i, xi. such that f(x, t) + Y(t)x is V-uniformly decreasing. Then the

. A collection D = {dij} of edge specifications, defining network of coupled dynamical systems in (11) synchronizescontrol objectives (setpoints) for each j: (v1,vj) C E for in the sense that xi - x - > 0 as t --> - for all i, j if thesome vi C V. following two conditions are satisfied:

For agent j, the tails of all incoming edges to vertex (i) limt- ui-uj = 0 for all i, j.represent leaders of j, and their set is denoted by Lj c V. (ii) There exists an N by N symmetric irreducible zero rowFormation leaders (vertices of in-degree zero) regulate their sums matrix U with nonpositive off-diagonal elements suchbehavior so that the formation may achieve some group thatobjectives, such as navigation in obstacle environments or (U ® V) (C(t) ® D(t) -I® Y(t)) < 0 (12)tracking reference paths.

Given a specification dkj on edge (vk,vj) C E, a setpoint for all t.for agent j can be expressed as x = xk - dkj. For agents* * * J- y III111 BASIC FORMATION TRACKING CONTROLLERwith multiple leaders, the specification redundancy can be

resolved by projecting the incoming edges specifications into The control objective is to solve a formation tackingorthogonal components problem for N vehicles. This implies that each vehicle must

converge to and stay at their designation in the formationXj =Y, Skj (Xk- dk-j) (I10) while the formation as a whole follows a virtual vehicle.

kaL1 Equipped with the results presented in the previous sec-tion, we first construct a basic formation tracking controller

where Skj are projection matrices with Ykrank(Skj) = n- (FTC) from (7). Let dri=[dx d ] denote the formationThen the error for the closed-loop system of vehicle j is .r.dr,

deoe h frato

Tefnedtobetheerrorvforatheclosed-loop sythemofveriledsspecification on edge (vr, vi). In virtue of linear structuresdefined to be the deviation from the prescribed setpoint of (7), we propose:

i = xj- xj, and the formation error vector is constructed Basic FTC for vehicle i:by stacking the errors of all followers

| vi V vr+C2xei, (13)

x [x ] Vj e V\LF- ti= (trcOz

where cl > 0, C2 > 0 andC. Synchronization in networks of nonlinear dynamical sys- Ttems Pei [Xei Yei OejlT

FCos01 sin0, 0 iF -xirX dxr1Definition 5: Given a matrix V e Rn,n>, a function f(y, t): ri

Rtn+1 SR is V-uniformly decreasing if (y z)TV(f(y,t) sin 0O cos 0, 0 Iy -yi - dyr, (14)f(z, t)) < -g Y-z 12 for some M > 0 and all y, z C IRP and [ 0 0 iJL or0i It C R. Remark 1: It is not required to have constraints for every

Note that a differentiable function f(y,t) is V-uniformly pair of vehicles. We need only a sufficient number ofdecreasing if and only if V(df(y)/dy) +31I for some a > 0 constraints which uniquely determine the formation.and all y, t. Consider the following synchronization result Theorem 3. The basic FTC (13) and (14) solves thefor the coupled network of identical dynamical systems with formation tracking problem.state equations: Proof By Theorem 1, every vehicle i follows the virtual (or

leader) vehicle (thus the desired trajectory) with a formationx=(f(xi,t),... ,f(x,q,t))T +(C(t) ®D(t))x-+u(t), (11) constraint dri on edge (vr, vi). Thus all vehicles tracks the

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Lei Fang, Panos Antsaklis, “Decentralized Formation Tracking of Multi-vehicle Systems with Nonlinear Dynamics,” 14th Mediterranean Conference on Control and Automation (MED’06), Università Politecnica delle Marche, Ancona, Italy, June 28-30, 2006.

