Flocks of Autonomous Mobile Agents

1 Flocks of Autonomous Mobile Agents Herbert G. Tanner, Ali Jadbabaie and George J. Pappas Department of Electrical and Systems Engineering University of Pennsylvania Abstract—The motion of a group of nonholonomic mobile agents is synchronized using local control laws. This synchronization strategy is inspired by the early flocking model proposed by Reynolds [17] and following work [22, 8]. The control laws presented ensure that all agent headings and speeds converge asymptotically to the same value and collisions between the agents are avoided. The stability of this type of motion is closely related to the connectivity properties of the underlying interconnection graph. Proof techniques are based on LaSalle’s invariant principle and algebraic graph theory and the results are verified in numerical simulations. Keywords—Cooperative control, multi-agent systems, flocking motion. I. Introduction Recent technological advances offer more efficient computation and less expensive communication. The ability to compute locally and share information has facilitated the development of new multi-agent systems. Such type of systems promise increased performance, efficiency and robustness, at a fraction of the cost compared to their centralized counterparts, utilizing distributed coordination sensing and actuation. The question arising is how to achieve coordination in multi-agent systems. Nature is abundant in marvelous examples of coordinated behavior. Across the scale, from biochemical cellular networks, up to ant colonies, schools of fish, flocks of birds and herds of land animals, one can find systems that exhibit astonishingly efficient and robust coordination schemes [1, 15, 23, 7, 5]. At the same time, several researchers in the area of statistical physics and complexity theory have addressed flocking and schooling behavior in the context of non-equilibrium phenomena in many- degree-of-freedom dynamical systems and self organization in systems of self-propelled particles [22, 21, 20, 13, 11, 18, 9]. Similar problems have become a major thrust in systems and control theory, in the context of cooperative control, distributed control of multiple vehicles and formation control; see for example [10, 2, 16, 4, 12, 6, 19, 8, 14, 3]. The main goal of the above papers is to develop a decentralized control strategy, such that a global objective, such as a tight formation with fixed pairwise inter vehicle distances is achieved. In 1986 Craig Reynolds [17] made a computer model of coordinated animal motion such as bird flocks and fish schools. He called the generic simulated flocking creatures “boids”. The basic flocking model consists of three simple steering behaviors which describe how an individual boid maneuvers based on the positions and velocities its nearby flockmates:separation, alignment, and cohesion. In 1995, a similar model was proposed by Vicsek et al. [22]. Under an alignment rule, a spontaneous development of coherent collective motion is observed, resulting in the headings of all agents to converge to a common value. A proof of convergence for Vicsek’s model (in the noise-free case) was given in [8]. In this paper provide a system theoretic justification for the flocking phenomenon in [17]. In our flocking model, we consider dynamic mobile agents steered by local control laws. The mobile agents are described by nonholonomic differential equations of the form: ẋi = vi cos θi ẏi = vi sin θi (1a) (1b) θ̇i = ωi v̇i = ai , (1c) (1d) FLOCKS OF AUTONOMOUS MOBILE AGENTS for i ranging from 1 to N, which is the total number of agents in the group. In the above, ri = (xi , yi)T is the position vector of vehicle i, θi its orientation (Figure 1), vi its translational speed and ai , ωi its control inputs. The relative positions between the vehicles are denoted rij
ri − rj . vi ωi θi 2 noted Ni . We can represent the interconnection topology by means of an undirected graph G = {V, E}, where V is the set of vertices indexed by the agents in the group, and E is the set of edges, with one edge between each pair of interconnected agents. For a pair of neighboring agents, (i, j) ∈ E we define an artificial potential function Vij that depends on the distance between these two agents. An example of such function, given here for illustration purposes, is the following: Vij (rij ) = ŷ ri ai ẑ x̂ Fig. 1. Agent i and its control inputs. For this system of mobile agents we show that all agent headings converge to the same value, and pairwise relative speeds are stabilized. This results in an overall motion in which the group maintains its shape and moves uniformly in a particular direction. An artificial potential function ensures that collisions between the agents are avoided and cohesion in the group is maintained. Our analysis is based on Lyapunov stability and algebraic graph theory. Central to this analysis is the connectivity of the graph that represents control interactions between the agents, a property which allows propagation of information. While the proof techniques are totally different from those in [8], the end result is similar, suggesting that addition of cohesion and separation forces in addition to alignment as well as addition of dynamics, does not affect the stability of the flocking motion. 1 rij  2 + log rij 2 . The potential of an agent i, is now defined as the sum of all artificial potentials associated with every one of its neighbors:
