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Multi-V ehicle Flocking: Scalability of Cooperati Contr ol Algorithms using airwise otentials ao-Li Chuang uan R. Huang Maria R. D’Orsogna Andrea L. Bertozzi Abstract In this paper we study cooperati contr ol algo- rithms using pairwise interactions, or the pur pose of contr ol- ling ﬂocks of unmanned ehicles. An important issue is the ole the potential plays in the stability and possible collapse of the gr oup as agent number incr eases. model set of interacting Dubins ehicles with ﬁxed tur ning angl and speed. perf orm simulations or lar ge number of agents and we sho experimental ealizations of the model on testbed with small number of ehicles. In both cases, critical thr esholds exist between coher ent, stable, and scalable ﬂocking and dispersed or collapsing motion of the gr oup. ODU ON A. Motivation Social aggre ation is remarkable aspect of animal beha vior Lar ge numbers of indi vidual agents interacting with each other are able to self-or anize into comple yet coordinated patterns such as insect sw arms, ﬁsh schools and bird ﬂocks [1]. These systems ha recently become of great interest for the athematical [2], ph ysical [3], [4] and biological sciences [5] with promising applications for he de elopment and control of autonomous, multi-v ehicular ensembles [6], [7]. One of the main goals of this nascent ﬁeld of research is to progra interactions among indi viduals so that desired collecti beha viors may arise. The emer gence of spatial patterns ho we er can be dram atically af fected by en small parameter changes in interactions among indi viduals, in constituent number or speed [8]. In this paper we formulate criteria, alid for ener al pairwise interactions, to ensure local group cohesion of ﬁrst order model. When inter actions are controlled by Morse potential, we in estig ate stability and scalability through numerical simulations and practical testbed applications, demonstrating the xistence of thresholds and cutof fs for dif ferent re gimes of aggre ation. B. Related work and outline Sw arming ehicular systems are ofte modeled as tw o- dimensional point particles in which members may interact with one another hrough attracti e-repulsi pairwise in- teractions. Speciﬁc potential choices lead self-propelled or kinematic particles to self-or anize into coherent patterns [4], [9], [10], [11], [12]. More recently sw arm stabilization or collapse with increasing constituent number has been Dept. of Mathematics, Uni ersity of California Los Angeles, Los Angeles, CA 90095 chuang, dorsogna, bertozzi @math.ucla.edu Dept. of Ph ysics, Duk Uni ersity Durham, NC 27708 Dept. of Electrical Engineering, Uni ersity of California Los Angel es, Los Angeles, CA 90095 yuanh@seas.ucla.edu corresponding author predicted [8]. irtual leaders [6] and structural potential functions [13], [14] can be introduced to direct and stabilize ehicles into desired formations or to oid obstacles. The rob ustness of arious algorithms in the presence of noise, communication delays and other non-idealities, ha been tested on se eral testbeds, both for single and multi-v ehicular systems [15], [16]. Acti vities such as spatial dispersion, gradient na vig ation, and cluster formation ha also been reported [17] as well as single-v ehicle path follo wing, sta- tionary obstacle oidance, and cooperati searching [18], [19]. The subject of ﬂock cohesion for ﬁrst order systems has been analyzed in detail in Refs. [9], [10], [11], where the attracti e-repulsi interac tion is speciﬁed and al ays has the unph ysical feature of being unbounded for lar ge distances. The proof that agents con er ge to ﬁnite re gion in space depends hea vily on this assumption. In the present ork, on the other hand, we present ener al theory applicable to any ﬁrst order kinematic system subject to interactions, and ﬁnd local conditions for ﬂock cohesion. apply our theory to the speciﬁc case of the Morse potential, which decays xponentially at lar ge distances and represents much more realistic description of natural and artiﬁcial sw arming agents. The theory is presented in section II where we also compare our results with kno wn properties of second order dynamic descriptions. In section III we adapt our model to group of Dubins ehicles [20], [21] with speciﬁc attracti and repul- si interactions. discuss stability and scalability of the system for certain parameter ranges, and we also in estig ate the ef fects of virtual leaders. Finally in section IV, results from numerical simulations and xperimental realizations of the model for small ehicle numbers are sho wn. II A. irst or der models consider general potential ﬂo for particle at position at distance from the origin, subject to dissipation and to pairwise interactions ij (1) Here ij denotes the distance between nts i, or simplicity in the remainder of this paper we will set The potential has an attracti and repulsi part denoted by respecti ely Then, with The center of mass =1 is stationary for 2007 IEEE International Conference on Robotics and Automation Roma, Italy, 10-14 April 2007 ThB9.1 1-4244-0602-1/07/$20.00 2007 IEEE. 2292

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an interaction potential that depends solely on the distance between agents. ithout loss of generality we let free agent is deﬁned as one whose dist ance to all other members of the sw arm is greater than the repulsi length scale of the potential. In Re f. [9], free agents interact through an ad-hoc potential that, at lar ge enough distances, is essentially spring. This unrealistic attraction increases with distance, so that two free agents inﬁnitely ar from each other are also inﬁnitely attracted to each other It is not surprising then, that such free agents con er ge to an absorbing ball around the center of mass with ﬁnite con- er gence time: the crucial point in the proofs is the strongly attracti e, yet unph ysical, nature of the interactions at inﬁnite distances. In particular agents are sho wn to collapse inside the absorbing re gion, re ardless of constituent number and initial condition. The radius of the abs orbing ball is independent of so that the density of the ﬁnal resting state di er ges as In this paper we ﬁnd the conditions on ener al inter action for which this collapsing beha vior can be pro en locally that is if all agents start inside ﬁx ed set. will later particularize this theorem to the case of Morse potential, that has the much more realistic feature of decaying to zero as the interparticle distance becomes lar ge. mak the follo wing deﬁnition: Deﬁnition Dif fused state ﬂock is in dif fused state if ij where is the repulsi range such that for all Note that in order to be in dif fused state, the potential must yield only attraction outside of certain radius. The follo wing Lemma sho ws that, re ardless of the speciﬁc form of the potential, dif fused state al ays shrinks. Lemma eak maximum principle Deﬁne the ﬂock radius as sup or ﬂock in the dif fused state, Pr oof Let and deﬁne ij ij /r ij From Eq. then: ij ij (2) ij ij (3) since and in the dif fused state. Thus and are decreasing functions and corollary to the abo Lemma is that the sw arm size decreases en if only the outermost agents are in dif fused state. This is due to the act that the proof only uses an estimate for the arthest agents of the sw arm. no pro local stability limit for general interactions and ﬁnd conditions for particles initially constrained to local re gion of radius to olv into more compact ball of radius The proof uses yapuno function discussed in [9], [10]. Theor em Existence of bound states. Consider particles located at with If ﬁnite constant alue xists such that max then asymptotically with Pr oof choose the yapuno function Its time deri ati obe ys the follo wing ij (4) ij ij (5) 1) (6) where max In going from Eqn. to Eqn. we ha added and subtracted in the sum and where is an arbitrary constant. also used the act that ij Also note that ij since by assumption Asymptotically then: (7) and we require for this bound be more stringent than the initial radius Cor ollary 1:Existence of collapsed states. If Theorem holds for all then as the system will collapse with all particles con er ging at Pr oof: This follo ws from the act that for /K the yapuno function The imit is thus reached in time: ma max ln (0) (8) where (0) is the yapuno function at time After ma is reached, Theorem can be applied ag ain, and the iteration process can be repeated until the limit is reached. Theorem 1, applied to the parabolic potential of Ref. [9] is the global con er gence theorem there sho wn Our control algorithm adopts generalized Morse poten- tial that decays at inﬁnite distances, as ould be xpected for systems of ehicles with limited communication range: ij ij /` ij /` (9) Here, represent the strengt of the attracti and re- pulsi potentials, and their length scales, respecti ely Deﬁne /` /C suf ﬁcient condition for Theorem is (10) which can be satisﬁed only so that can be chosen as ln `/C Thus, with the proper alues of `, ThB9.1 2293

