Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seille, Spain, December 12-15, 2005 MoB032 Collectie circular motion of multi-ehicle systems with sensory limitations Nicola Ceccarelli, Mauro Di Marco, Andrea Garulli, Antonio Giannitrapani Abstract Collectie motion of a multi-agent system composed of nonholonomic ehicles is addressed The aim of the ehicles is to achiee rotational motion around a irtual reference beacon A control law is proposed, which guarantees global asymptotic stability of the circular motion with a prescribed direction of rotation, in the case of a single ehicle Equilibrium configurations of the multi-ehicle system are studied and sufficient conditions for their local stability are gien, in terms of the control law design parameters Practical issues related to sensory limitations are taken into account The transient behaior of the multi-ehicle system is analyzed ia numerical simulations I INTRODUCTION Multi-agent systems hae receied an increased interest in recent years, due to their enormous potential in seeral fields: collectie motion of autonomous ehicles, exploration of unknown enironments, sureillance, distributed sensor networks, biology, etc (see eg [1], [2] and references therein) Although a rigorous stability analysis of multi-agent systems is generally a ery difficult task, nice theoretical results hae been obtained in the case of linear motion models One of the first contributions in this respect was gien in [3], where a multi-ehicle system with a irtual reference beacon is considered Leader following, leaderless coordination and cyclic pursuits hae been studied in [1], [4] Stability analysis becomes een more challenging when kinematic constraints are taken into account, as in the case of wheeled nonholonomic ehicles Unicycle-like motion models hae been recently considered in seeral papers [5], [6], [2], [7] In particular, in [6], [5], [7] different control laws are proposed for circular and parallel motion of planar multi-ehicle systems, with complete stability analysis of the single-ehicle case In [2], equilibrium formations of multi-ehicle systems in cyclic pursuit are studied, and local stability is discussed in detail In this paper, the objectie of a team of nonholonomic ehicles is to achiee collectie circular motion around a irtual reference beacon As a first contribution, a control law is proposed, which is shown to guarantee global asymptotic stability of the counterclockwise circular motion around a fixed beacon, in the single-ehicle case This turns out to be a useful property also in the multi-agent case, because each ehicle tries to rotate around the beacon in the same direction of rotation Then, the control law is suitably modified to cope with the multi-ehicle case Equilibrium configurations of the multi-ehicle system under the proposed control law Authors are with the Dipartimento di Ingegneria dell Informazione, Uniersità di Siena, Via Roma 56, 53100 Siena, Italy E-mail: ceccarelli,dimarco,garulli,giannitrapani}@diiunisiit are discussed and sufficient conditions for local asymptotic stability are deried Sensory limitations are explicitly taken into account; in particular: i) each agent can perceie only ehicles lying in a limited isibility region; ii) a ehicle cannot measure the orientation of another ehicle, but only its relatie distance; iii) ehicles are indistinguishable Finally, simulation results are presented to show the effectieness of the proposed control law in the multi-ehicle case The paper is organized as follows In Section II, the control law is formulated for the single-ehicle case Global asymptotic stability of the counterclockwise circular motion around a fixed beacon is proed Section III concerns the multi-ehicle scenario: the modified control law is introduced and the resulting equilibrium configurations are studied Sufficient conditions for local stability are gien Simulation results are