Vi
Vij (rij ). j∈Ni The neighboring relations of each agent determine its control law:
(vi − vj ) − [(∇ri Vi )x cos θi ai = − k j∈Ni    
  (2a) (vi − vj ) + (∇ri Vi )y sin θi ]    j∈Ni
ωi = − k (θi − θj ) − [(∇ri Vi )y cos θi j∈Ni    
  (θi − θj ) − (∇ri Vi )x sin θi ]    (2b) j∈Ni II. Coordination Strategy with k being a constant positive parameter (control gain). In (2a)-(2b) the first terms are responsible for stabilization of relative headings and speeds, while the second terms reduce the potential of each boid. For the closed loop system we have the following: Let us now assume a fixed control interconnection topology on the group of vehicles, where interconnections express control dependences. The agents that are interconnected with i, are called neighbors of i. The set of neighbors of i is de- Proposition II.1 Consider the system of N mobile agents with dynamics (1). Then, for a sufficiently large control gain k, the agent headings and speeds converge to the same value while collisions between the agents are avoided. FLOCKS OF AUTONOMOUS MOBILE AGENTS 3 Consider the positive semi-definite ensures that V will be decreasing. Any level set of V is invariant and from (3) all speed and heading differences have to remain bounded. Given 1 T 1 T ⊥ ⊥ V = θ Lc θ + v Lc v, (3) that v 1 = θ1 = 0 is an equilibrium configu2 2 ration for (1), application of LaSalle’s invariant where θ and v are the stack vectors of headings principle shows that this configuration is asympand speeds, respectively and Lc is the Laplacian totically stable. of the complete graph with N vertices. Taking III. Numerical Validation the derivative of V we get: Consider a group of 10 mobile agents. At first V̇ = − θT Lc (kLθ + diag{(∇ri Vi )⊥ } |Lθ|) assume that the interconnection graph is complete, that is every agent has every other agent − v T Lc (kLv + diag{(∇ri Vi ) } |Lv|) as a neighbor. Let the agents be initiated with = − kNθT Lθ − θLc diag{(∇ri Vi )⊥ } |Lθ| (x, y) positions selected randomly in the interval − kNv T Lv − vLc diag{(∇ri Vi ) } |Lv| , [−5, 5] m, speeds in the interval [−1, 1] m/s and headings in [−π, π]. Control gain k is set to 1. where (∇ri Vi ) and (∇ri Vi )⊥ are the components We depict the evolution of a set of N − 1 = 9 of each ∇ri Vi along the direction of the velocity of heading differences that span the relative headagent i and its orthogonal direction, respectively. ings space in Figure 2. The Figure shows that the It turns out that components of v and θ that are heading differences have converged exponentially along the direction of the vector of ones, 1, pre- to zero after approximately 10 simulation secserve the value of V . To this end, decompose onds, resulting in all agents moving in the same ⊥ ⊥ these vectors as θ = θ1 ⊕ θ1 and v = v 1 ⊕ v 1 , direction. where superscripts 1 and 1⊥ identify the components of the vectors along the direction of the vector of ones and its orthogonal, respectively. Then, the derivative of V becomes Proof: function heading differences 2 θ −θ 1 2 θ2−θ3 θ −θ 3 4 θ −θ 4 5 θ −θ 5 6 θ6−θ7 θ −θ 7 8 θ −θ 8 9 θ −θ 1.5 1 ⊥ ⊥ ⊥ 1⊥ 1⊥ 0.5 V̇ = − k(θ1 )T (NLθ1 + Lc diag{(∇ri Vi )⊥ }|Lθ1 |) − (v )T (kNLv 1⊥ ≤ − kNλ2 θ  2 1⊥ + Lc diag{(∇ri Vi ) }|Lv |) ⊥ 10 −0.5 −1 √ 1⊥ + θ  Lc  diag{(∇ri Vi )⊥ } N Lθ  − kNλ2 v 1 2 9 0 rad 1⊥ (4) −1.5 −2 √  ⊥ ⊥ + v 1  Lc  diag{(∇ri Vi ) } N Lv 1  √ ⊥ ≤ −N(kλ2 − N Nfmax )θ1 2 √ ⊥ (5) Fig. 2. Convergence of headings when the interconnec− N(kλ2 − N Nfmax )v 1 2 tion graph is complete. (6) Figure 3 shows the convergence of speeds. In where fmax is the magnitude of the maximum Figure 3 the (absolute) speeds of the 10 agents potential force. Choosing converge exponentially to the same value, which √ is approximately 0.5m/s. N Nfmax Figures 4-5 refer to the case where the interk≥ , λ2 connection graph is not complete, i.e. it does −2.5 −3 0 1 2 3 4 5 time 6 7 8 9 10 FLOCKS OF AUTONOMOUS MOBILE AGENTS 4 speeds 1 v 1 v 2 v3 v 4 v5 v 6 v 7 v 8 v 9 v10 0.9 0.8 0.7 Figure ??. Speeds converge exponentially to a value close to 0.7m/s. Note once again that due to the reduced value of λ2 , the agent speeds take longer to reach their steady state compared to the case of the complete interconnection graph. 0.6 m/s speeds 1.8 v 1 v 2 v 3 v 4 v 5 v6 v 7 v 8 v9 v 0.5 1.6 0.4 1.4 0.2 1.2 0.1 0 1 2 3 4 5 6 7 8 9 m/s 0.3 10 10 1 time 0.8 Fig. 3. Convergence of speeds when the interconnection graph is complete. 0.6 0.4 not have all possible edges, but it is nevertheless connected. The evolution of same set of heading differences as in the previous case is now depicted in Figure 4. The Figure shows that the headings still converge to the same value, however the rate of convergence is smaller and it takes almost 15 simulation seconds to reach steady state. This is due to the fact that in this case, the second eigenvalue of the graph Laplacian, λ2 is smaller. Recall from (6) that λ2 is related to the convergence rate of the Lyapunov-like function. heading differences 4 θ −θ 1 2 θ2−θ3 θ −θ 3 4 θ −θ 4 5 θ5−θ6 θ −θ 6 7 θ7−θ8 θ8−θ9 θ −θ 3 2 9 10 rad 1 0 −1 −2 −3 0 5 10 15 0.2 0 5 10 15 time Fig. 5. Convergence of speeds when the interconnection graph is connected but not complete. IV. Conclusions In this paper we show how a group of nonholonomic mobile agents can synchronize their headings and speeds using local control laws. 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