Scaling the Size of a Multiagent Formation via Distributed Feedback

Transcription

1 Scaling the Size of a Multiagent Formation via Distributed Feedback Samuel Coogan, Murat Arcak, Magnus Egerstedt Abstract We consider a multiagent coordination problem where the objective is to steer a group of agents into a formation that translates along a predefined trajectory velocity. Unlike previous control strategies that require a static desired formation or set of desired formations, we introduce a strategy in which one agent assigns a scale for the formation the remaining agents adjust to the new scale. Thus, the formation can dynamically adapt to changes in the environment in group objectives or respond to perceived threats. We introduce two strategies: one that requires agents to communicate estimates of the desired formation scale along edges of a communication network one that only requires relative position sensing among agents. We show that the former strategy guarantees stability for any desired connected formation. For the latter strategy, we present a geometric constraint which can be used with the small gain theorem. I. INTRODUCTION Group coordination of mobile agents using distributed feedback laws has been an active area of control theory research in recent years. The main emphasis of such coordination problems is to achieve a desired group behavior using only local feedback rules [], [], [], []. In particular, formation control as a desired group behavior has received considerable attention, such as in [5], [6], [7]. Use of artificial potential fields for formation control is explored in [8], formation control strategies with communication topology switching are investigated in [9], strategies allowing the desired formation to switch among various configurations is presented in []. In this paper, we consider the problem of formation control along a velocity trajectory when only one agent knows the desired formation size. The remaining follower agents estimate the desired formation size proceed with a cooperative control law utilizing this estimate. By allowing the size of the formation to change, the group can dynamically adapt to changes in the environment such as unforeseen obstacles along the trajectory path, adapt to changes in group objectives, or respond to threats. We present two strategies for updating the desired formation size estimates. In the first strategy, we assume a communication topology exists among the agents, allowing the agents to share formation size estimates. Rather than permitting the leader to simply dictate the desired formation scale, which requires a communication structure heavily dependent on the identity of the leader, we present an update rule utilizing the internal estimates of multiple neighbors Samuel Coogan Murat Arcak are with the Department of Electrical Engineering Computer Sciences, University of California, Berkeley {scoogan,arcak}@eecs.berkeley.edu Magnus Egerstedt is with the School of Electrical Computer Engineering, Georgia Institute of Technology magnus@ece.gatech.edu while still guaranteeing stability for any desired formation. The advantage to this approach is that the follower agents do not need to know which agent is the leader. For example, the leader could be changed during operation without communicating this change to the remaining agents. In some scenarios, we are interested in the minimum amount of information required to allow formation scaling control as it may be undesirable or impossible for the agents to actively communicate with one another. In the second strategy, we assume that only relative position information defined by a sensing topology is available to each agent. In this strategy, no interagent communication is needed. This paper is organized as follows: In Section II, we introduce the problem statement. In Section III we introduce a cooperative control strategy that requires interagent communication guarantees stability for any connected formation. In Section IV we introduce a strategy that does not require interagent communication. In this case, we use the small gain theorem to derive a sufficient geometric condition for stability. In Section V, we present simulation results, in Section VI, we summarize our results provide directions for future research. II. PROBLEM DEFINITION Consider a team of n agents each represented by the vector x i R p, i =,..., n, assume each agent is modeled with double integrator dynamics: ẍ i = f i. () We define x = [x T x T x T... x T n ] T v = ẋ to denote the stacked vector containing the velocities of all of the agents. We assume there is a position sensing topology by which agents are capable of inferring the relative position of other agents. We represent this topology with a sensing graph with n nodes representing the n agents. We assume relative position sensing is bidirectional, if agent i agent j have access to the quantity x i x j, then the ith jth nodes of the sensing graph are connected by an edge. Suppose there are m edges in the sensing graph. We arbitrarily assign a direction to each edge of the sensing graph define the n m incidence matrix D as: + if node i is the head of edge j d ij = if node i is the tail of edge j otherwise where d ij is the ijth entry of D. Since the graph is bidirectional, this choice of direction does not affect any of the following results.