proided in Section IV, while some concluding remarks and future research directions are outlined in Section V Due to space limitations, the proofs of most technical results are omitted; the interested reader is referred to [8] II CONTROL LAW FOR A SINGLE VEHICLE Consider the planar unicycle model ẋ(t) = cos θ(t) (1) ẏ(t) = sin θ(t) (2) θ(t) = u(t); (3) where [x yθ] R 2 [ π, π) represents the ehicle pose, is the forward speed (assumed to be constant) and u(t) is the angular speed, which plays the role of control input The following control law, based on the ehicle relatie pose with respect to a reference beacon, is proposed k g((t)) αdist ((t)) (t) > 0 u(t) = (4) 0 (t) =0 with ( (c 1) + 0 ) g() =ln (5) c 0 and if 0 ψ α dist () = (6) 2π if ψ<<2π In (4)-(6), is the distance between the ehicle position r =[x y] and the beacon position r b =[x b y b ] ; [0, 2π) represents the angular distance between the heading of the ehicle and the direction of the beacon (see Fig 1); k>0, c>1, 0 > 0 and ψ ( 3 2π, 2π) are gien constants Remark 1: The terms in equation (4) hae different motiations The term g() in (5) assures that the control law 0-7803-9568-9/05/$2000 2005 IEEE 740
y b y Fig 1 r b x b r = e iγ r Single ehicle (triangle) and beacon (cross) steers the ehicle towards the beacon if 0 and steers it away from the beacon if 0 Thetermα dist in (6) is chosen to priilege the counterclockwise rotation with respect to the clockwise one, and it is critical for stability analysis The threshold ψ is introduced so that, when is large and is close to 2π, the ehicle goes straight towards the beacon instead of making useless circular motions (which would slow down conergence, especially in the multi-ehicle case) The choice of ψ> 3 2π is necessary to guarantee a unique direction of rotation about the beacon (as it will be shown in the following) Let us introduce the following change of ariables (see [6]) Γ r = r b r = e iγ (7) = (x x b ) 2 +(y y b ) 2 (8) = (Γ θ)mod(2π) (9) where Γ [0, 2π) denotes the angular distance between r and the x-axis (see Fig 1) By differentiating (7) with respect to time, one obtains x ṙ = e iγ + i Γe iγ (10) By using (9), ṙ = e iθ = e i e iγ, and hence for 0 one has = cos (11) Γ = sin (12) By differentiating (9) with respect to time, and using (4)-(6), one gets = Γ θ sin kg() if 0 ψ = sin kg()( 2π) if ψ<<2π (13) θ Let us consider now the system = cos() sin kg() if 0 ψ = sin kg()( 2π) if ψ<<2π (14) The first aim is to guarantee that (14) has a unique equilibrium point, corresponding to counterclockwise rotation of the ehicle around the beacon To this end, let us select the parameters, k, c, 0 so that min g() > 2 3 πk (15) This choice guarantees that for = 3 2π it holds < 0, ie clockwise rotation is not a limit cycle for system (1)-(6) Let D R ++ (0, 2π), where R ++ denotes the set of strictly positie real numbers The following result is straightforward [ ] e Proposition 1: The point p e = π where e is such 2 that: k g( e ) π =0 (16) e 2 is the only equilibrium of system (14), for (, ) D A consequence of Proposition 1 is that the counterclockwise circular motion with radius e and angular elocity Γ = e is a limit cycle for system (1)-(6) In order to perform stability analysis of the equilibrium p e in Proposition 1, let us introduce the following Lyapuno function V (, ) = A(ˆ)dˆ + B(ˆ)dˆ (17) π e 2 where A() = 2 ( kg() π π 2 ) (18) and cos if 0 < ψ B() = cos 2π if ψ<<2π (19) Hence ( ) V 2 π sin if 0 < ψ (, ) = ( ) cos 2 π sin 2π if ψ<<2π Define the following sets: D R ++ (0, 3 2π], (20) ˆD D \ D, (21) K R ++ (ψ, 2π) (22) It can be shown that V (, ) 0, (, ); V (, ) 0 for (, ) D, and V (, ) < 0 for (, ) KMoreoer V (, ) =0only for = e, = π 2,andV(, ) is radially unbounded on D Since the ector field (14) is discontinuous, one cannot use directly the LaSalle s Inariance Principle to proe asymptotic stability of p e on D Notice howeer that it can 741