60 20

401 0

5-/20

0-

0 -50 10 20 30 40 50 60 70 80 90 100

-20 0 f-50.4-

-40-0.3

0.2- - --60-

0.1

-80 0-100 -80 -60 -40 -20 0 20 40 0 10 20 30 40 50 60 70 80 90 100

Fig. 4. Circular motion of three vehicles with a triangle formation. Initial Fig. 5. Control signals v and w) for Virtual vehicle: solid line; Vehicle 1:vehicle postures are: [-8 -9 3zr/5]T for vehicle 1 (denoted as *); [-15 - dotted line; Vehicle 2: dashed line; and Vehicle 3: dot-dash line.20 zr/2]T for vehicle 2 (El); [-10 - 15 w/l3]T for vehicle 3 ( ).

reference trajectory while staying in formation, which is is that it requires every vehicle to get access to the referencespecified by formation constraints dri's as shown in Fig. 3 velocities vr and )r. This further implies that the referenceF] vehicle needs to establish direct communication links with

Corollary 1: Suppose only vehicle 1 follows the virtual all other vehicles in the group, which may not be practicalvehicle. The composite system with inputs vr and W)r and in some applications.states xi = [Xe1 Ye, Oe1 ] T is globally K-exponentially stable In a more general setting, we assume that only a subsetand therefore formation input-to-state stable (see the defini- of vehicles (leaders) have direct access to the referencetion in [22]). velocities. Other vehicles (followers) use their neighboringExample 1 (Basic FTC): Assume that we have a system leaders' information to accomplish the formation tracking

consisting of three vehicles, which are required to move in task. In this case, formation tracking controllers operatesome predefined formation pattern. First, as in [5], we will in a decentralized fashion since only neighboring leaders'consider the case of moving in a triangle formation along a information has been used.circle. That is, the virtual (or reference) vehicle dynamics are Consensus-based FTC for vehicle igiven by: Xr = VrCOs(oJrt) +XrO, Yr vrsin(OJrt) +Yro, whereVr is the reference forward velocity, (Or the reference angular r vi Vr1 + C2Xe1 + , EL1 a1 (Xe1-Xe1),velocity, and [x,0 y,o]T the initial offsets. )i = )r, + Cl Oei + j L aij( ei - Oe) (15)Assume that the parameters vr = 10, cor = 0.2, [Xro yro =T Vri = EL,i aij(vr -vri)

[-25 o]T. In our simulations we used an isosceles right 0ri = eLi ai1 (o), - (ri)triangle with sides equal to 3X2, 3X2, and 6. Also, fix the whereposition of the virtual leader at the vertex with the rightangle. Then, from the above constraints the required (fixed) [ Xe 1 cos 0i sin Oi 0 1 l-x iformation specifications for the vehicles are given by Pei Yei = -sinO cos0, O i -y Yi|

0 3 3 LOei J L 0 0 1 JLoir -oiJdr,= L 1, dr2 31 dr3 L 3 [

and aij represents relative confidence of agent i in theFor FTC we chose the parameters as cl =0.3 and C2 information state of agent j.

0.5. Fig. 4 shows the trajectories of the system for about Remark 2. As one can see from (15). the communication100 seconds. Initially the vehicles are not in the required between vehicles is local and distributed, in the sense thatformation; however, they form the formation quite fast (K- each vehicle receives the posture and velocity informationexponentially fast) while following the reference trajectory only from its neighboring leaders.(solid line in the figure). Fig. 4 shows the control signals v

We have the following theorem regarding the stability ofand w for each vehicle.the consensus-based FTC.

Theorem 4. The consensus-based FTC (15) solves theIV. CONSENSUS-BASED FORMATION TRACKING formation tracking problem if the formation graph C has

CONTROLLER a spanning tree and the controller parameters c l, c2 > 0 areThe basic FTC has the advantage that it is simple and leads large enough. Lower bounds for c land c2 are related to the

to globally stabilizing controllers. A disadvantage, however, Laplaican matrix for C.

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Lei Fang, Panos Antsaklis, “Decentralized Formation Tracking of Multi-vehicle Systems with Nonlinear Dynamics,” 14th Mediterranean Conference on Control and Automation (MED’06), Università Politecnica delle Marche, Ancona, Italy, June 28-30, 2006.