2 Suppose the agents are to maintain a certain formation by maintaining the relative interagent distances defined by the sensing topology. Suppose, in addition, it is allowable for the formation size to change, one agent (for notational simplicity without loss of generality, agent n) chooses a constant target formation scale λ. Each other agent possesses a local formation scalar λ i representing its estimate of λ. We define λ := [λ λ... λ n ] T. Our goal is to achieve the following three desired group behaviors: B) Each agent reaches in the limit a common velocity vector v(t) for the group, i.e. lim ẋ i v(t) =, i =,..., n. () t B) Each agent s formation scaling factor reaches λ in the limit, i.e. lim λ i λ =, i =,..., n. () t B) The relative position of two agents connected by an edge k n z k := d ik x i () i= converges to a prescribed target value λ zk d for each k =,..., m. Note that z d = [z d T... z dt ] T must be designed such that z d R(D T I p ) where R denotes range space, is the Kronecker product operator, I p denotes the p p identity matrix. III. FORMATION SCALING WITH COMMUNICATION BETWEEN AGENTS We begin by defining a linear controller for the input f i to each agent: m f i = d ij (z j λ i zj d ) k i (ẋ i v(t)) + v(t) (5) j= where k i > is a damping coefficient. Let n denote the length n column vector of all ones define Γ := ṽ(t) := ẋ(t) n v(t) (6) m j= d jzj d... m j= d njz d j (7) which can be written compactly as Γ := diag{(d I p )z d } (I n p ) where the diag{ } operation converts a vector into a diagonal matrix with the elements of the vector along the diagonal. Then (5) can be written in matrix form as f = (D I p )z + Γλ (K I p )ṽ + n v(t) (8) where K := diag{ [ ] k... k n } f = [ ] f T... fn T T. Given that λ n = λ is constant, we now seek to design an update rule for λ,..., λ n. Suppose agents are capable of communicating a scalar quantity to other agents. One obvious approach allows the leader agent to communicate the desired formation scale to its neighbors, who in turn communicate the desired scale to their neighbors, etc. This approach requires each agent to know a priori which neighboring agent forms a path to the leader, may not be robust to dropped communication links, does not adapt well to a reassignment of the leader since a different communication tree is required for each possible leader agent. We now present an alternative approach in which a communication structure update rule are designed to address these shortcomings. Suppose each agent can share its formation scaling factor with neighboring agents as defined by a communication graph with n nodes labeled ν,..., ν n where an edge points from agent i to agent j if i has access to j s scaling factor, this edge is denoted by (ν i, ν j ). We will assume there exists a directed path from any node ν i to the leader for all i n (in the undirected case, this condition is equivalent to connectedness). We denote the communication graph by G = (V, E) where V = {ν,..., ν n } is the vertex set of the graph E V V is the edge set of the graph. Let Γ i = (D i I p )z d be the p block of Γ corresponding to agent i s control input where D i indicates the ith row of the incidence matrix D, define tunable parameters α ij for i, j =,..., n such that α ij > iff (ν i, ν j ) E. We propose the following control strategy: λ i = α ij (λ i λ j ) βp i Γ T i (v i v(t)) (9) j N c i for i =... n where β is a tunable, scalar parameter such that β, N c i = {j : (ν i, ν j ) E}, p i is a positive scalar designed below to ensure stability. Note that λn =. This control strategy is intuitively appealing because it includes an agreement expression among the scaling parameters an additional term dependent on the velocity error. Define z := z λ z d () λ := (λ λ n ). () Let λ λ be λ λ with the last element removed, i.e. λ = [λ λ... λ n ] T () λ = (λ λ n ). () Then objectives B B amount to asymptotic stability of the origin for the closed loop system with state X = [ṽ T z T λt ] T. () Let L be an n n matrix defined by { n k=,k i l ij = α ik, i = j α ij, i j. (5)