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C = C / C l = l / l C l = 1 collapsed collapsed dispersive cohesive cohesive Fig. 1. Phase diagram for particles interacting according to the ﬁrst order model of Eqn. 1. The re gion with is guaranteed to gi rise to collapsed structures for an choice of if agents are initially suf ﬁci ently close to each other In the dispersi mode particles will form an unbound system, re ardless of initial conditions. In the re gion min cohesi structures form. Their shape and scaling with depends on further details of the potential and on the dimensionality of the system. and he proper nitial condition fre agents subject to the non diver ging Morse interactions are guaranteed to coll apse to ball of radius Also note that Corollary holds here, since the latter condition holds for all The abo condition is suf ﬁcient ut not necessary one, and other combinations of `, could gi rise to acceptable alues without resulting in state where all agents collapse to point. or other speciﬁc choices of the potential parameters, numeric estimates can determine whether alues xist that satisfy Theorem 1. can also pro that the system is dispersi for the same Morse potential in the re gion where Lemma Disper sion under the Mor se potential or of the Morse potential bounded state at where will olv into an unbounded one as Pr oof Of the bounded particles, let be the one furthest ay from the origin. let for all so hat at time Not that to simplify the analysis we let only one particle be on the boundary the results do not change by considering multiple particles at for Consider the distance between the -th particle and the center of mass of the remaining particles. This distance is 1)) 1) since the stationary center of mass is assumed to be ﬁx ed at the origin. The distance of the -th particle from the center of mass and from the center of mass of the remaining particles therefore dif fer only by multiplicati actor The olution of obe ys the follo wing: ik ik ik (11) as long as ik ik and where we ha used the act that This result indicates that C = C / C l = l / l C l = 1 l = 1 catastrophic H−stable catastrophic C = 1 catastrophic H−stable H−stable Fig. 2. Phase diagram for the second order model of Eqn. 12. Note that the system is self-propelling and asympt otically each particle will ha e a ﬁnite elocity gi ving rise to circular or ﬂocking structures both in the H-stable and catastrophic re gimes. H-stability permits to further characterize the details of the cohesi re gime: for cohesi structures originate that shrink in size with while patterns that are xtensi with are formed in the re gion and are increasing functions in time. The -th particle will thus mo e a ay from the center of mass of the other particles, and increase its distance from the origin as long as for all other particles. If the inequality ceases to hold, at time for some the outer bound of the system will be since has increased. Let at we can then apply Lemma ag ain with and with the initial condition The system size will thus increase in an unlimited ashion. can adapt this result to the remai nd of the phase space where min through the follo wing: Cor ollary Cohesion under the Mor se potential. or min lo wer and an upper bound on the asymptotic ij for all xist so that the system is cohesi e. Pr oof: From Eqn. 11 it is vident that for min the distance is an increasing function of time whene er ik ln for all This implies that an bound state of radius 2( ln will increase its size and will not be compacted further On the other hand, when ik ln for all the distance between the -th particle and the center of mass of the other particles will decreas and particles will reside into more compact ball. The ystem is thus of cohesi ype. In this param eter re gion min the potential consists of short range repulsion and of long range attraction. Based on the abo observ ations, we may con- clude that the sw arm will be formed by particles separated by distances ik such that the repulsion felt by the ‘closer particles is balanced by the attraction xperienced by the ‘f arther ones. The ﬁnal size of the cohesi sw arm will depend on the total number of constituents. In the limit of lar ge cohesi sw arm may gi rise either to an xtended state, with ﬁnite density or to collapsed one where the density is di er ging. As we shall see in the analysis for the second order model, other features of the total potential, and the dimensionality of the system play major role in determining such asymptotic sw arm conﬁgurations. or ThB9.1 2294

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instance, consider the follo wing qualitati ar guments for the yapuno function i,tot ij In the limit of lar ge and ﬁnite sw arm area the sums can be replaced with tw o-dimensional inte grals so that i,tot or this yapuno function will increase with so that the collapsed limit in ﬁnite re gion of space cannot be asymptotically reached. The system thus stays cohesi ut does not collapse, possibility that may occur for where the pre vious ar guments do not hold. Similar considerations can be found in Ref. [2]. The phase diagram for the parameters is sho wn in Fig. 1. B. Second or der models and H-stability In Ref. [8] we ha studied the same Morse potential in dynamic, second order system. It wil be useful to compare the results of the ﬁrst and second order approaches to further characterize the Morse interaction. Our second order model reads: ij (12) Here, self propulsion and drag of an indi vidual are introduced through and the potential is as abo e. The system is conserv ati if and is chosen so that there xists special alue for which As pumping and dissipation occur through it is reasonable to xpect that the steady state conﬁgurations of Eqn. 12 are minimizers of the ener gy ij and zeroes of Dra wing on analogies with statistical ensembles [22], in Ref. [8] we sho that an im po tant indicator of the xpected morphology is the H-stability of the interaction potential system is said to be H-stable if the ener gy per particle is bounded from belo as the number of particles goes to inﬁnity Mathematically system is H-stable if constant xists such that: lim ij (13) In the limit H-stable interactions result in particles either occup ying the entire space at their disposal in as- lik manner and with zero density or eeping interparticle distances ﬁx ed, so that the density remains constant. In the language of the purely dissipati model of Eqn. 1, H-stable interactions correspond to dispersed or cohesi agent beha vior In the lat ter case, ﬁnite nearest-neighbor distance emer ges as Non H-stable potentials, on the other hand, are called ‘catastrophic as the typically result in systems that collapse to localized re gion in space with di er ging density in the limit. or ﬁnite catastrophic potentials gi rise to cohesi motions of agent groups. As nearest-neighbor distances become anishingly small, and the group entually collapses. The potentials analyzed in Ref. [9] are all xamples of catas- trophic potentials for the dynamic system. compare the results for the ﬁrst order model of Eqn. to the the phase diagram arising from the second order model of Eqn. 12 in Fig. 2. The re gion with is classiﬁed as catastrophic in Ref. [8], with particles con er ging to ards their center of mass and becoming denser as This is consistent wit the results pro en here that particles initially in ball of radius get ‘squeezed into tighter one. On the other hand, the re gion with is classiﬁed as stable in Ref. [8], with no possible squeezing ef fects in the long time limit. This can be understood as follo ws. In the re gion the pairwise potential has positi e, local minimum for ij and barrier at ij ma ln `/C before decaying to zero as ij The ﬁrst order system (1) is purely dissipati and there are no ﬂuctuations in the total ener gy which can only decrease in time. or second order systems of the type described in Eqn. 12 ho we er en if the local ener gy minimum is reached, with all particles simultaneously at ij ﬂuctuations due to xchange with the en vironment as imposed by can entually dri the system ay to ards the dispersed, global ener gy minimum at ij III D ADA ON The model described in Eqns. and 12 cannot be directly applied to the speciﬁc platform of autonomous ehicles we are equi pped with, due to me chanical constraints that limit speed and turning radii capabilities. The real ehicles we use are described in Ref. [16] and consist of Dubins micro-cars wit ﬁx ed speed and ﬁx ed left and right turning radii. The ﬁrst constraint implies our dynamical system must be described as ﬁrst order The only independent ariable denoting agent is its heading angle with respect to ﬁx ed orientation we deﬁne as The Dubins ehi cles interact with each other by means of the Morse potential of Eqn. with ari able parameters Due to the ﬁx ed turning radii, the interactions cannot directly control and an appropriate control algorithm must be de vised. or each ehicle then, we measure the angle between ehicle heading and the total force it xperiences, as gi en by the right hand ide of Eqn. and as sho wn in Fig. 3. ehicle then changes direction only if where is an angular threshold The equations of motion are as follo ws: cos sin (14) if left turn if right turn otherwise (15) Here, is the speed of the ehicle, and are the left and right turning radii, respecti ely is the de viation radius. In the ideal case and so that ehicle direction is unaf fected for Because of alignment asymmetries, in gene ral and is lar ge ut ﬁnite number ehicular motion proceeds along ThB9.1 2295