be shown that one solution always exists, in the Filippo s sense (see eg [9]) Neertheless, the Lyapuno function (17) will be useful to proe the main result, stated below Theorem 1: The counterclockwise circular motion around the beacon of fixed position r b, with rotational radius e defined in (16) and angular elocity e, is a globally asymptotically stable limit cycle for the system (1)-(6) In order to proe Theorem 1, two preliminary Lemmas are needed First, Lemma 1 shows that for any initial condition in ˆD, there exists a finite time t such that (( t),( t)) D Then, Lemma 2 shows that, for initial ehicle poses outside D (ie, when =0or =0), there exists a finite time ˆt such that ((ˆt),(ˆt)) D From these two lemmas, one can conclude that eery trajectory will end up in D in finite time Finally, Theorem 1 will be proed using Lyapuno arguments in D Lemma 1: For any trajectory of system (14) with initial condition ((0),(0)) ˆD, there exists a finite time t >0 such that (( t),( t)) D Now, let us consider all the initial ehicle poses such that the ehicle points towards the beacon, or the ehicle lies exactly on the beacon, ie B (r (0),θ(0)) : =0, 0 << } (r (0),θ(0)) : =0} (23) Lemma 2: For any trajectory of system (1)-(6) with initial conditions in B, there exists a finite time ˆt such that ((ˆt),(ˆt)) D Now Theorem 1 can be proed Proof: From Lemmas 1-2 it follows that for any initial condition (r (0),θ(0)) R 2 [0, 2π), there exists a finite time t 0 such that ((t ),(t )) D Nowwewant to proe that any trajectory starting in D conerges to the equilibrium p e defined in Proposition 1 Notice that in D system (14) boils down to = cos = sin kg() (24) Consider any initial condition ((0),(0)) η = V ((0),(0)) and the set D Define S η (, ) D : V (, ) η } (25) Since V (, ) is radially unbounded, the set S η is compact and its boundary has the following form S η (, ) : (V (, ) =η < 3 2 π) (V (, ) η = 3 2 π)} (26) Moreoer, being V (, ) =[A() B()], it can be obsered that S η is connected We want to show that the set S η is a iability domain for the system (24), ie the ector field always points inside, or is tangent to, the contingent cone to any point on S η (see Figure 2, and [10, p 25-26] for a rigorous definition) Consider (, ) S η such that < 3 2π V(, ) =η S η 3 2 π Fig 2 Examples of contingent cones for a set S η The dashed circular sectors represent the contingent cones associated to different points on S η The arrows represent the corresponding ector field (24) Since for such (, ) one has V (, ) < 0, one can conclude that the ector field points inside the contingent cone to S η Consider now, if there exists, any (, ) S η such that = 3 2 π V(, ) <η Because of (15) the ector field defined by system (24) is of the form [ ] [ ] 0 = (27) α with α>0 Hence, it points inside the contingent cone Finally consider, if there exists, any (, ) S η such that = 3 2 π V(, ) =η Notice that for such (, ), one has V (, ) =[β,0], with β R Hence, one of the two edges of the contingent cone to S η is orthogonal to the line = 3 2π (see Figure 2) Again, from (24) one has that the ector field is of the form (27) and hence it is tangent to the contingent cone Now, notice that the ector field (24) is Lipschitz on S η Hence, by applying Nagumo theorem (see eg, [10, Theorem 124, p 28]), for any ((0),(0)) S η the unique solution of (24) will not leae S η for any t 0 Inotherwords,S η is a positiely inariant set with respect to system (24) By recalling that V (, ) 0, (, ) D, one can apply LaSalle s Inariance Principle (see eg [11]) to conclude that the trajectories starting in S η conerge asymptotically to the largest inariant set M such that M E (, ) S η : V (, ) =0} It is triial to show that, for any η 0, the set M contains only the equilibrium point p e Being the choice of ((0),(0)) D arbitrary, conergence to p e occurs for any trajectory starting in D Because the equilibrium point p e corresponds to counterclockwise circular motion around the beacon r b with radius e, it can be concluded that such motion is a globally asymptotically stable limit cycle for the system (1)-(6) Remark 2: The stability analysis of this section applies to a broad class of functions g() In particular global asymptotic stability holds for any locally Lipschitz g() satisfying (15), g(0) <, and such that there exists a 742