3 Since agent n does not update its scaling factor via feedback, we set the last row of L to. L is called the graph Laplacian of the communication graph, when we refer to the graph Laplacian of a given graph with edge weights, we imply the above construction. Define L to be L with the last row removed, define L to be L with the last row the last column removed. Also, let Γ be equal to Γ with the last column removed, let P = diag{ [ p... p n ] }. Then (9) can be written in matrix form as λ = λ = Lλ βp Γ T ṽ (6) = L λ + Lλ n βp Γ T ṽ (7) = L λ βp Γ T ṽ, (8) where we use the fact that L n = n hence L n = n where n denotes the length n column vector of all zeros. It is apparent from (8) that f = (D I p )z + Γλ (K I p )ṽ + n v(t) (9) This gives = (D I p ) z λ (D I p )z d + Γ λ + λ Γ n (K I p )ṽ + n v(t). () ṽ = (D I p ) z + Γ λ (K I p )ṽ. () Because λ n =, () can be written as ṽ = (D I p ) z + Γ λ (K I p )ṽ. () Let n m denote the n m matrix all zeros. When the dimensions are clear, the subscript is omitted. The dynamics of (8) (8) can then be formulated as a homogenous linear system K I p D I p Γ Ẋ = D T I p X () βp Γ T L evolving on the following invariant subspace of R (n+m)p+n : S X = {(ṽ, z, λ) : ṽ R np, z R(D T I p ), λ R n }. () To facilitate the proof of the stability of (), we now introduce a generalization of Lemma.6 in [] to the case of a directed communication topology. Lemma. All eigenvalues of L are in the open right half plane. Proof: Let B(a, R) be the closed disk of radius R centered at a. The Gershgorin circle theorem states that all eigenvalues of L are located in the union of the following n disks: ( B i l ii, ) n j=,j i l ij, i =,..., n. (5) But l ii n j=,j i l ij, so the union of the n disks lie in the closed right half plane may only intersect the jωaxis at the origin. It remains to show that L does not have a zero eigenvalue. As a proof by contradiction, assume L has a nontrivial nullspace such that Lw = for some nontrivial w. Because the last row of L is T n, we have [ w T ] T N (L) where N denotes null space. By our assumption that the communication graph contains a directed spanning tree, the rank of L is n []. Therefore, {} forms a basis for the nullspace of L, forming a contradiction. We require the p i s in (9) to be chosen such that L T P + P L > (6) where P is a diagonal matrix with the p i s along the diagonal. A diagonal solution P > to (6) exists because L has nonnegative diagonal nonpositive off-diagonal entries eigenvalues in the open right half plane ([], Chapter 6, Theorem.). In many practical cases, p i can be taken to be for all i. In particular, if the subgraph induced by removing the leader agent its edges is balanced (i.e., n j=,j i α ji = n j=,j i α ij), then L T +L is positive definite P can be taken to be the identity as shown in the following corollary. This encompasses any case in which the communication graph is bidirectional. Corollary. Let G l be the subgraph of the communication graph induced by removing the leader agent. If G l is balanced, then L T + L >. Proof: Let E be the set of edges connected to node n in G with the same edge weighting but with the head tail of these edges reversed. Define G := (V, E E), i.e. G is the communication graph with additional edges pointing towards the leader node. By construction, G is balanced. Let L be the graph Laplacian corresponding to G. Note that L is also equal to L with the last row column removed. Because G is balanced, L s = L + (L ) T []. Note that L s = L + L T is a principle subminor of L s, hence L s. Also note that, because G is balanced, T is a left eigenvector of L s. As a proof by contradiction, suppose L s has a nontrivial nullspace such that Lw = for some nontrivial w. Because T L s =, we see that the last row of L s is a linear combination of the remaining rows, hence y = [ w T ] T N (L s ). But {} forms a basis for the nullspace of L s, a contradiction. Theorem. The origin of () () with P selected as in (6) is asymptotically stable. Proof: Case. (β > ) We construct a Lyapunov function V = (ṽt ṽ + z T z + β λt P λ). (7) Let Q := L T P + P L. Then V = ṽ T (K I p )ṽ λ T ( β Q ) λ. (8)