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vehicle Testbed vehicle Fig. 3. Deﬁnition of ariables for ehicle The heading is denoted by the angle between its direction of motion and the axis of the testbed. is the interaction force it xperiences due to all other ehicles. This direction deﬁnes an angle with the heading direction. ehicle is at distance ij from ehicle and the angles and here sho wn are used in the collision oidance scheme described in the te xt. The origin of the reference coordinate system is ﬁx ed at the left-lo wer corner of the testbed. All ehicular angles, are deﬁned in the direction speciﬁed by the heading parameter until the turning commands are gi en. crucial point is that the interaction potential in Eqn. is soft-core and does not pre ent ehicles from colliding. In act, en hard-core potentials cannot oid collisions due to communication delays, errors in position information, and the ﬁnite turning radii of the ehicles. The repulsi range may be increa sed to initiate turning at lar ger inter -v ehicle distances. This ho we er ould signiﬁcantly af fect pattern formation and the emer gence of cooperati aggre ates ould be unlik ely Instead, we add an additional collision oidance algorithm to address short range interactions. use ‘w ait and go scheme for ehicles closer than cutof distance or ehicles i, at distance ij such that ij we deﬁne the angles between their main axis and ij as sho wn in Fig. 3. If ehicle will pause while ehicle eers ay until ij The cutof distance in the control algorithm acts as an ef fecti hard- core potential. If an one of the ehicles (in our simulations the one with higher labeling inde x) will pause and let the other proceed. When the ’w ait and go scheme cannot oid collision as sho wn in Fig. 4, and an alternate algorithm is in vok ed. or ehicles and we deﬁne the angle ij between ij and the se gment joining their opposite front edges measured from max as sho wn in Fig. 4. If max where is an angular threshold then the ehicle closer to the center of the testbed is eered to ards the center and the other in the opposite direction. LT In this section we study the beha vior and performance scaling of set of Dubins ehicles controlled by the ﬁrst order la ws based on the model in the pre vious section. consider both testbed implementation and numerical simulations for small and lar ge numbers of ehicles, respecti ely The computer model is alidated ag ainst the testbed in the case of fe ehicles. It is also possible to incorporate the presence of ij ij vehicle vehicle Testbed Fig. 4. Collision oidance ailure: The angles and are too small and ehicles and collide en if one of them should pause. An additional algorithm is required to steer the ehicles ay from each other and is described in the te xt. It relies on the angle ij here depicted. man virtual ehicles in practical testbed applications and study the ef fects of lar ger ehicle numbers on the actual ones. A. estbed Simulations The testbed has three orking ehicles. virtual leader mo es around an ellipse with semimajor axis approximately 15 times the ehicle length. There is some ariability in ehicle speed. address this issue, the position of the leader is check ed ag ainst the distance to the closest ehicle. If the distance becomes lar ger than certain threshold the leader will pause; otherwise, it will mo at its intrinsic speed, select our parameters as follo ws: cm, 95 cm, 10 er and 10 er g. so that 67 and 06 Note that these parameters correspond to potential in the ‘catastrophic re gime of Ref. [8]. or potential parameters in the H-stable re gime we ha not been able to realize stable conﬁgurations of ehicular aggre ation due, in part, to the constant speed of the ehicles. The leader interacts with the ehicles according to the same Morse potential used for ehicle-v ehicle interaction. When leading more than one ehicle, the leader contrib ution to the potential is increased 1.1 ti mes and 2.1 times the ehicular potential for the tw o- ehicle and the three-v ehicle xperiments, respecti ely 1) One vehicle follows leader: The parameters men- tioned abo pro vide short-range repulsion and long-range attraction resulting in an equilibrium se ration. Figure sho ws results for near the equilibrium eq calculated to be eq 20 cm. Running tests with 20 cm, 20 cm, and 20 cm, we note that leader follo wing becomes inef fecti for belo eq 2) wo vehicles follow leader: The ehicles are found to alternate between snak e-lik compe ting beha vior as sho wn in Fig. 6-top and stable gliding beha vior as sho wn in Fig. 6- middle. The stable beha vior emer ges when one ehicle trails the other and the form rather ﬂat triangle with the leader that glides around the ellipse as sho wn in Fig. 6-bottom. 3) Thr ee vehicles follow leader: The ehicles still alternate between competing and gliding beha viors as in the tw o-v ehicle case as sho wn in Fig. 7-top. When stable motion emer ges, the ehicles and the leader form stretched quadrilateral that glides around the ellipse as sho wn in Figs. 7-middle and bottom. note that fragmentation can ThB9.1 2296

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70 140 210 70 140 Y (cm) X (cm) 70 140 210 70 140 Y (cm) X (cm) 70 140 210 70 140 Y (cm) X (cm) Fig. 5. ehicular motion: These panels sho fragments of the ehicle trajectory when it tries to follo virtual leader along an elliptical path. The ehicle is unstable when is decreased belo eq 20 cm. op left: 20 cm; op right: 20 cm; Bottom: 20 cm. 70 140 210 70 140 Y (cm) X (cm) 70 140 210 70 140 Y (cm) X (cm) 70 140 210 70 140 Y (cm) X (cm) Fig. 6. o v ehicles try to follo virtual leader along an elliptical path. op: o v ehicles xhibit snak e-lik motion as the compete for the optimal spot behind the virtual leader; middle and bottom: The ehicles motion becomes stable when one trails the other and the form ﬂat triangle with the leader which glides along the path. 70 140 210 70 140 Y (cm) X (cm) 70 140 210 70 140 Y (cm) X (cm) 70 140 210 70 140 Y (cm) X (cm) Fig. 7. Three ehicles try to follo virtual leader along an elliptical path. op: ehicles xhibit snak e-lik motion when the le el with each other; Middle: The formation becomes stable when one trails anothe Bottom: The ehicles and the leader form stretched quadrilateral that glides along the path. sometimes occur due to the stretched formation, as the attraction between the two slo wer ehicles ov erwhelms the long-range attraction from the leader reduce such occurrences, we can enhance the leader attraction by increasing its weight. Also, both group cohesion and stabilization of the abo e e xamples can be realized by imposing rigid formations for the ehicle group as in Ref. [6]. Note, ho we er that in the absence of rigid structure, en though the ehicles shift position with respect to each other the are able to maintain coherent group as the follo the leader around the track. B. Computer Simulations Computer simulations pro vide po werful tool to study scalability and statistical issues for lar ge numbers of ehicles. Figure sho ws two distinct formations observ ed in computer simulations of 100 ehicles. Aggre ates similar to the orte sho wn in the left-hand panel of Fig. are seen for weak or non-e xistent leaders. or strong, ef fecti leaders, ehicles align and follo as sho wn in the right hand panel. or the second-order model of Eqn. 12 as speciﬁed in Ref. [8] it is sho wn that as the number of agents increase, collapse, stability or dispersion of the agents depend on the parameters ThB9.1 2297