Proof Let LG be the Laplacian matrix induced by the 3formation graph G and it is defined by d d-3

(L(J),/=- {2k=: I,ki aik,d

L0dri

We will write Pe = [Pei, RPeNI e 23N [Vr Qr]T =3 Lo_[Vr ,VrN O)rl1... O)r]NCT 2N. The closed loop system 2(15)-(4) for all vehicles can be expressed in a compact formas

[ h(Pei) V1,Wr4 1Fig. 6. A formation graph with formation specifications on edges

Pe [ ( + (-LGO®D)Pe, (16) 10

h(peN) ivrN (t)N80-

[~ ] =(-LG 12)[OI ] (17) 60 K>

where-1 0 0 40

D [0 0 2J (18)

describes the specific coupling between two vehicles.It can be seen that (17) is in the form of linear consensus °

algorithms. Since the formation graph is acyclic and hasa rooted spanning tree (with the root corresponding to -20 0 0 2 3 4 5 6 0

-20 -10 o 10 20 30 40 50 60 70 80the virtual vehicle), the reference velocities (coordinationvariables) vr1(t) and Wr1(t) for any vehicle i in the group Fig. 7. Tracking a sinusoidal trajectory in a triangle formation. Initialwill approach to vr(t) and Wr(t), respectively [14], [17]. (It vehicle postures are: [12 12 O]T for vehicle 1 (denoted as *); [-15 -is important for the formation graph to be acyclic such that 20 zr/4]T for vehicle 2 (l); [-10 15 - zr/4]T for vehicle 3(1).each vehicle can follow arbitrary root reference velocities.For general graphs with loops, the consensus algorithms have and the proof is complete.band-limited properties [13].) In particular, an upper bound for g (-LG) is given byWe thus re-write (16) as 12(-LG) = minRe(2,) where Re(2,) is the real part of i,, the

[ h(pel) vr, Pi (t) eigenvalues of -LG that do not correspond to the eigenvectorPe= +( :LG D)Pe+ (19) e. It suffices to make min{ci,c2} > g2(-LG) - D

Example 2: In this example, we chose virtual vehicleh(peN) Vr,r 0LN(t) J dynamics of a sinusoidal form: (xr(t), yr(t)) = (t, sin(t)).

and N (t) -> 0 as t -*> c. The functions /i can be considered The acyclic formation graph with formation specifications isas residual errors that occurred when replacing vri and oJr1 shown in Fig. 6.in (16) with vr and Wr, respectively. The (unweighted) Laplacian matrix corresponds to Fig. 6Now (19) is in the same form of (12). We further set is given by:

Y = aD so that h(pe) + acDpe is V-uniformly decreasing (see -1 0 0Lemma 11 in [25]) provided that cI- a > 0 and C2-oc > 0 2 -1 -10. Theorem 2 says that (19) synchronizes if there exists a LG 0 0 1 -1 (22)symmetric zero row sums matrix U with nonpositive off- 0 0 0 0diagonal elements such that (U ® V) (-LG XD-IX Y) < 0.Since VD < 0 and Y = aD, this is equivalent to Since g2(-LG) =-2 we used consensus-based FTC (15)

with positive cl, c2, say cl = 0.3 and c2 = 0.5. As shown inU(-LGafcI) > 0. (20) Fig. 7, successful formation tracking with a desired triangle

Let g(-La) be the supremum ofall real numbers such that formation is achieved. Vehicle control signals vi's and coi's-(L 6cx) .0O. It was shown in [27] that p(-LG) exists are shown in Fig. 8.

for constant row sums matrices and can be computed by asequence of semidefinite programming problems. Choose cl V. CONCLUSIONS AND FUTURE WORKand c2 to be large enough such that

This paper addressed the formation tracking problem formin{c , c2} > gL(-Lc) (21) multiple mobile vehicles with nonholonomic constraints.