4 5 (a) Fig.. (a) -agent, 5-edge sensing graph with edge indexing based on the monitoring subgraph. (b) One possible monitoring graph constructed from the sensing graph. We now use LaSalle s invariance principle. Note that V means ṽ, λ. From (), this implies z N (D I p ) for all t. Recalling that z R(D T I p ) = N (D I p ), we see that z. Hence, V iff z, λ, z. Therefore, X = is asymptotically stable. Case. (β = ) The resulting transition matrix of () is then upper triangular, therefore the system is asymptotically stable iff the subsystems [ ] ṽ z = (b) [ K Ip D I p D T I p ] [ṽ ] z (9) λ = L λ () are asymptotically stable. () is clearly asymptotically stable, (9) can be seen to be asymptotically stable by a Lyapunov argument again invoking LaSalle s invariance principle as in Case. IV. FORMATION SCALING WITH NO COMMUNICATION We again consider agent dynamics of the form in (5). However, we now present an update rule for λ i that does not require agents to share local scaling estimates with other agents, i.e. only relative position information is shared along edges in the sensing graph. From the sensing graph, we construct a directed spanning tree (also known as an arborescence []) rooted at the leader with the following properties: ) The arborescence contains all the vertices a (directed) subset of the edges of the sensing graph. ) Each vertex except the leader is the head of exactly one edge. The leader agent is the head of no edge. ) A directed path exists from the leader node to all other nodes. We call this graph the monitoring graph. Without loss of generality for notational purposes, assume the leader is again agent n. To simplify notation analysis, for each edge in the monitoring graph, we number the corresponding edge in the sensing graph to match the head node index of that edge in the monitoring graph, we arbitrarily index the remaining edges of the sensing graph. Fig. shows an example of a formation with the sensing graph, one possible construction of the monitoring graph, the induced edge indexing scheme. The proposed update rule for other agents is ( ) λ i = λ i (zd i )T z i zi d for i =,..., n. () This means that each agent monitors one edge to update its local formation scaling factor. For notational simplicity, we let (z d )T z d T =... () (z d n )T z d n Ψ = diag{(d I p )z d } (I n p ), i.e. Ψ is equal to Γ with the last row column removed. We can then write the dynamics of (8) () as a homogenous linear system ṽ K I p D I p Γ ṽ z = D T I p z () λ ˆT In λ where ˆT = [T (n ) p(m n+) ]. Objectives B B amount to asymptotic stability of the origin of (). We can represent this cooperative control system as the interconnection of two subsystems as in Fig. : ṽ, z subsystem: [ ] ṽ = z [ K Ip D I p D T I p y = [ p(n ) pn λ subsystem: ] [ṽ ] + z I p(n ) p p(n ) pm p(n ) u () ] [ ] ṽ I p(n ) p(n ) p(m n+). z (5) λ = I n λ + T y (6) u = Ψ λ. (7) Note that both subsystems are stable. The stability of the composite system can be checked using a number of techniques, including the small gain theorem. Despite its potential conservatism, the small gain theorem gives a stability condition that can be verified based on the geometry of the formation. Because both subsystems are linear stable, asymptotic stability of the origin of () follows from the small gain theorem stability condition. We first present a geometric interpretation of the L gain of the λ subsystem: Theorem. The L gain of the λ subsystem is: m j= d ijz d j γ = max i=,...,n z d i. (8) Proof: Let H(s) be the transfer function of the λ subsystem, i.e. H(s) = s+ ΨT. To calculate γ = sup ω H(jω), we note that the supremum occurs when

