Collectie circular motion of multi-ehicle systems with sensory limitations Nicola Ceccarelli, Mauro Di Marco, Andrea Garulli, Antonello Giannitrapani Dipartimento di Ingegneria dell Informazione Uniersità di Siena Via Roma 56, 53100 Siena, Italy ceccarelli,dimarco,garulli,giannitrapani@dii.unisi.it 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 e.g. [1], [] 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], [], [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 [], 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 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. II. CONTROL LAW FOR A SINGLE VEHICLE Consider the planar unicycle model ẋ(t) = cos θ(t) (1) ẏ(t) = sin θ(t) () θ(t) = u(t); (3) where [x y θ] R [ 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 with u(t) = k g((t)) α dist ((t), (t)) (4) ( (c 1) + 0 ) g() = ln c 0 (5)
PSfrag replacements y b y Fig. 1. r b x b r = e iγ r Single ehicle (triangle) and beacon (cross). and > 0 0 ψ α dist (, ) = π > 0 ψ < < π 0 = 0 Γ x θ. (6) In (4)-(6), is the distance between the ehicle position r =[x y] and the beacon position r b = [x b y b ] ; [0, π) 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 π, π) are gien constants. Remark 1: The terms in equation (4) hae different motiations. The term g() in (5) assures that the control law steers the ehicle towards the beacon if 0 and steers it away from the beacon if 0. The term α 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 π, 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 π 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]) r = r b r = e iγ (7) = (x x b ) + (y y b ) (8) = (Γ θ)mod(π) (9) where Γ [0, π) denotes the angular distance between r and the x-axis (see Fig. 1). By differentiating (7) with respect to time, one obtains ṙ = e iγ + i Γe iγ. (10) Since, by using (9), ṙ = e iθ = e i e iγ, one has that for 0 = cos (11) Γ = sin. (1) By differentiating (9) with respect to time, and using (4)-(6), one gets = { Γ θ sin k g() if 0 ψ. = sin k g() ( π) if ψ < < π Let us consider now the system = cos() = { sin k g() if 0 ψ. sin k g() ( π) if ψ < < π (13) (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() > 3 π k. (15) This choice guarantees that for = 3 π it holds that < 0, i.e. clockwise rotation is not an equilibrium for system (14), or equialently clockwise rotation is not a limit cycle for the system (1)-(6). Let D R ++ (0, π), where R ++ denotes the set of strictly positie real numbers. The following result is straightforward. ] that: Proposition 1: The point p e = [ e π where e is such k g( e ) π e = 0 (16) is the only equilibrium of system (14), for (, ) D. The consequence of Proposition 1 is that 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 p e, let us introduce the following Lyapuno function where V (, ) = B() = e A(ˆ)dˆ + π { cos if 0 < ψ cos π if ψ < < π B(ˆ)dˆ (17) (18) and A() = ( k g() π π ). (19) Hence ( ) V cos π sin if 0 < ψ (, ) = ( ). cos π sin π if ψ < < π Define the following sets: D R ++ (0, 3 π], (0) K R ++ (ψ, π), (1) ˆD D \ D. ()
rag replacements e 0, 0 D ˆD K π 3 ψ π π Fig.. Vector field (14) on D. It can be shown that V (, ) 0, and V (, ) 0 for (, ) D, V (, ) < 0 for (, ) K. Moreoer V (, ) = 0 only for ( e, π ) and V (, ) is radially unbounded on D i.e. lim V (, ) = + lim 0 V (, ) = + lim lim 0 V (, ) = + V (, ) = +. π 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 be shown that one solution always exists, in the Filippo s sense (see e.g. [8], [9]). Neertheless, the Lyapuno function (17) will be useful to proe the main result, stated below. Theorem 1: For the kinematic system (1)-(6), 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. In order to proe Theorem 1, two preliminary Lemmas are needed. First, it is proed that for any initial condition in ˆD, there exists a finite time t such that ((t ), (t )) D (Lemma 1). Then, Lemma shows that, for initial ehicle poses outside D (i.e., when = 0 or = 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 (a phase portrait of ector field (14) is depicted in Figure ). Finally, Theorem 1 will be proed using Lyapuno arguments in D. Lemma 1: For any initial condition ( 0, 0 ) ˆD, there exists t > 0 such that ((t ), (t )) D. (3) Proof: Consider the following sets: ˆD 1 R ++ ( 3 π, β] ˆD R ++ (β, π) where β is such that < f < 0 for any (, ) ˆD 1. Notice that such a β can always be found because of (15). Obsere that for any (, ) ˆD, one has < 0 but, despite the discontinuity in (14), also lim (, ) = 0 + ( 3 π, π), and hence the trajectories cannot leae the set ˆD through the line = 0. Moreoer, because V (, ) is radially unbounded on D and V (, ) < 0 on the set K in (1), then the trajectories cannot leae the set ˆD through the line = π. Now consider any initial condition ( 0, 0 ) ˆD. In such set, one has cos β < 0. Since there cannot exist trajectories such that (t) = 0 or (t) = π, as shown aboe, one can always find a time t 1 0 cos β such that (t 1 ) = β and hence ((t 1 ), (t 1 )) ˆD 1. Choose now any initial condition ( 0, 0 ) ˆD 1. Since < f, and there cannot exist a trajectory such that (t) = 0, one can always find a time t 0 3 π f such that (t ) = 3 π with (t ) > 0. Therefore, we hae proed that for any initial condition in ˆD, there exists a finite time t 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, i.e. B {(r b, r (0), θ(0)) : = 0, 0 < < } {(r b, r (0), θ(0)) : = 0}. (4) Lemma : For any trajectory of system (1)-(6) with initial conditions in B, there exists a finite time ˆt such that ((ˆt), (ˆt)) D. Proof: Define the following sets of initial conditions B 1 = {(r b, r (0), θ(0)) : = 0, 0 < < }, B = {(r b, r (0), θ(0)) : = 0}. Consider first (r b, r (0), θ(0)) B 1. Then, system (14) boils down to (t) = (t) = 0 t [0, t) where t = (0). Hence the ehicle proceeds straight towards the beacon until it reaches it, i.e. ( t) = 0 and hence (r b, r ( t), θ( t)) B. Consider now an initial condition (r b, r (0), θ(0)) B (notice that in this case is not defined). We want to proe that in finite time ((t), (t)) D. W.l.o.g. set r b = r (0) = [0, 0] and θ(0) = 0. Obsere that system (1)-(6) doesn t hae any equilibrium point, and that θ obtained from (3)-(4) is bounded in compact sets. This means that there exists δ > 0 such that θ(t) M δ =: ɛ, t δ (one can choose e.g., M = k ln(c) ψ). Hence, it follows that [ ] [ ] δ cos ɛ δ r 0 (δ), (5) δ sin ɛ where and hae to be interpreted as componentwise operators. By geometric arguments it can be shown that δ cos ɛ (δ) δ π ɛ (δ) π + ɛ
PSfrag replacements S c 3 π Fig. 3. Examples of contingent cones for a set S c. The dashed circular sectors represent the contingent cones associated to different points on S c. The arrows represent the corresponding ector field (6). which means ((δ), (δ)) D. This concludes the proof. Now Theorem 1 can be proed. Proof: From Lemmas 1- it follows that for any initial condition (r b, r (0), θ(0)) R R [0, π), there exists a finite time t 0 such that ((t ), (t )) D. Now we want 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 k g(). (6) Consider any initial condition ( 0, 0 ) D. Define c = V ( 0, 0 ) and the set S c { (, ) D : V (, ) c }. (7) Since V (, ) is radially unbounded, the set S c is compact and its boundary has the following form S c { (, ) : (V (, ) = c < 3 π) (V (, ) c = 3 π)}. (8) We want to show that the set S c is a iability domain for the system (6), i.e. the ector field always points inside, or is tangent to, the contingent cone to any point on S c (see Figure 3, and [10, p. 5-6] for a rigorous definition). Consider, if there exists, any (, ) S c such that = 3 π V (, ) < c. Because of (15) the ector field defined by system (6) is of the form [ ] [ ] 0 = (9) α with α > 0. Hence, it points inside the contingent cone. Consider now (, ) S c such that < 3 π V (, ) = c. Since for such (, ) one has V (, ) < 0, the ector field points inside the contingent cone to S c. Finally consider, if there exists, any (, ) S c such that = 3 π V (, ) = c. Notice that for such (, ), one has V (, ) = [β, 0], with β R. Hence, one of the two edges of the contingent cone to S c is orthogonal to the line = 3 π (see Figure 3). Again, from (6) one has that the ector field is of the form (9) and hence it is tangent to the contingent cone. Now, notice that the ector field (6) is Lipschitz on S c. Hence, by applying Nagumo theorem (see e.g., [10, Theorem 1..4, p. 8]), for any ( 0, 0 ) S c there exists a unique solution for t [0, ], and such solution will not leae S c. In other words, S c is a positiely inariant set with respect to system (6). By recalling that V (, ) 0, (, ) D, one can apply LaSalle s Inariance Principle (see e.g. [11]) to conclude that the trajectories starting in S c conerge asymptotically to the largest inariant set M such that M E {(, ) S c : V (, ) = 0}. It is triial to show that, for any c 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 : The stability analysis of this section holds for a larger class of functions g(), than the one proposed in (5). In particular for any locally Lipschitz g() such that - the inequality (15) holds -! e s.t A( e ) = 0 - A() > 0 for > e - A() < 0 for < e, with A() defined in (19), the counterclockwise circular motion is globally asymptotically stable for the system (1)- (6). It is triial to show that the function (5) belongs to this class. Moreoer notice that global stability is presered also using simpler functions, e.g. g() = constant > 0. 