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Lei Fang, Panos Antsaklis, “Decentralized Formation Tracking of Multi-vehicle Systems with Nonlinear Dynamics,” 14th Mediterranean Conference on Control and Automation (MED’06), Università Politecnica delle Marche, Ancona, Italy, June 28-30, 2006.

25 25

r W r 1 Virtual Vehicle [7] Z.-P. Jiang and H. Nijmeijer, "A recursive technique for tracking control20 _... Vehicle of nonholonomic systems in chained form," IEEE Trans. Automat.

15 Vehicle 3 Contr., vol. 44, pp. 265-279, 1999.10 1 [8] Y. Kanayama, Y Kimura, F. Miyazaki, and T. Noguchi, "A stable

tracking control method for an autonomous mobile robot," in Proc. IEEE5 - Int. Conf Robotics & Automat., Cincinnati, OH, 1990, pp. 384-389.0 = [9] T.-C. Lee, K.-T. Song, C.-H. Lee, and C.-C. Teng, "Tracking control of5. unicycle-modeled mobile robots using a saturation feedback controller,"0 5 10 15 20 25 30 35 40 45 50 IEEE Trans. Contr. Syst. Technol., vol. 9, pp. 305-318, 2001.

[10] Z. Lin, B. Francis, and M. Maggiore, "Necessary and sufficientgraphical conditions for formation control of unicycles," IEEE Trans.Automat. Contr., vol. 50, no. 1, pp. 121-127, 2005.

0.5 [11] A. De Luca and M. D. Di Benedetto, "Control of nonholonomicsystems via dynamic compensation," Kybernetica, vol. 29, no. 6, pp.

3 0 593-608!, 1993.[12] A. De Luca, G. Oriolo, and C. Samson, "Feedback control of a

-0.5 -nonholonomic car-like robot,," in Robot Motion Planning and Control,, J.-P. Laumond, Ed. London, U.K.: Springer-Verlag, 1998, vol. 229, Lecture

-10 Notes in Computer and Information Sciences, pp. 171-253.0 5 10 15 20 25 30 35 40 45 50 [13] R. Olfati-Saber and J. S. Shamma. "Consensus filters for sensor

networks and distributed sensor fusion," in Proc. 44th IEEE CDC-ECCFig. 8. Vehicle control signals vi's and w1's. Seville, Spain, 2005, pp. 6698-6703.

[14] R. Olfati-Saber, J. A. Fax, and R. M. Murray. "Consensus andcooperation in multi-agent networked systems," Proc. of the IEEE, Feb.

We deveoped abasic ormatin tracing cotrolle (FTC) 2006 (under review).We developed a basic formation tracking controller (FTC) [15] G. Oriolo, A. De Luca, and M. Vendittelli, "WMR control via dynamicas well as a consensus-based one using only neighboring feedback linearization: Design, implementation, and experimental vali-leaders information. The stability properties of the multiple dation," IEEE Trans. Contr. Syst. Technol., vol. 10, no. 6, pp. 835-852,vehicle system in closed loop with these FTCs were studied 2002.[16] E. Panteley, E. Lefeber, A. Loria, and H. Nijmeijer,, "Exponentialusing cascaded systems theory and nonlinear synchronization tracking of a mobile car using a cascaded approach," in Proc. IFACtheory. In particular, we established connections between Workshop Motion Contr., Grenoble, France, 1998, pp. 221-226.stability of consensus-based FTC and Laplaican matrices for [17] Wei Ren, Randal W. Beard, Ella Atkins,"Information consensus in

multi-vehicle cooperative control: A tutorial survey of theory andformation graphs. Our simple formation tracking strategy applications" submitted to IEEE Contr. Syst. Mag., 2005.holds great potential to be extended to the case of air and [18] W. Ren and E. Atkins, "Distributed multi-vehicle coordinated controlmarine vehicles. via local information exchange," submitted to Int. J Robust and Nonlin.

Contr., 2005.We did not discuss collision avoidance and formation error [19] C. Samson and K. Ait-Abderrahim, "Feedback control of a nonholo-

propagation problems. Our FTC does not guarantee avoid- nomic wheeled cart in Cartesian space," in Proc. 1991 IEEE Int. Confance of collisions and there is a need to take care of them in Robot. Automat., Sacramento, CA, 1991, pp. 1136-1141.