θ j j ji j ij r j r b i i ij V j j d 0 d l θ i 2α V i r i i d s Fig 3 Two ehicles (triangles) and a beacon (cross) Fig 4 Visibility region of i-th and j-th ehicle unique solution e of A() =0and e A(σ)dσ is radially unbounded for D III MULTI-VEHICLE SYSTEMS In this section the control law (4) is modified in order to deal with a multi-ehicle scenario Consider a group of n agents whose motion is described by the kinematic equations ẋ i (t) = cos θ i (t) (28) ẏ i (t) = sin θ i (t) (29) θ i (t) = u i (t), (30) with i =1nLet: i and i be defined as in Section II; ij and ij denote respectiely the linear and angular distance between ehicle i and ehicle j (see Figure 3); g(, c, 0 ) be equal to g() in (5) In the control input u i (t) a new additie term is introduced which depends on the interaction between the i-th ehicle and any other perceied ehicle j u i (t) =f ib ( i, i )+ f ij ( ij, ij ) (31) j i j N i where f ib is the same as in the right hand side of (4), ie while f ib ( i, i )=k b g( i,c b, 0 ) α dist ( i ), (32) f ij ( ij, ij )=k g( ij,c,d 0 ) β dist ( ij ), (33) with k > 0, c > 1, d 0 > 0 and ij 0 β dist ( ij )= ij π (34) ij 2π π < ij < 2π The set N i contains the indexes of the ehicles lying inside the isibility region V i associated with the i-th ehicle In this paper, the isibility region has been chosen as the union of the following two sets (see Figure 4): Circular sector of ray d l and angular amplitude 2α, centered at the ehicle pose and orientation It represents a long range sensor with limited angular isibility (eg, a laser range finder) Circular region around the ehicle of radius d s,which represents a proximity sensor (eg, a ring of sonars) Remark 3: The motiation for the control law (31)-(34) relies in the fact that each agent i is drien by the term f ib ( ) towards the counterclockwise circular motion about the beacon, while the terms f ij ( ) (and the isibility regions V i ) faour collision free trajectories trying to keep distance ij = d 0 for all the agents j N i This follows intuitiely from the fact that ehicle i is attracted by any ehicle j N i if ij > d 0, and repulsed if ij < d 0 The expected result of such combined actions is that the agents reach the counterclockwise circular motion in a number of platoons, in which the distances between consecutie ehicles is d 0 Asit will be shown in Section IV, this is confirmed by simulations (see for example Figure 5) In the following, local stability analysis of system (28)- (30) under the control law (31)-(34), is presented A Equilibrium configurations From Section II and by using a coordinate transformation similar to the one adopted in [2], one obtains the equations i = cos i (35) i = sin i u i i (36) ij = (cos ij +cos ji ) (37) ij = (sin ij +sin ji ) u i (38) ij i, j =1n, j i, andu i defined in (31) Notice that there are algebraic relationships between the state ariables i, i, ij, ij, which will be taken into account in the stability analysis Let us choose the parameters d 0, d l and the number of ehicles n so that ( d0 ) ( dl ) (n 1) arcsin +arcsin <π (39) 2 e 2 e This choice guarantees that the n ehicles can lie on a circle of radius e, with distance d 0 between two consecutie 743
