1 Distributed Motion Coordination: From Swarming to Synchronization Ali Jadbabaie Department of Electrical and Systems Engineering and GRASP Laboratory University of Pennsylvania Hamilton Institute Seminar 06/24/2005 SWARMS 2 Distributed Coordination in nature Flocks, swarms and schools exhibit coordinated group behavior although each animal acts completely autonomously How do these behaviors emerge? How are they sustained? How do individual decisions lead to collective group behavior? SWARMS 3 Complexity, Statistical Physics, emergence of collective behavior SWARMS 4 r agent i neighbors of agent i Vicseks kinematic model How can a group of moving agents collectively decide on direction, based on nearest neighbor interaction? How does global behavior emerge from local interactions? SWARMS 5 = heading = speed MAIN QUESTION : MAIN QUESTION : Under what conditions do all headings converge to the same value and agents reach a consensus on where to go? Distributed consensus algorithm SWARMS 6 Multi-agent Representations: Proximity Graphs We use graphs to model neighboring relations V: A set of vertices indexed by the set of mobile agents. E: A set of edges the represent the neighboring relations. W: A set of weights over the set of edges. Agent is neighborhood The neighboring relation is represented by a fixed graph G, or a collection of graphs G 1, G 2,G m SWARMS 7 Vicseks model switching signal, adjacency matrix Valence matrix finite set of indices corresponding to all graphs over n vertices. SWARMS 8 Conditions for reaching consensus Theorem (Jadbabaie et al. 2003): If there is a sequence of bounded, non-overlapping time intervals T k, such that over any interval of length T k, the network of agents is jointly connected , then all agents move in a formation. This happens to be both necessary and sufficient for exponential coordination, boundedness of intervals not required for asymptotic coordination. (Moreau 04, Ren & Beard 05) SWARMS 9 Extensions Asynchronous update (Tsitsiklis et al. 84,Cao and Morse 04) Switching, directed graphs (Moreau 04, Ren & Beard04, Zhu, Francis 04) Gossip in networks ( Boyd et a. 04) Balanced, directed graphs, no switching. Olfati & Murray 04 Consensus +quantization (Savkin 04). Consensus on random graphs (Hatano and Mesbahi 04) No quadratic Lyapunov function exists, but max j i min j j is a valid Lyapunov function, if connectivity holds. (products of length n-1 of F i s are pseudo-contractive with respect to a subspace norm. ) Analysis extends to Choose x = tan( ) SWARMS 10 Motion Coordination with Dynamic Models Double integrator model Neighbors of i: SWARMS 11 The control law The control law has two components: cohesion/separation input synchronization input SWARMS 12 Pair wise attraction and repulsion potentials Each V ij is required to be: increasing as unbounded as unique minimum The potential energy V i of boid i depends on its neighbors Example (Ogren et al 04, Fiorelli and Leonard 02) SWARMS 13 The Laplacian of the graph The graph Laplacian encodes structural properties of the graph Some properties of the Laplacian: It is positive semi-definite The multiplicity of the zero eigenvalue is the number of connected components One corresponding eigenvector is the vector of ones, 1. The second smallest eigenvalue quantifies connectivity. SWARMS 14 Dynamic Topology Local sensing/communication Graph changes with time Control is discontinuous Non smooth Lyapunov theory Topology dictates analysis Fixed Topology Fixed (logical) network Graph is constant Control is smooth Classic Lyapunov theory SWARMS 15 For both fixed and dynamic topology: If the neighboring graph stays connected, all agent velocity vectors become asymptotically the same, collisions between interconnected agents are avoided and the system approaches a configuration that minimizes all agent potentials. Conditions for coordination We could shape potentials for any desired configuration, and also update it as the objective changes. SWARMS 16 History of the coupled oscillators History of the coupled oscillators Study of Mutual synchronization of biological oscillators goes back to Weiner in 1950s. Examples: pacemaker cells in the heart and nervous system, collective synchronization of pancreatic beta cells, synchronously flashing fire flies. Synchronization of oscillators has also been studied in the context of injection locking in RF circuits Good abstraction for studying networks of loads and generators in the power grid. All-to-all case with infinite oscillators characterized, finite case and arbitrary topologies open See the book by Steven Strogatz SWARMS 17 Synchronization of coupled oscillators Consider a group of N oscillators coupled nonlinearly as It is the simplest model of coupled oscillators, simple enough for analysis, but complicated enough to have interesting non-trivial behavior. The degree of synchronization, is measured with the magnitude of the average phasor: r(t) close to 1 means synchronization, and r(t) close to zero means asynchrony. How can we