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of the potential. It is interesting to in estig ate ho these results compare to the ﬁrst-order model of Eqns.14 and 15. In particular in Ref.[8] it is sho wn that for range of parameter alues deﬁned by and coherent beha vior is xpected. In Fig. we sho the steady state formation radius as function of ehicle number in the catastrophic re gime, where coherent structures are xpected to collapse as the number constituents increases. In the present model, the size of catastrophic ﬂock remains steady as ehicle number increases, consistent with an increasing ehicle density On the other hand, for parameter alues in the H-stable re gime, where aggre ation is xtensi in lar ge number limit, the ﬂock size xpands with increasing ehicle number Repulsion is more accentuated in the H-stable re gime: for parameters that are close to the stable-catastrophic threshold ﬂocking is still possible, ut as the parameters are chosen further and further ay into the H-stable re gime, cooperati ﬂocks no onger occur and ehicle groups loose coherence. Figure 10 sho ws that the critical be yond which the ﬂock dis- inte grates is located deeper into the H-stable re gime as the number of ehicles increases. ON ON e c ns ider well-kno wn ﬁrst order gradient ﬂo model for robot interactions in sw arm. pro ne results on cohesion and collapse for general class of potentials. In particular we ﬁnd conditions under which the system is guaranteed to con er ge inside ball of ﬁx ed radius, pro vided it started from ball of pre-deﬁned lar ger radius. These radii are independent of number of agents and result in state in which sw arm density goes to inﬁnity as ehicle number increases. Such scaling results are ery important in designing lar ge agent sw arming algorithms. adapt the model to system of Dubins ehicles and consider both testbed and numerical simulations for the sw arm. include virtual leader which allo ws for continued motion of the sw arm in conﬁned geometry or small numbers of agents, the estbed eriﬁes some simple acts about stability of the algorithm under certain parameter of the virtual leader potential. or lar ge numbers of agents we sho in computer simulations ho the size of the sw arm scales as the agent number increases. In our model, as the number of agents gro ws, the sw arm is able to maintain its cohesion using potentials with parameters that ould lead to instability at smaller numbers. KNO LE DG This research as supported by ONR grant N000140610059 and AR grants W911NF-05-1-0112 and 50 363-MA-MUR. [1] S. Camazine, J. L. Deneubour g, N. R. Franks, J. Sne yd, G. Theraulaz, and E. Bonabeau, Self or ganization in biolo gical systems Princeton Uni ersity Press, Princeton, NJ, 2003. [2] A. Mogilner L. Edelstein-K eshet, L. Bent and A. Spiros, “Mutual in- teractions, potentials, and indi vidual distance in social aggre ation”, Math. Biol. ol. 47, pp. 353-389, 2003. Fig. 8. ehicular formations in the presence of leader: The formation to the left occurs when the ehicles all out of the leader path and self- aggre ate into orte x-lik formation. The formation to the right occurs when the ehicles successfully follo the leader 10 100 Number of vehicles 20 200 Flock radius (cm) Fig. 9. Scaling in the H-stable and catastrophic re gimes. The potential parameters are set at 95 cm, 10 er and 10 er g. ith these parameter choices, H-stability is guaranteed for 73 cm. In the op curv 76 cm, in the middle one, 69 cm, just belo the transition threshold. The bottom curv e, for which 35 cm, alls deeply into the catastrophic re gime. Straight lines are po wer la ﬁts with po wers 10 10 for the top and middle et. ithin ﬁtting errors, the catastrophic curv deﬁnes constant ﬂocking radius. 100 200 300 Number of vehicles 70 75 80 Critical (cm) Fig. 10. Criti cal ersus ehicle number The dat points indicate the threshold be yond which the cooperati ﬂock disinte grates. are the same as in Fig. [3] icsek, A. Czirok, E.B. Jacob, I. Cohen, and O. Schochet, “No el type of phase transitions in system of self-dri en particles”, Phys. Re Lett. ol. 75, pp. 1226-1229, 1995. [4] H. Le vine, J. Rappel, and I. Cohen “Self-or anization in systems of self-propelled particles”, Phys. Re ol. 63, pp. 017101, 2000. [5] I. D. Couzin, J. Krause, R. James, G. D. Ruxton, and N. R. Franks, “Collecti memory and spatial sorting in animal groups”, Theor Biol. ol. 218, pp. 1-11, 2002. [6] N. E. Leonard and E. Fiorelli, “V irtual leaders, artiﬁcial potentials, and coordinated control of groups”, in Pr oc. Conf Decision Contr Orlando, FL, pp. 2968-2973, 2001. [7] A. Jadbabaie, J. Lin, and A. S. Morse, “Coordination of groups of mobile autonomous agents using nearest neighbor rules”, IEEE ans. utom. Contr ol. 48, pp. 988-1001, 2003. [8] M. R. D’Orsogna, L. Chuang, A. L. Bertozzi and L. Chayes, “Self- ThB9.1 2298

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propelled particles with soft -core interactions: patterns, stability and collapse”, Phys. Re Lett. ol. 96, 104302, 2006. [9] Gazi and K. assino, “Stability analysis of sw arms”, in IEEE ans. utom. Contr ol. 48, pp. 692-697, 2003. [10] Gazi and K. assino, class of attractions/repulsion functions for stable sw arm aggre ations”, in Pr oc. Conf Decision Contr Las as, NV pp. 2842-2847, 2002. [11] Gazi and K. assino, “Stability analysis of social foraging sw arms: combined ef fects of attractant/repellent proﬁles”, in Pr of Conf Deci- sion Contr Las as, NV pp. 2848-2853, 2002. [12] N. Shimo yama, K. Sug ara, Mizuguchi, Hayaka and M. Sano, “Collecti motion in system of motile elements”, Phys. Re Lett. ol. 76, 3870-3873, 1996 [13] R. Olf ati-Saber and R. M. Murray “Distrib uted cooperati con- trol of multiple ehicle formations using structural potential functions”, in IF orld Congr ess Barcelona, Spain, 2002. http://thayer .dartmouth.edu/ olf ati/papers/if ac02 ros rmm.pdf [14] R. Olf ati-Saber “Flocking for multi-agent dynamic systems: algo- rithms and theory”, IEEE ans. on utom. Contr ol. 51, 2006, to appear [15] L. Cremean, Dunbar D. Gogh, J. Kick E. Kla vins, J. Meltzer R. M. Murray “The Caltech Multi-V ehicle ireless estbed”, in Pr oc. Conf Decision Contr Las as, NV pp.86-88, 2002; Z. Jin, S. aydo, E. B. ildanger M. Lammers, H. Scholze, ole D. Held, and R. M. Murray “MVWT -II: The second generation Caltech Multi- ehicle ireless estbed”, in Pr oc. Amer Contr Conf Boston, MA, pp. 5321-5326, 2004. [16] C. H. Hsieh, L. Chuang, Huang, K. K. Leung, A. L. Bertozzi, and E. Frazzoli, An economical micro-car testbed for alidation of coop- erati control strate gies”, in Pr oc. Amer Contr Conf Minneapolis, MN, pp. 1446-1451, 2006. [17] J. McLurkin, MIT Computer Science and Artiﬁcial Intellig ence Lab- or atory http://people.csail.mit.edu/jamesm/ [18] D. J. Lee and M. Spong, ”Stable Flocking of Inertial Agents on Balanced Communication Graphs”, in Amer Contr Conf Minneapo- lis, MN, pp. 2136-2141, 2006. [19] B. Q. Nguyen, L. Chuang, D. ung, C. Hsieh, Z. Jin, L. Shi, D. Marthaler A. L. Bertozzi, and R. M. Murray “V irtual attracti e- repulsi potentials for cooperati control of second order dynamic ehicles on the Caltech MVWT”, in Pr oc. Amer Contr Conf Port- land, OR, pp. 1084-1089, 2005. [20] L. E. Dubins, “On curv es of minimal length with constraint on erage curv ature and with prescribed initial and terminal positions and tangents”, Amer Math. ol. 79, pp. 497-516, 1957. [21] A. M. Shk el and J. Lumelsk “Classiﬁcation of the Dubins set”, Robotics and utonomous Systems ol. 34, pp. 179-202, 2001. [22] D. uelle, Statistical Mec hanics, rigor ous esults A. Benjam in Inc, Ne w Y ork, NY 1969. ThB9.1 2299