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) (30) ẏ i (t) = sin θ i (t) (31) θ i (t) = u i (t), (3) with i = 1... n. Let: 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 4); 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 ) (33) j i j N i
Sfrag replacements j r b i i ij Fig. 4. PSfrag replacements Fig. 5. r i r j ji j ij Two ehicles (triangles) and a beacon (cross). V j i α d s j d 0 θ i V i d l θ j Visibility region of i-th and j-th ehicle. where f ib is the same as in the right hand side of (4), i.e. while f ib ( i, i ) = k b g( i, c b, 0 ) α dist ( i, i ), (34) f ij ( ij, ij ) = k g( ij, c, d 0 ) β dist ( ij ), (35) with k > 0, c > 1, d 0 > 0 and { ij 0 β dist ( ij ) = ij π ij π π < ij < π. (36) 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 5): Circular sector of ray d l and angular amplitude α, centered at the ehicle pose and orientation. It represents long range sensor with limited angular isibility (e.g., a laser range finder). Circular region around the ehicle of radius d s, which represents a proximity sensor (e.g., a ring of sonar). Remark 3: The intuitie motiation of such a control law 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. As it will be shown in Section IV, this is confirmed by simulations (see for example Figure 8). In the following, local stability analysis of system (30)- (3) under the control law (33)-(36), is presented. A. Equilibrium configurations From Section II and by using a coordinate transformation similar to the one adopted in [], one obtains the equations i = cos i (37) i = i sin i u i (38) ij = (cos ij + cos ji ) (39) ij = ij (sin ij + sin ji ) u i (40) i, j = 1... n, j i, and u i defined in (33). Notice that there are algebraic relationships between the state ariables i, i, ij, ij, which must 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 (n 1) arcsin ( d0 ) ( dl ) + arcsin < π. (41) This choice guarantees that the n ehicles can lie on a circle of radius e, with distance d 0 between two consecutie ehicles and distance longer than d l between the first and the last. Proposition : 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 = 1... n and j N i, corresponds to an equilibrium point of system (37)-(40). Proof: First, notice that by assumption (41), there is at least one configuration that satisfies the statement of the proposition. The counterclockwise circular motion implies i = π, and hence i = 0. Now, notice that ij = d 0 for any j N i implies that f ij ( ) = 0. Hence u i in (33) is equal to (4). From (16), it can be concluded that for i = e one has i = 0. If all the ehicles lie on the same circle of radius e, with the same direction of rotation, by triial geometric considerations it can ( be shown that ij + ji = π ij and min{ ij, ji } = arcsin ). Hence ij = 0 and, by exploiting (16) again ij = sin ij = 0. ij e
PSfrag replacements e 1 3 ariables in system (43)-(47) are sufficient to describe completely the n-ehicles system. Indeed, the remaining state ariables in system (37)-(40) can be obtained ia algebraic relationships form the aboe 3n 1 ones. In particular one has Fig. 6. A possible equilibrium configuration of n ehicles: N i = {i 1} and i(i 1) = d 0 for i =,..., n, while N 1 is empty. which concludes the proof. Remark 4: Consider isibility regions with d s < d 0 and α π/ (typical for range finders). Hence, the condition ij = d 0 i = 1... n and j N i in Proposition implies that, when lying on the circle of radius e, each ehicle perceies at most one ehicle, i.e. card(n i ) {0, 1}. Moreoer, due to assumption (41), there is at least one ehicle that does not perceie the others. If there are q ehicles with card(n i ) = 0 and n q ehicles with card(n i ) = 1, the equilibrium configuration is made of q separate platoons. The limit cases are obiously q = 1 (a unique platoon) and q = n (n ehicles