5 I + u = Ψ λ z [ ] I [ ] K D D T I p Ψ λ s+ [ṽ ] T y = z. z n Fig.. Interconnection of subsystems () (5) (6) (7). Dimension subscripts are removed for clarity. ω = is equal to ΨT. Observe that Ψ T are block diagonal. Therefore, it is clear that ΨT consists of n blocks of dimension p p along the diagonal. The singular values of ΨT are simply the singular values of each p p block. Let g i = m j= d ijzj d for i =,..., n. We have that γ is the largest singular value of ΨT, the p p blocks of (z ΨT take the form g d i )T d (z i. The singular values of g zi d i )T i { } zi d g are i zi d,. This leads to a geometric criterion ensuring stability when the small gain theorem is applied: if γ is the L gain of the ṽ, z subsystem, then γ < γ ensures stability of the composite system []. Note that γ depends on the damping coefficients on the incidence matrix D, but does not depend on the specifics of the formation geometry. We now present a bound on γ in the case when the damping coefficients are identical for all agents: Theorem. Assume the agents have uniform velocity damping (i.e. k := k =... = k n ). Let µ,..., µ n be the positive eigenvalues of the unweighted sensing graph Laplacian DD T. Let / µ i if k µ i ρ i = µ i if k < (9) µ i k µ i k for i =,..., n. The L gain of the ṽ, z subsystem, denoted by γ, satisfies γ max{ρ i }. () i Proof: Define the following: [ ] [ ] A A A := K Ip D I = p A A D T () I p pm pm [ ] I B := n I p () (m+) (n ) C := [ (n ) n I n (n ) (m n+) ] Ip. () We have that the transfer matrix from u to y of the ṽ, z subsystem is G(s) = C ( si (n+m)p A ) B. () If we view (si (n+m)p A) as a block matrix, we see from the structure of B C that G(s) is a submatrix of the lower left block of (si (n+m)p A). We can compute the lower left block using block matrix inversion techniques obtain an equivalent expression for G(s): G(s) = [ [ ] ] Ip(n ) I p(n ) p(n ) p(m n+) F (s) p p(n ) where (5) F (s) = ((si mp A ) + A (si np A ) ) (6) ( A )(si np A ) = [ ((s + ks)i m + D T D) D T ] I p. (7) We are interested in the value of F (s) evaluated along the jω-axis. First suppose that s = jω where ω. Let D = UΣ V T be a singular value decomposition of D with U R n n, V R m m orthonormal, Σ R n m diagonal. Note that the nonzero diagonal elements of Σ are µ i for i =,..., n. It follows that (s + ks)i m + D T D = V ((s + ks)i + Σ T Σ )V T, therefore we can write ((s + ks)i m + D T D) = V Σ V T (8) where Σ is a diagonal matrix with the values /(s + ks + µ i ), i =,..., n along the diagonal. Thus we see that for s = jω, ω, F (s) = (V Σ Σ T U T ) I p. (9) Since V U are orthonormal, the nonzero singular values of F (s) are the nonzero singular values of Σ Σ T, which are µi / s +ks+µ i, i =,..., n. In the case when s =, we have [ ((s + ks)i m + D T D) D T ] I p = D + I p (5) lim s where + denotes the Moore-Penrose pseudoinverse, the singular values of D + are / µ i [5]. Thus the p(n ) nonzero singular values of F (s) when s = jω, ω R are the second order transfer functions µi σ i (s) = s, i =,..., n, (5) + ks + µ i each occurring with multiplicity p due to the Kronecker product with the identity in (7). We have thatsup ω σ i (jω) = ρ i. We also have that G(s) is a submatrix of F (s). It follows that G(s) F (s) for all s, hence γ max i {ρ i }. Corollary. Assume the agents have uniform velocity damping k. Let µ denote the smallest positive eigenvalue of DD T, known as the Fiedler eigenvalue of the graph Laplacian. For sufficiently large k, we have γ µ. (5) Proof: Let µ i be the nonzero eigenvalues of DD T. If k µ i for all i, then ρ i = µi for all i, the result follows.