Huang Maria R DOrsogna Andrea L Bertozzi Abstract In this paper we study cooperati contr ol algo rithms using pairwise interactions or the pur pose of contr ol ling 64258ocks of unmanned ehicles An important issue is the ole the potential plays in t ID: 23309

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Multi-V ehicle Flocking: Scalability of Cooperati Contr ol Algorithms using airwise otentials ao-Li Chuang uan R. Huang Maria R. D’Orsogna Andrea L. Bertozzi Abstract In this paper we study cooperati contr ol algo- rithms using pairwise interactions, or the pur pose of contr ol- ling ﬂocks of unmanned ehicles. An important issue is the ole the potential plays in the stability and possible collapse of the gr oup as agent number incr eases. model set of interacting Dubins ehicles with ﬁxed tur ning angl and speed. perf orm simulations or lar ge number of agents and we sho experimental ealizations of the model on testbed with small number of ehicles. In both cases, critical thr esholds exist between coher ent, stable, and scalable ﬂocking and dispersed or collapsing motion of the gr oup. ODU ON A. Motivation Social aggre ation is remarkable aspect of animal beha vior Lar ge numbers of indi vidual agents interacting with each other are able to self-or anize into comple yet coordinated patterns such as insect sw arms, ﬁsh schools and bird ﬂocks [1]. These systems ha recently become of great interest for the athematical [2], ph ysical [3], [4] and biological sciences [5] with promising applications for he de elopment and control of autonomous, multi-v ehicular ensembles [6], [7]. One of the main goals of this nascent ﬁeld of research is to progra interactions among indi viduals so that desired collecti beha viors may arise. The emer gence of spatial patterns ho we er can be dram atically af fected by en small parameter changes in interactions among indi viduals, in constituent number or speed [8]. In this paper we formulate criteria, alid for ener al pairwise interactions, to ensure local group cohesion of ﬁrst order model. When inter actions are controlled by Morse potential, we in estig ate stability and scalability through numerical simulations and practical testbed applications, demonstrating the xistence of thresholds and cutof fs for dif ferent re gimes of aggre ation. B. Related work and outline Sw arming ehicular systems are ofte modeled as tw o- dimensional point particles in which members may interact with one another hrough attracti e-repulsi pairwise in- teractions. Speciﬁc potential choices lead self-propelled or kinematic particles to self-or anize into coherent patterns [4], [9], [10], [11], [12]. More recently sw arm stabilization or collapse with increasing constituent number has been Dept. of Mathematics, Uni ersity of California Los Angeles, Los Angeles, CA 90095 chuang, dorsogna, bertozzi @math.ucla.edu Dept. of Ph ysics, Duk Uni ersity Durham, NC 27708 Dept. of Electrical Engineering, Uni ersity of California Los Angel es, Los Angeles, CA 90095 yuanh@seas.ucla.edu corresponding author predicted [8]. irtual leaders [6] and structural potential functions [13], [14] can be introduced to direct and stabilize ehicles into desired formations or to oid obstacles. The rob ustness of arious algorithms in the presence of noise, communication delays and other non-idealities, ha been tested on se eral testbeds, both for single and multi-v ehicular systems [15], [16]. Acti vities such as spatial dispersion, gradient na vig ation, and cluster formation ha also been reported [17] as well as single-v ehicle path follo wing, sta- tionary obstacle oidance, and cooperati searching [18], [19]. The subject of ﬂock cohesion for ﬁrst order systems has been analyzed in detail in Refs. [9], [10], [11], where the attracti e-repulsi interac tion is speciﬁed and al ays has the unph ysical feature of being unbounded for lar ge distances. The proof that agents con er ge to ﬁnite re gion in space depends hea vily on this assumption. In the present ork, on the other hand, we present ener al theory applicable to any ﬁrst order kinematic system subject to interactions, and ﬁnd local conditions for ﬂock cohesion. apply our theory to the speciﬁc case of the Morse potential, which decays xponentially at lar ge distances and represents much more realistic description of natural and artiﬁcial sw arming agents. The theory is presented in section II where we also compare our results with kno wn properties of second order dynamic descriptions. In section III we adapt our model to group of Dubins ehicles [20], [21] with speciﬁc attracti and repul- si interactions. discuss stability and scalability of the system for certain parameter ranges, and we also in estig ate the ef fects of virtual leaders. Finally in section IV, results from numerical simulations and xperimental realizations of the model for small ehicle numbers are sho wn. II A. irst or der models consider general potential ﬂo for particle at position at distance from the origin, subject to dissipation and to pairwise interactions ij (1) Here ij denotes the distance between nts i, or simplicity in the remainder of this paper we will set The potential has an attracti and repulsi part denoted by respecti ely Then, with The center of mass =1 is stationary for 2007 IEEE International Conference on Robotics and Automation Roma, Italy, 10-14 April 2007 ThB9.1 1-4244-0602-1/07/$20.00 2007 IEEE. 2292

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an interaction potential that depends solely on the distance between agents. ithout loss of generality we let free agent is deﬁned as one whose dist ance to all other members of the sw arm is greater than the repulsi length scale of the potential. In Re f. [9], free agents interact through an ad-hoc potential that, at lar ge enough distances, is essentially spring. This unrealistic attraction increases with distance, so that two free agents inﬁnitely ar from each other are also inﬁnitely attracted to each other It is not surprising then, that such free agents con er ge to an absorbing ball around the center of mass with ﬁnite con- er gence time: the crucial point in the proofs is the strongly attracti e, yet unph ysical, nature of the interactions at inﬁnite distances. In particular agents are sho wn to collapse inside the absorbing re gion, re ardless of constituent number and initial condition. The radius of the abs orbing ball is independent of so that the density of the ﬁnal resting state di er ges as In this paper we ﬁnd the conditions on ener al inter action for which this collapsing beha vior can be pro en locally that is if all agents start inside ﬁx ed set. will later particularize this theorem to the case of Morse potential, that has the much more realistic feature of decaying to zero as the interparticle distance becomes lar ge. mak the follo wing deﬁnition: Deﬁnition Dif fused state ﬂock is in dif fused state if ij where is the repulsi range such that for all Note that in order to be in dif fused state, the potential must yield only attraction outside of certain radius. The follo wing Lemma sho ws that, re ardless of the speciﬁc form of the potential, dif fused state al ays shrinks. Lemma eak maximum principle Deﬁne the ﬂock radius as sup or ﬂock in the dif fused state, Pr oof Let and deﬁne ij ij /r ij From Eq. then: ij ij (2) ij ij (3) since and in the dif fused state. Thus and are decreasing functions and corollary to the abo Lemma is that the sw arm size decreases en if only the outermost agents are in dif fused state. This is due to the act that the proof only uses an estimate for the arthest agents of the sw arm. no pro local stability limit for general interactions and ﬁnd conditions for particles initially constrained to local re gion of radius to olv into more compact ball of radius The proof uses yapuno function discussed in [9], [10]. Theor em Existence of bound states. Consider particles located at with If ﬁnite constant alue xists such that max then asymptotically with Pr oof choose the yapuno function Its time deri ati obe ys the follo wing ij (4) ij ij (5) 1) (6) where max In going from Eqn. to Eqn. we ha added and subtracted in the sum and where is an arbitrary constant. also used the act that ij Also note that ij since by assumption Asymptotically then: (7) and we require for this bound be more stringent than the initial radius Cor ollary 1:Existence of collapsed states. If Theorem holds for all then as the system will collapse with all particles con er ging at Pr oof: This follo ws from the act that for /K the yapuno function The imit is thus reached in time: ma max ln (0) (8) where (0) is the yapuno function at time After ma is reached, Theorem can be applied ag ain, and the iteration process can be repeated until the limit is reached. Theorem 1, applied to the parabolic potential of Ref. [9] is the global con er gence theorem there sho wn Our control algorithm adopts generalized Morse poten- tial that decays at inﬁnite distances, as ould be xpected for systems of ehicles with limited communication range: ij ij /` ij /` (9) Here, represent the strengt of the attracti and re- pulsi potentials, and their length scales, respecti ely Deﬁne /` /C suf ﬁcient condition for Theorem is (10) which can be satisﬁed only so that can be chosen as ln `/C Thus, with the proper alues of `, ThB9.1 2293