rotating independently around the beacon). B. Local stability analysis Let us choose d s < d 0 and α π/, so that card(n i ) {0, 1} in the equilibrium configurations defined by Proposition. W.l.o.g. consider n ehicles in counterclockwise circular motion about a beacon and renumber them as in Figure 6. The equilibrium point defined by Proposition is therefore i = e i = π i = 1... n (i 1)i = π d0 + arccos( ) i =... n. n (4) Being card(n 1 ) = 0 and card(n i ) = 1 for i, 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 assumptions that there is a neighborhood of the equilibrium configuration in which the sets N i does not change (this can be seen as a further mild constraint on the size of the isibility region, i.e. arcsin( d l ) < arcsin( d0 )). The total kinematic system is described by the equations 1 = cos 1 (43) 1 = 1 sin 1 u 1 (44) i = cos i (45) i = sin i u i (46) i (i 1)i = (i 1)i (sin (i 1)i + sin i(i 1) ) u i 1 (47) for i =... n. The control inputs are gien by u 1 = f 1b and u i = f ib + f i(i 1) for i. Notice that the 3n 1 state (i 1)i = (i 1) cos( (i 1)i (i 1) ) + + i (i 1) sin ( (i 1)i (i 1) ) (48) ( ) (i 1) i(i 1) = i arcsin sin( (i 1)i (i 1) ) (49) i for i =,..., n. By linearizing system (43)-(47) around the equilibrium point (4), one gets a system of the form 1 1 1.. n n (n 1)n A 0 0 = B............. 0 B 1 1 1.. n n (n 1)n (50) where A R 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 local stability of system (43)-(47). In order to compute the expression of matrix B, one has to take into account the dependence of the right hand side of (46) and (47) on (i 1)i and i(i 1), which in turn depend on i, i and (i 1)i ia (48)-(49). Through some tedious calculations, it can be shown that matrix B has the form where b 1 = b = b 3 = b 31 = e ( c k 1 π c d 0 π e c k 1 e 0 0 b 1 b b 3 b 31 b 3 b 33 + k b c b 1 (c b 1) e+ 0 π + arccos( d0 ) ) d 0 c d 0 4 e d 0 ( π b 3 = d 0 1 d 0 4 e b 33 = 0. (51) arccos( d0 )) By noticing that b ij 0, and applying the Routh-Hurwithz criterion to the characteristic polynomial of matrix B, it can be seen that the only condition that must be checked is b (b 1 + b 3 b 3 ) + b 31 b 3 > 0. Moreoer it can be
shown that the following inequality holds b (b 1 + b 3 b 3 ) + b 31 b 3 > { c b 1 + k b 1 π e c b k e c 1 c }. (5) Therefore, it suffices to enforce positiity of the right hand side in (5) to guarantee local asymptotic stability of the considered equilibrium. This holds for example wheneer k k b c c b c b 1 c 1. On the other hand, (5) 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, or for k = 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 (30)-(3) under the control law (33)- (36). In all the examples, the considered isibility region V i has been chosen with d l = 1, α = π 4, d s = 3. The control law parameters are set to: ψ = 7 4 π, k b = 0.07, c b =, k = 0.1, c = 3, d 0 = 8. A 4-ehicle system with a static reference beacon is considered in Figure 7, with = 0.3, 0 = 6. It can be seen that the multi-ehicle system conerges to a single-platoon formation, which satisfies Proposition with e 9.84 gien by (16). A 8-ehicle system is considered in Figure 8. In this case, = 1 and 0 = 10. The equilibrium configuration reached by the multi-ehicle system consists of 3 separate platoons of cardinality 5, and 1 respectiely (i.e., q = 3 according to Remark 4), with radius e 0.9. In Figure 9, the same control law has been applied to the case of 6 ehicles following a non-static beacon, to ealuate the tracking behaior of the multi-ehicle system. The beacon moes through four sequential way-points. A similar scenario was considered in [7]. In particular, the beacon trajectory is r b1 = [0, 0] if t 600 r b = [100, 0] if 600 t 100 r b3 = [100, 100] if 100 t 1800 r b4 = [0, 100] if t 1800 Notice that when the current target changes 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, see Figure 9. The trajectories show that the configuration in rotational motion is reached for each way-point. Repeated runs hae been performed to analyze the role of the initial configuration of the multi-ehicle system. In Figure 10, the estimated conergence times for 100 simulations with random initial conditions are reported, for the 4-ehicle system considered aboe. The conergence test is based on the difference between the rotational radius of each ehicle PSfrag replacements t PSfrag replacements Fig. 7. A 4-ehicle scenario with static