[20] D. P. Scharf, F. Y Hadaegh, S. R. Ploen, "A survey of spacecraftthe future work. Furthermore, Corollary 1 showed that two formation flying guidance and control (Part II): Control," in Proc. ACCvehicles with a cascaded interconnection is formation ISS. 2004, Boston, MA, pp. 2976-2985, 2004.

[21] R. Skjetne, I.-A. F. Ihle, and T. I. Fossen, "Formation control byIts invariance properties under cascading could be explored synchronizing multiple maneuvering systems," in Proc. 6th IFAC Confto quantify the formation errors when individual vehicle's Maneuvering and Control of Marine Craft, Girona, Spain, 2003.tracking errors are bounded. [22] H. G. Tanner, G. J. Pappas, and V. Kumar, "Input-to-state stability on

formation graphs," in Proc. 41st IEEE Conf CDC(, Las Vegas, Nevada,REFERENCES 2002, pp. 2439-2444.

[23] H. G. Tanner, G. J. Pappas, and V. Kumar, "Leader-to-formation[1] G. Lafferriere, A. Williams, John S. Caughman, and J. J. P. Veerman, stability" IEEE Trans. R&A, vol. 20, no. 3, pp. 443-455, 2004.

"Decentralized control of vehicle formations" Systems & Control Let- [24] C. Canudas de Wit, H. Khennouf, C. Samson, and 0. J. Sordalen,ters, vol. 54, no. 9, pp. 899-910, 2005. "Nonlinear control design for mobile robots," in Recent Trends in Mobile

[2] Y Chen and Z. Wang, "Formation control: A review and a new Robots, Y F. Zheng, Ed. Singapore: World Scientific, 1993, vol. 11, pp.consideration," in 2005 IEEEIRSJ Int. Conf Intellig. Robots and Syst., 121-156.pp. 3181-3186, Edmonton, Alberta, Canada, 2005. [25] C. W. Wu and L. 0. Chua, "Synchronization in an array of linearly

[3] B. d'Andre-Novel, G. Bastin, and G. Campion, "Control of nonholo- coupled dynamical systems," IEEE Trans. Circuits & Systems-I: Fun-nomic wheeled mobile robots by state feedback linearization," Int. J damental Theory and Applications, vol. 42, no. 8, pp. 430-447, 1995.Robot. Res., vol. 14, no. 6, pp. 543-559, 1995. [26] C. W. Wu, "Synchronization in networks of nonlinear dynamical

[4] J. A. Fax and R. M. Murray, "Information flow and cooperative control systems coupled via a directed graph," Nonlinearity, vol. 18, pp. 1057-of vehicle formations," IEEE Trans. Automat. Contr., vol. 49, no. 9, pp. 1064, 2005.1465-1476, 2004. [27] C. W. Wu, "on a matrix inequlity and its application to the synchro-

[5] V. Gazi, "Formation control of mobile robots using decentralized nization in coupled chaotic systems," Complex Computing-Networks:nonlinear servomechanism," in Proc. 12th Mediterranean Conf Contr. Brain-like and Wave-oriented Electrodynamic Algorithms, Springer Pro-and Automat., Kusadasi, Turkey, 2004. ceedings in Physics, vol. 104, pp. 279-288, 2006.

[6] R. Ghabcheloo, A. Pascoal, C. Silvestre, and I. Kaminer, "Coordinatedpath following control of multiple wheeled robots with directed com-munication links," in Proc. IEEE CDC-ECC'05, Seville, Spain, 2005,pp. 7084-7089.

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Lei Fang, Panos Antsaklis, “Decentralized Formation Tracking of Multi-vehicle Systems with Nonlinear Dynamics,” 14th Mediterranean Conference on Control and Automation (MED’06), Università Politecnica delle Marche, Ancona, Italy, June 28-30, 2006.