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C = C / C l = l / l C l = 1 collapsed collapsed dispersive cohesive cohesive Fig. 1. Phase diagram for particles interacting according to the ﬁrst order model of Eqn. 1. The re gion with is guaranteed to gi rise to collapsed structures for an choice of if agents are initially suf ﬁci ently close to each other In the dispersi mode particles will form an unbound system, re ardless of initial conditions. In the re gion min cohesi structures form. Their shape and scaling with depends on further details of the potential and on the dimensionality of the system. and he proper nitial condition fre agents subject to the non diver ging Morse interactions are guaranteed to coll apse to ball of radius Also note that Corollary holds here, since the latter condition holds for all The abo condition is suf ﬁcient ut not necessary one, and other combinations of `, could gi rise to acceptable alues without resulting in state where all agents collapse to point. or other speciﬁc choices of the potential parameters, numeric estimates can determine whether alues xist that satisfy Theorem 1. can also pro that the system is dispersi for the same Morse potential in the re gion where Lemma Disper sion under the Mor se potential or of the Morse potential bounded state at where will olv into an unbounded one as Pr oof Of the bounded particles, let be the one furthest ay from the origin. let for all so hat at time Not that to simplify the analysis we let only one particle be on the boundary the results do not change by considering multiple particles at for Consider the distance between the -th particle and the center of mass of the remaining particles. This distance is 1)) 1) since the stationary center of mass is assumed to be ﬁx ed at the origin. The distance of the -th particle from the center of mass and from the center of mass of the remaining particles therefore dif fer only by multiplicati actor The olution of obe ys the follo wing: ik ik ik (11) as long as ik ik and where we ha used the act that This result indicates that C = C / C l = l / l C l = 1 l = 1 catastrophic H−stable catastrophic C = 1 catastrophic H−stable H−stable Fig. 2. Phase diagram for the second order model of Eqn. 12. Note that the system is self-propelling and asympt otically each particle will ha e a ﬁnite elocity gi ving rise to circular or ﬂocking structures both in the H-stable and catastrophic re gimes. H-stability permits to further characterize the details of the cohesi re gime: for cohesi structures originate that shrink in size with while patterns that are xtensi with are formed in the re gion and are increasing functions in time. The -th particle will thus mo e a ay from the center of mass of the other particles, and increase its distance from the origin as long as for all other particles. If the inequality ceases to hold, at time for some the outer bound of the system will be since has increased. Let at we can then apply Lemma ag ain with and with the initial condition The system size will thus increase in an unlimited ashion. can adapt this result to the remai nd of the phase space where min through the follo wing: Cor ollary Cohesion under the Mor se potential. or min lo wer and an upper bound on the asymptotic ij for all xist so that the system is cohesi e. Pr oof: From Eqn. 11 it is vident that for min the distance is an increasing function of time whene er ik ln for all This implies that an bound state of radius 2( ln will increase its size and will not be compacted further On the other hand, when ik ln for all the distance between the -th particle and the center of mass of the other particles will decreas and particles will reside into more compact ball. The ystem is thus of cohesi ype. In this param eter re gion min the potential consists of short range repulsion and of long range attraction. Based on the abo observ ations, we may con- clude that the sw arm will be formed by particles separated by distances ik such that the repulsion felt by the ‘closer particles is balanced by the attraction xperienced by the ‘f arther ones. The ﬁnal size of the cohesi sw arm will depend on the total number of constituents. In the limit of lar ge cohesi sw arm may gi rise either to an xtended state, with ﬁnite density or to collapsed one where the density is di er ging. As we shall see in the analysis for the second order model, other features of the total potential, and the dimensionality of the system play major role in determining such asymptotic sw arm conﬁgurations. or ThB9.1 2294

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instance, consider the follo wing qualitati ar guments for the yapuno function i,tot ij In the limit of lar ge and ﬁnite sw arm area the sums can be replaced with tw o-dimensional inte grals so that i,tot or this yapuno function will increase with so that the collapsed limit in ﬁnite re gion of space cannot be asymptotically reached. The system thus stays cohesi ut does not collapse, possibility that may occur for where the pre vious ar guments do not hold. Similar considerations can be found in Ref. [2]. The phase diagram for the parameters is sho wn in Fig. 1. B. Second or der models and H-stability In Ref. [8] we ha studied the same Morse potential in dynamic, second order system. It wil be useful to compare the results of the ﬁrst and second order approaches to further characterize the Morse interaction. Our second order model reads: ij (12) Here, self propulsion and drag of an indi vidual are introduced through and the potential is as abo e. The system is conserv ati if and is chosen so that there xists special alue for which As pumping and dissipation occur through it is reasonable to xpect that the steady state conﬁgurations of Eqn. 12 are minimizers of the ener gy ij and zeroes of Dra wing on analogies with statistical ensembles [22], in Ref. [8] we sho that an im po tant indicator of the xpected morphology is the H-stability of the interaction potential system is said to be H-stable if the ener gy per particle is bounded from belo as the number of particles goes to inﬁnity Mathematically system is H-stable if constant xists such that: lim ij (13) In the limit H-stable interactions result in particles either occup ying the entire space at their disposal in as- lik manner and with zero density or eeping interparticle distances ﬁx ed, so that the density remains constant. In the language of the purely dissipati model of Eqn. 1, H-stable interactions correspond to dispersed or cohesi agent beha vior In the lat ter case, ﬁnite nearest-neighbor distance emer ges as Non H-stable potentials, on the other hand, are called ‘catastrophic as the typically result in systems that collapse to localized re gion in space with di er ging density in the limit. or ﬁnite catastrophic potentials gi rise to cohesi motions of agent groups. As nearest-neighbor distances become anishingly small, and the group entually collapses. The potentials analyzed in Ref. [9] are all xamples of catas- trophic potentials for the dynamic system. compare the results for the ﬁrst order model of Eqn. to the the phase diagram arising from the second order model of Eqn. 12 in Fig. 2. The re gion with is classiﬁed as catastrophic in Ref. [8], with particles con er ging to ards their center of mass and becoming denser as This is consistent wit the results pro en here that particles initially in ball of radius get ‘squeezed into tighter one. On the other hand, the re gion with is classiﬁed as stable in Ref. [8], with no possible squeezing ef fects in the long time limit. This can be understood as follo ws. In the re gion the pairwise potential has positi e, local minimum for ij and barrier at ij ma ln `/C before decaying to zero as ij The ﬁrst order system (1) is purely dissipati and there are no ﬂuctuations in the total ener gy which can only decrease in time. or second order systems of the type described in Eqn. 12 ho we er en if the local ener gy minimum is reached, with all particles simultaneously at ij ﬂuctuations due to xchange with the en vironment as imposed by can entually dri the system ay to ards the dispersed, global ener gy minimum at ij III D ADA ON The model described in Eqns. and 12 cannot be directly applied to the speciﬁc platform of autonomous ehicles we are equi pped with, due to me chanical constraints that limit speed and turning radii capabilities. The real ehicles we use are described in Ref. [16] and consist of Dubins micro-cars wit ﬁx ed speed and ﬁx ed left and right turning radii. The ﬁrst constraint implies our dynamical system must be described as ﬁrst order The only independent ariable denoting agent is its heading angle with respect to ﬁx ed orientation we deﬁne as The Dubins ehi cles interact with each other by means of the Morse potential of Eqn. with ari able parameters Due to the ﬁx ed turning radii, the interactions cannot directly control and an appropriate control algorithm must be de vised. or each ehicle then, we measure the angle between ehicle heading and the total force it xperiences, as gi en by the right hand ide of Eqn. and as sho wn in Fig. 3. ehicle then changes direction only if where is an angular threshold The equations of motion are as follo ws: cos sin (14) if left turn if right turn otherwise (15) Here, is the speed of the ehicle, and are the left and right turning radii, respecti ely is the de viation radius. In the ideal case and so that ehicle direction is unaf fected for Because of alignment asymmetries, in gene ral and is lar ge ut ﬁnite number ehicular motion proceeds along ThB9.1 2295