6 λi z j z j d Evolution of formation scaling factors over time without interagent communication 6 8 Evolution of edge lengths over time without interagent communication 6 8 Fig.. Simulation results for a -agent, 5-edge formation with no interagent communication, rom initial positions a desired formation scaling factor of. λi z j z j d Evolution of formation scaling factors over time with interagent communication 6 8 Evolution of edge lengths over time with interagent communication 6 8 Fig.. Simulation results for a -agent, 5-edge formation with interagent communication, rom initial positions a desired formation scaling factor of. V. SIMULATION RESULTS Consider the formation in the plane depicted in Fig. (a) where each edge is assigned an arbitrary direction, D corresponds to the chosen node edge numbering direction assignment, agent is the leader, the monitoring subgraph is shown in Fig. (b). We let z d = [ ] T.9, (5) which is consistent with the shape depicted in Fig. (a). We have that µ = is the smallest positive eigenvalue of DD T µ = is the largest eigenvalue of DD T. Suppose k := k = k = k = k. If k 8, then, by Corollary, γ. (5) Using the geometric condition presented in Theorem we have that γ =.. Thus γ γ =. =.9 < (55) therefore the system with no interagent communication is stable. Fig. shows simulation results in the case of no interagent communication with λ =, λ i () = for i =,..., n, the agents initial positions initialized romly. Fig. shows simulation results for the same formation desired formation size, but interagent communication is allowed the communication graph is equivalent to the unweighted sensing graph with edges pointing towards the leader agent removed. The plots demonstrate that the control strategy with communication converges to the desired formation more quickly, yet the improvement is modest. VI. CONCLUSIONS In this paper, we introduce increased flexibility adaptivity to the stard formation maintenance problem by presenting cooperative control strategies that allow a multiagent team to dynamically alter its formation size. We have presented a strategy that relies on interagent communication produces a stable cooperative control system given any desired connected formation, we have presented a strategy that relies only on relative position information with stability results that are dependent on the desired formation velocity damping. In the latter case, we have introduced a simple geometric criterion which may be used with the small gain theorem to determine stability. In both cases, these strategies can be combined with other objectives such as collision avoidance using stard algorithms. REFERENCES [] W. Ren, R. W. Beard, E. M. Atkins, A survey of consensus problems in multi-agent coordination, in Proceedings of the American Control Conference, vol., pp , 5. [] R. Olfati-Saber, J. A. Fax, R. M. Murray, Consensus cooperation in networked multi-agent systems, Proceedings of the IEEE, vol. 95, no., pp. 5, 7. [] R. Olfati-Saber R. Murray, Consensus problems in networks of agents with switching topology time-delays, Automatic Control, IEEE Transactions on, vol. 9, pp. 5 5, Sept.. [] A. Jadbabaie, J. Lin, A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, Automatic Control, IEEE Transactions on, vol. 8, no. 6, pp. 988,. [5] H. Tanner, A. Jadbabaie, G. Pappas, Stable flocking of mobile agents, part i: fixed topology, in Decision Control,. Proceedings. nd IEEE Conference on, vol., pp. 5, Dec.. [6] J. Lawton, R. Beard, B. Young, A decentralized approach to formation maneuvers, Robotics Automation, IEEE Transactions on, vol. 9, pp. 9 9, Dec.. [7] J. Desai, J. Ostrowski, V. Kumar, Controlling formations of multiple mobile robots, in Robotics Automation, 998. Proceedings. 998 IEEE International Conference on, vol., pp , May 998. [8] P. Ogren, E. Fiorelli, N. E. Leonard, Cooperative control of mobile sensor networks: Adaptive gradient climbing in a distributed environment, Automatic Control, IEEE Transactions on, vol. 9, no. 8, pp. 9,. [9] J. Fax R. Murray, Information flow cooperative control of vehicle formations, Automatic Control, IEEE Transactions on, vol. 9, pp , Sept.. [] H. Axelsson, A. Muhammad, M. Egerstedt, Autonomous formation switching for multiple, mobile robots, in IFAC Conference on Analysis Design of Hybrid Systems,. [] M. Mesbahi M. Egerstedt, Graph Theoretic Methods in Multiagent Networks. Princeton University Press,. [] A. Berman R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences. Society for Industrial Applied Mathematics, 99. [] C. Berge, The Theory of Graphs. Dover Publications, Inc.,. [] H. K. Khalil, Nonlinear Systems. Prentice Hall, third ed.,. [5] A. J. Laub, Matrix Analysis for Scientists Engineers. Society for Industrial Applied Mathematics, 5.