ehicles and distance longer than d l between the first and the last The following result proides equilibrium configurations for the considered multi-ehicle system Proposition 2: Eery configuration of n ehicles in counterclockwise circular motion about a fixed beacon, with rotational ray i = e defined in (16), and ij = d 0 i =1n and j N i, corresponds to an equilibrium point of system (35)-(38), under the control law (31)-(34) B Stability analysis Let us choose d s < d 0 and α π/2, so that card(n i ) 0, 1} in the equilibrium configurations defined by Proposition 2 Wlog consider n ehicles in counterclockwise circular motion about a beacon and renumber them so that ehicle 1 cannot perceie any other ehicle, while any ehicle i 1sees only ehicle i 1 The equilibrium point defined by Proposition 2 is such that i = e i = π 2 i =1n (i 1)i = π d0 (40) 2 + arccos( 2 e ) i =2n Being card(n 1 ) = 0 and card(n i ) = 1 for i 2, the kinematic of the i-th ehicle is locally affected only by the position of the beacon and that of ehicle i 1, except for ehicle 1 whose kinematic is locally determined only by the beacon position We make the assumption that there is a neighborhood of the aboe equilibrium configuration in which the sets N i do not change (this can be seen as a further mild constraint on the size of the isibility region, ie arcsin( d l 2 e ) < 2arcsin( d0 2 e )) The total kinematic system is described by the equations 1 = cos 1 (41) 1 = sin 1 u 1 (42) 1 i = cos i (43) i = sin i u i i (44) (i 1)i = (sin (i 1)i +sin i(i 1) ) u i 1 (45) (i 1)i for i =2n The control inputs are gien by u 1 = f 1b and u i = f ib + f i(i 1) for i 2 Notice that the 3n 1 state ariables in system (41)-(45) are sufficient to describe completely the n-ehicles system Indeed, the remaining state ariables in system (35)-(38) can be obtained ia algebraic relationships form the aboe 3n 1 ones By linearizing system (41)-(45) around the equilibrium point (40), one gets a system of the form 1 1 2 2 12 n n (n 1)n A 0 0 = B 0 B 1 1 2 2 12 n n (n 1)n (46) where A R 2 2 and B R 3 3 The eigenalues of A are strictly negatie; in fact the kinematic of ehicle 1 is decoupled by that of the other ehicles, and hence the stability analysis performed in Section II for the single ehicle case holds Therefore, it is sufficient to show that matrix B is Hurwitz to guarantee asymptotic stability of system (41)-(45) The following Lemma gies a sufficient condition in this respect Lemma 3: If the parameters k b,c b,k,c in (31)-(34) satisfy k 2 c c b 1 (47) k b c b c 1 then matrix B in (46) is Hurwitz Therefore, a control law satisfying (47) guarantees asymptotic stability of the considered equilibrium configuration of the n-ehicle system On the other hand, (47) gies a guideline to find alues of the parameters for which the considered equilibrium is unstable: examples can easily be found for c = c b and k k b,orfork = k b and c c b This is in good agreement with intuition, as it basically says that the beacon-drien control term should not be excessiely reduced with respect to the control input due to interaction with the other agents IV SIMULATION RESULTS In this section, simulation studies are proided for the multi-ehicle system (28)-(30) under the control law (31)- (34) In the examples, the considered isibility region V i has been chosen with d l =12, α = π 4, d s =3 The control law parameters are set to: ψ = 7 4 π, k b =007, c b =2, k =01, c =3, d 0 =8, =1and 0 =10 A typical run for an 8-ehicle system is reported in Figure 5 The equilibrium configuration reached by the multi-ehicle system consists of 3 separate platoons of cardinality 5, 2 and 1 respectiely, with radius e 209 The same control law has been applied to the case of 6 ehicles following a non-static beacon which jumps through four sequential way-points (a similar scenario was considered in [7]) In particular, beacon positions are set as follows: r b1 =[0, 0] if t 600 r b2 = [100, 0] if 600 t 1200 r b3 = [100, 100] if 1200 t 1800 r b4 =[0, 100] if t 1800 Figure 6 shows the trajectories of the 6 ehicles Notice that when the beacon switches from r bi to r bj, the ehicles moe from a rotational configuration about r bi towards r bj in a sort of parallel motion At the same time, the desired rotational motion is reached for each way-point It is worth remarking that this is obtained without switching between two different control laws Repeated runs hae been performed to analyze the role of the initial configuration of the multi-ehicle system Figure 7 reports the estimated conergence times for 100 simulations with random initial conditions, for a 4-ehicle system The conergence test is based on the difference between the 744