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vehicle Testbed vehicle Fig. 3. Deﬁnition of ariables for ehicle The heading is denoted by the angle between its direction of motion and the axis of the testbed. is the interaction force it xperiences due to all other ehicles. This direction deﬁnes an angle with the heading direction. ehicle is at distance ij from ehicle and the angles and here sho wn are used in the collision oidance scheme described in the te xt. The origin of the reference coordinate system is ﬁx ed at the left-lo wer corner of the testbed. All ehicular angles, are deﬁned in the direction speciﬁed by the heading parameter until the turning commands are gi en. crucial point is that the interaction potential in Eqn. is soft-core and does not pre ent ehicles from colliding. In act, en hard-core potentials cannot oid collisions due to communication delays, errors in position information, and the ﬁnite turning radii of the ehicles. The repulsi range may be increa sed to initiate turning at lar ger inter -v ehicle distances. This ho we er ould signiﬁcantly af fect pattern formation and the emer gence of cooperati aggre ates ould be unlik ely Instead, we add an additional collision oidance algorithm to address short range interactions. use ‘w ait and go scheme for ehicles closer than cutof distance or ehicles i, at distance ij such that ij we deﬁne the angles between their main axis and ij as sho wn in Fig. 3. If ehicle will pause while ehicle eers ay until ij The cutof distance in the control algorithm acts as an ef fecti hard- core potential. If an one of the ehicles (in our simulations the one with higher labeling inde x) will pause and let the other proceed. When the ’w ait and go scheme cannot oid collision as sho wn in Fig. 4, and an alternate algorithm is in vok ed. or ehicles and we deﬁne the angle ij between ij and the se gment joining their opposite front edges measured from max as sho wn in Fig. 4. If max where is an angular threshold then the ehicle closer to the center of the testbed is eered to ards the center and the other in the opposite direction. LT In this section we study the beha vior and performance scaling of set of Dubins ehicles controlled by the ﬁrst order la ws based on the model in the pre vious section. consider both testbed implementation and numerical simulations for small and lar ge numbers of ehicles, respecti ely The computer model is alidated ag ainst the testbed in the case of fe ehicles. It is also possible to incorporate the presence of ij ij vehicle vehicle Testbed Fig. 4. Collision oidance ailure: The angles and are too small and ehicles and collide en if one of them should pause. An additional algorithm is required to steer the ehicles ay from each other and is described in the te xt. It relies on the angle ij here depicted. man virtual ehicles in practical testbed applications and study the ef fects of lar ger ehicle numbers on the actual ones. A. estbed Simulations The testbed has three orking ehicles. virtual leader mo es around an ellipse with semimajor axis approximately 15 times the ehicle length. There is some ariability in ehicle speed. address this issue, the position of the leader is check ed ag ainst the distance to the closest ehicle. If the distance becomes lar ger than certain threshold the leader will pause; otherwise, it will mo at its intrinsic speed, select our parameters as follo ws: cm, 95 cm, 10 er and 10 er g. so that 67 and 06 Note that these parameters correspond to potential in the ‘catastrophic re gime of Ref. [8]. or potential parameters in the H-stable re gime we ha not been able to realize stable conﬁgurations of ehicular aggre ation due, in part, to the constant speed of the ehicles. The leader interacts with the ehicles according to the same Morse potential used for ehicle-v ehicle interaction. When leading more than one ehicle, the leader contrib ution to the potential is increased 1.1 ti mes and 2.1 times the ehicular potential for the tw o- ehicle and the three-v ehicle xperiments, respecti ely 1) One vehicle follows leader: The parameters men- tioned abo pro vide short-range repulsion and long-range attraction resulting in an equilibrium se ration. Figure sho ws results for near the equilibrium eq calculated to be eq 20 cm. Running tests with 20 cm, 20 cm, and 20 cm, we note that leader follo wing becomes inef fecti for belo eq 2) wo vehicles follow leader: The ehicles are found to alternate between snak e-lik compe ting beha vior as sho wn in Fig. 6-top and stable gliding beha vior as sho wn in Fig. 6- middle. The stable beha vior emer ges when one ehicle trails the other and the form rather ﬂat triangle with the leader that glides around the ellipse as sho wn in Fig. 6-bottom. 3) Thr ee vehicles follow leader: The ehicles still alternate between competing and gliding beha viors as in the tw o-v ehicle case as sho wn in Fig. 7-top. When stable motion emer ges, the ehicles and the leader form stretched quadrilateral that glides around the ellipse as sho wn in Figs. 7-middle and bottom. note that fragmentation can ThB9.1 2296

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70 140 210 70 140 Y (cm) X (cm) 70 140 210 70 140 Y (cm) X (cm) 70 140 210 70 140 Y (cm) X (cm) Fig. 5. ehicular motion: These panels sho fragments of the ehicle trajectory when it tries to follo virtual leader along an elliptical path. The ehicle is unstable when is decreased belo eq 20 cm. op left: 20 cm; op right: 20 cm; Bottom: 20 cm. 70 140 210 70 140 Y (cm) X (cm) 70 140 210 70 140 Y (cm) X (cm) 70 140 210 70 140 Y (cm) X (cm) Fig. 6. o v ehicles try to follo virtual leader along an elliptical path. op: o v ehicles xhibit snak e-lik motion as the compete for the optimal spot behind the virtual leader; middle and bottom: The ehicles motion becomes stable when one trails the other and the form ﬂat triangle with the leader which glides along the path. 70 140 210 70 140 Y (cm) X (cm) 70 140 210 70 140 Y (cm) X (cm) 70 140 210 70 140 Y (cm) X (cm) Fig. 7. Three ehicles try to follo virtual leader along an elliptical path. op: ehicles xhibit snak e-lik motion when the le el with each other; Middle: The formation becomes stable when one trails anothe Bottom: The ehicles and the leader form stretched quadrilateral that glides along the path. sometimes occur due to the stretched formation, as the attraction between the two slo wer ehicles ov erwhelms the long-range attraction from the leader reduce such occurrences, we can enhance the leader attraction by increasing its weight. Also, both group cohesion and stabilization of the abo e e xamples can be realized by imposing rigid formations for the ehicle group as in Ref. [6]. Note, ho we er that in the absence of rigid structure, en though the ehicles shift position with respect to each other the are able to maintain coherent group as the follo the leader around the track. B. Computer Simulations Computer simulations pro vide po werful tool to study scalability and statistical issues for lar ge numbers of ehicles. Figure sho ws two distinct formations observ ed in computer simulations of 100 ehicles. Aggre ates similar to the orte sho wn in the left-hand panel of Fig. are seen for weak or non-e xistent leaders. or strong, ef fecti leaders, ehicles align and follo as sho wn in the right hand panel. or the second-order model of Eqn. 12 as speciﬁed in Ref. [8] it is sho wn that as the number of agents increase, collapse, stability or dispersion of the agents depend on the parameters ThB9.1 2297