beacon (black triangles represent initial poses). t Fig. 8. A 8-ehicle scenario with static beacon. and e (namely i (t) e ). All the simulations terminated with success. The transient behaior of the multi-ehicle system is clearly affected by the choice of the control law parameters. In particular, the following aspects must be taken into account: The clockwise rotation about the beacon must not be an equilibrium point, i.e. the parameters hae to erify inequality (15). The equilibrium radius e must be consistent with the number of ehicles and d 0. This means that the configuration of a single platoon in cyclic pursuit must be feasible with respect to inequality (41). The choice of k is influenced by a trade-off between collision aoidance and the effect of the beacon attraction. Indeed, notice that the multi-ehicle system with k = 0 is globally asymptotically stable by the triial consideration that the oerall system is the composition of globally asymptotically stable decoupled subsystems. This means that a possible source of instability could
PSfrag replacements t Fig. 9. 000 r b4 r b3 r b1 rb A 6-ehicle team tracking a moing beacon (asterisks). labeling of the ehicles is not required. This aspects are critical for multi-ehicle systems with equipment limitations, hampering long range measurements or estimation of other ehicles orientation. Moreoer, there is no need of smart communication protocols to identify ehicles or to exchange information, because the required measurements can be easily obtained from range sensors. This work is still at a preliminary stage and seeral interesting deelopments can be foreseen. From a theoretical point of iew, it would be desirable to proide sufficient conditions for the conergence of the multi-ehicle system to the desired equilibrium configurations. A deeper understanding of the role of the design parameters in the control law and how they affect the behaior of the multi-ehicle system would be useful. Moreoer, the tracking performance in the presence of a moing beacon must be thoroughly inestigated. In this respect, a nice feature of the proposed control law is that, when the beacon is faraway from the ehicles, each agent points straight towards it. Hence, a smooth transition between circular and parallel motion is expected. t 1500 1000 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, useful suggestions and comments, especially Prof. Bruce Francis, Prof. Mireille Broucke and Prof. Manfredi Maggiore. Sfrag replacements 500 0 10 0 30 40 50 60 70 80 90 100 Fig. 10. runs Conergence times oer 100 runs come only from the cross terms in (35). In this respect, the choice of a small k is recommended for good performance in term of conergence time. On the contrary, a large k faours a safe collision-free motion. V. CONCLUSIONS AND FUTURE WORK The main features of the control law proposed in this work are: i) it guarantees global stability in the singleehicle case; ii) control parameters can be easily selected to achiee local stability of the equilibrium configurations of interest in the multi-ehicle scenario; iii) simulatie studies show promising results in terms of conergence rate and tracking performance. With respect to similar approaches presented in the literature, quite restrictie assumptions hae been remoed, e.g.: total isibility is not required, i.e. the sensors perceie only the agents in the isibility regions V i ; exteroceptie orientation measurements are not performed: each ehicle has to measure only distances to other ehicles lying in its isibility region; 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 003. [] 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 004. [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, 001, pp. 968 973. [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. 6 69, April 004. [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 003. [6] E. W. Justh and P. S. Krishnaprasad, Equilibria and steering laws for planar formations, Systems and Control Letters, ol. 5, pp. 5 38, 004. [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, 004, pp. 193 1936. [8] A. Filippo, Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publisher, 1988. [9] J. P. Aubin and A. Cellina, Differential Inclusions. Springer-Verlag Berlin Heidelberg New York Tokyo, 1984. [10] J. P. Aubin, Viability theory. Birkhauser, 1991. [11] H. Khalil, Nonlinear Systems. New York: Macmillan Publishing Co., 199.