2000 1500 t 1000 500 0 10 20 30 40 50 60 70 80 90 100 runs Fig 7 Conergence times oer 100 runs Fig 5 A 8-ehicle scenario with a static beacon r b4 r b3 communication protocols to identify ehicles or to exchange information are not required This work is still at a preliminary stage and seeral interesting deelopments can be foreseen: to proide sufficient conditions for global asymptotic stability of the multi-ehicle system, to analyze the role of the design parameters in the control law and to study tracking performance in the presence of a moing beacon are the subject of ongoing research VI ACKNOWLEDGMENTS The authors would like to thank people at the Systems Control Group of the Department of Electrical and Computer Engineering, Uniersity of Toronto, for fruitful discussions and useful comments, especially Prof Bruce Francis, Prof Mireille Broucke and Prof Manfredi Maggiore Fig 6 r b1 rb2 A 6-ehicle team tracking a moing beacon (asterisks) rotational radius of each ehicle and e (namely i (t) e ) All the simulations terminated successfully V CONCLUSIONS AND FUTURE WORK The problem of collectie circular motion for a team of nonholonomic ehicles has been addressed The main features of the proposed control law are: i) it guarantees global stability in the single-ehicle case; ii) control parameters can be easily selected to achiee local stability of the equilibrium configurations of interest in the multi-ehicle scenario; iii) simulation studies show promising results in terms of conergence rates and tracking performance With respect to similar approaches presented in the literature, quite restrictie assumptions hae been remoed, eg: total isibility is not required, exteroceptie orientation measurements are not performed, labelling of the ehicles is not necessary, REFERENCES [1] A Jadbabaie, J Lin, and A S Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Transactions on Automatic Control, ol 48, no 6, pp 988 1001, June 2003 [2] J A Marshall, M E Broucke, and B A Francis, Formations of ehicles in cyclic pursuit, IEEE Transactions on Automatic Control, ol 49, no 11, pp 1963 1974, Noember 2004 [3] N E Leonard and E Fiorelli, Virtual leaders, artificial potentials and coordinated control of groups, in Proceedings of the IEEE Conference on Decision and Control, Orlando, 2001, pp 2968 2973 [4] Z Lin, M E Broucke, and B A Francis, Local control strategies for groups of mobile autonomous agents, IEEE Transactions on Automatic Control, ol 49, no 4, pp 622 629, April 2004 [5] R Sepulchre, D Paley, and N E Leonard, Collectie motion and oscillator syncrhronization, in Proc Block Island Workshop on Cooperatie Control, V Kumar, N Leonard, and A Morse, Eds, June 2003 [6] E W Justh and P S Krishnaprasad, Equilibria and steering laws for planar formations, Systems and Control Letters, ol 52, pp 25 38, 2004 [7] D Paley, N E Leonard, and R Sepulchre, Collectie motion: bistability and trajectory tracking, in Proceedings of the IEEE Conference on Decision and Control, Nassau, Bahamas, 2004, pp 1932 1936 [8] N Ceccarelli, M D Marco, A Garulli, and A Giannitrapani, Stability analysis of collectie circular motion for nonholonomic multi-ehicle systems, Dipartimento di Ingegneria dell Informazione, Uniersità di Siena, Tech Rep 02/05, 2005 [9] A Filippo, Differential Equations with Discontinuous Righthand Sides Kluwer Academic Publisher, 1988 [10] J P Aubin, Viability theory Birkhauser, 1991 [11] H Khalil, Nonlinear Systems New York: Macmillan Publishing Co, 1992 745