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of the potential. It is interesting to in estig ate ho these results compare to the ﬁrst-order model of Eqns.14 and 15. In particular in Ref.[8] it is sho wn that for range of parameter alues deﬁned by and coherent beha vior is xpected. In Fig. we sho the steady state formation radius as function of ehicle number in the catastrophic re gime, where coherent structures are xpected to collapse as the number constituents increases. In the present model, the size of catastrophic ﬂock remains steady as ehicle number increases, consistent with an increasing ehicle density On the other hand, for parameter alues in the H-stable re gime, where aggre ation is xtensi in lar ge number limit, the ﬂock size xpands with increasing ehicle number Repulsion is more accentuated in the H-stable re gime: for parameters that are close to the stable-catastrophic threshold ﬂocking is still possible, ut as the parameters are chosen further and further ay into the H-stable re gime, cooperati ﬂocks no onger occur and ehicle groups loose coherence. Figure 10 sho ws that the critical be yond which the ﬂock dis- inte grates is located deeper into the H-stable re gime as the number of ehicles increases. ON ON e c ns ider well-kno wn ﬁrst order gradient ﬂo model for robot interactions in sw arm. pro ne results on cohesion and collapse for general class of potentials. In particular we ﬁnd conditions under which the system is guaranteed to con er ge inside ball of ﬁx ed radius, pro vided it started from ball of pre-deﬁned lar ger radius. These radii are independent of number of agents and result in state in which sw arm density goes to inﬁnity as ehicle number increases. Such scaling results are ery important in designing lar ge agent sw arming algorithms. adapt the model to system of Dubins ehicles and consider both testbed and numerical simulations for the sw arm. include virtual leader which allo ws for continued motion of the sw arm in conﬁned geometry or small numbers of agents, the estbed eriﬁes some simple acts about stability of the algorithm under certain parameter of the virtual leader potential. or lar ge numbers of agents we sho in computer simulations ho the size of the sw arm scales as the agent number increases. In our model, as the number of agents gro ws, the sw arm is able to maintain its cohesion using potentials with parameters that ould lead to instability at smaller numbers. KNO LE DG This research as supported by ONR grant N000140610059 and AR grants W911NF-05-1-0112 and 50 363-MA-MUR. [1] S. Camazine, J. L. Deneubour g, N. R. Franks, J. Sne yd, G. Theraulaz, and E. Bonabeau, Self or ganization in biolo gical systems Princeton Uni ersity Press, Princeton, NJ, 2003. [2] A. Mogilner L. Edelstein-K eshet, L. Bent and A. Spiros, “Mutual in- teractions, potentials, and indi vidual distance in social aggre ation”, Math. Biol. ol. 47, pp. 353-389, 2003. Fig. 8. ehicular formations in the presence of leader: The formation to the left occurs when the ehicles all out of the leader path and self- aggre ate into orte x-lik formation. The formation to the right occurs when the ehicles successfully follo the leader 10 100 Number of vehicles 20 200 Flock radius (cm) Fig. 9. Scaling in the H-stable and catastrophic re gimes. The potential parameters are set at 95 cm, 10 er and 10 er g. ith these parameter choices, H-stability is guaranteed for 73 cm. In the op curv 76 cm, in the middle one, 69 cm, just belo the transition threshold. The bottom curv e, for which 35 cm, alls deeply into the catastrophic re gime. Straight lines are po wer la ﬁts with po wers 10 10 for the top and middle et. ithin ﬁtting errors, the catastrophic curv deﬁnes constant ﬂocking radius. 100 200 300 Number of vehicles 70 75 80 Critical (cm) Fig. 10. Criti cal ersus ehicle number The dat points indicate the threshold be yond which the cooperati ﬂock disinte grates. are the same as in Fig. [3] icsek, A. Czirok, E.B. Jacob, I. Cohen, and O. Schochet, “No el type of phase transitions in system of self-dri en particles”, Phys. Re Lett. ol. 75, pp. 1226-1229, 1995. [4] H. Le vine, J. Rappel, and I. Cohen “Self-or anization in systems of self-propelled particles”, Phys. Re ol. 63, pp. 017101, 2000. [5] I. D. Couzin, J. Krause, R. James, G. D. Ruxton, and N. R. Franks, “Collecti memory and spatial sorting in animal groups”, Theor Biol. ol. 218, pp. 1-11, 2002. [6] N. E. Leonard and E. Fiorelli, “V irtual leaders, artiﬁcial potentials, and coordinated control of groups”, in Pr oc. Conf Decision Contr Orlando, FL, pp. 2968-2973, 2001. [7] A. Jadbabaie, J. Lin, and A. S. Morse, “Coordination of groups of mobile autonomous agents using nearest neighbor rules”, IEEE ans. utom. Contr ol. 48, pp. 988-1001, 2003. [8] M. R. D’Orsogna, L. Chuang, A. L. Bertozzi and L. Chayes, “Self- ThB9.1 2298

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propelled particles with soft -core interactions: patterns, stability and collapse”, Phys. Re Lett. ol. 96, 104302, 2006. [9] Gazi and K. assino, “Stability analysis of sw arms”, in IEEE ans. utom. Contr ol. 48, pp. 692-697, 2003. [10] Gazi and K. assino, class of attractions/repulsion functions for stable sw arm aggre ations”, in Pr oc. Conf Decision Contr Las as, NV pp. 2842-2847, 2002. [11] Gazi and K. assino, “Stability analysis of social foraging sw arms: combined ef fects of attractant/repellent proﬁles”, in Pr of Conf Deci- sion Contr Las as, NV pp. 2848-2853, 2002. [12] N. Shimo yama, K. Sug ara, Mizuguchi, Hayaka and M. Sano, “Collecti motion in system of motile elements”, Phys. Re Lett. ol. 76, 3870-3873, 1996 [13] R. Olf ati-Saber and R. M. Murray “Distrib uted cooperati con- trol of multiple ehicle formations using structural potential functions”, in IF orld Congr ess Barcelona, Spain, 2002. http://thayer .dartmouth.edu/ olf ati/papers/if ac02 ros rmm.pdf [14] R. Olf ati-Saber “Flocking for multi-agent dynamic systems: algo- rithms and theory”, IEEE ans. on utom. Contr ol. 51, 2006, to appear [15] L. Cremean, Dunbar D. Gogh, J. Kick E. Kla vins, J. Meltzer R. M. Murray “The Caltech Multi-V ehicle ireless estbed”, in Pr oc. Conf Decision Contr Las as, NV pp.86-88, 2002; Z. Jin, S. aydo, E. B. ildanger M. Lammers, H. Scholze, ole D. Held, and R. M. Murray “MVWT -II: The second generation Caltech Multi- ehicle ireless estbed”, in Pr oc. Amer Contr Conf Boston, MA, pp. 5321-5326, 2004. [16] C. H. Hsieh, L. Chuang, Huang, K. K. Leung, A. L. Bertozzi, and E. Frazzoli, An economical micro-car testbed for alidation of coop- erati control strate gies”, in Pr oc. Amer Contr Conf Minneapolis, MN, pp. 1446-1451, 2006. [17] J. McLurkin, MIT Computer Science and Artiﬁcial Intellig ence Lab- or atory http://people.csail.mit.edu/jamesm/ [18] D. J. Lee and M. Spong, ”Stable Flocking of Inertial Agents on Balanced Communication Graphs”, in Amer Contr Conf Minneapo- lis, MN, pp. 2136-2141, 2006. [19] B. Q. Nguyen, L. Chuang, D. ung, C. Hsieh, Z. Jin, L. Shi, D. Marthaler A. L. Bertozzi, and R. M. Murray “V irtual attracti e- repulsi potentials for cooperati control of second order dynamic ehicles on the Caltech MVWT”, in Pr oc. Amer Contr Conf Port- land, OR, pp. 1084-1089, 2005. [20] L. E. Dubins, “On curv es of minimal length with constraint on erage curv ature and with prescribed initial and terminal positions and tangents”, Amer Math. ol. 79, pp. 497-516, 1957. [21] A. M. Shk el and J. Lumelsk “Classiﬁcation of the Dubins set”, Robotics and utonomous Systems ol. 34, pp. 179-202, 2001. [22] D. uelle, Statistical Mec hanics, rigor ous esults A. Benjam in Inc, Ne w Y ork, NY 1969. ThB9.1 2299

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