extend this to arbitrary interconnections? SWARMS 18 Laplacian & Incidence Matrix Weighted Laplacian Some properties of the Laplacian: The e dimensional vector space of edges can be decomposed to an n-1 dimensional cut space (span of columns of B T ) and m-n+1 dimensional cycle space (Kernel of B). W is diagonal B is the (n x e) incidence matrix of graph G SWARMS 19 B is the incidence matrix of the graph representing the interconnection of oscillators Simple case: all oscillators are identical Theorem: Consider the unperturbed Kuramoto Model defined over an arbitrary connected graph with incidence matrix B. For any given 0 and any positive value of the coupling, the vector is a locally asymptotically stable equilibrium solution. Furthermore, the rate of approach to equilibrium is no worse than Kuramoto model with incidence matrices SWARMS 20 For | i|< /2 for a connected graph, all trajectories will converge to S Therefore, all velocity vectors will synchronize. A Special Case Fixed points: But, this stability result is not global. In the case of the ring topology is not the only equilibrium. This is due to the fact that B and B T have the same null space! is also stable: Thus SWARMS 21 When =0, is an asymptotically stable fixed point. The order parameter can be written as Where e is the number of edges in the graph is a Lyapunov function, measuring velocity misalignment. Using LaSalle, all trajectories converge to invariant sets. Can extend to the case of changing topology, if the graph is jointly connected. The speed of synchronization depends on the algebraic connectivity of the graph (2 nd smallest eigenvalue of the Laplacian). Properties of the model SWARMS 22 Onset of Synchronization When the frequencies are non zero, there is no fixed point for small values of coupling. Theorem: Bounds on the critical value of the coupling can be determined by maximum deviation of frequencies from the mean, and algebraic connectivity of the graph. When is random, Can develop a mean field model for general topologies by SWARMS 23 Want to globally minimize 1-r 2 over the whole network Let z= sin(B T ) Subject to: Dual decomposition and nonlinear network flow Supply at each node Sum of pair-wise potentials Kuramoto model is the Subgradient algorithm for solving the dual Subgradient algorithm Lagrangian Shor 87, Tsitsiklis 86 SWARMS 24 Kuramoto model with non- homogeneous delays Phase information from neighbors arrive with arbitrary time delay ij < 1. [A ij ] is the adjacency matrix. In case of degree regular graphs, linearized model with ij = was studies by Earl and Strogatz Using Lyapunov-Krasovskii functionals, can analyze the case of arbitrary connected graphs and non- homogeneous delays. We assume K is large enough that the oscillators are synchronized, and linearize the dynamics around the synchronized state, i (t) = , where i = t + i (t) with G ij = A ij cos( ij ) SWARMS 25 Theorem: Consider a network of N identical oscillators with linearized dynamics G ij = A ij cos( ij ) and G ij >0 when i and j are neighbors. Synchronized state is stable independent of delay. Proof sketch: Use the following V( ) as a Lyapunov/Krasovski functional : Delay Independent Stability Corollary: If [G ij ] is the adjacency matrix of a connected graph, then the continuous time, consensus problem is asymptotically stable with arbitrary time delays SWARMS 26 Biologically plausible coordination for kinematic robots The input Minimizes the misalignment potential The control law minimizes the potential by following its gradient. But we cant measure the headings of neighbors W/O communication SWARMS 27 Biologically plausible sensing Knowing relative heading would mean having binocular vision or solving structure from motion. This would require multiple visible features on each agent. Measured Quantities the projection of an agent (bearing) ij the speed of the projection (optical flow) The time-to-collision or Expansion rate (rate of approaching or receding of an object), measured as the relative rate of change of the projection area Pigeons and flies are capable of all 3 measurements!. Wang & Frost, Nature, 1992, Fry and Dickinson, Science 2003 ij l ij ii jj XwXw YwYw j i SWARMS 28 Theorem: with the distributed controller joint connectivity in time flocking A distributed control law Proof based on construction of Lyapunov function whose derivative is the quadratic form of a state dependent Laplacian, and the following lemma SWARMS 29 Simulations for 2d and 3d kinematic models SWARMS 30 Coordination in 3D Consider a group of N agents with different velocity vectors and constant, unit speed (extension to dynamic case possible). Agent is neighborhood is the heading and is the attitude. SWARMS 31 Geodesic Control Law XwXw YwYw ZwZw vivi vjvj ij Bullo, Murray and Sarti, Control on the Sphere and Reduced Attitude Stabilization, 1995 TiSTiS X i X i SWARMS 32 Geodesic Versor Geodesic versor Y ij shows the geodesic direction from v i to v j is the component of vj orthogonal to vi. XwXw YwYw ZwZw vjvj TiSTiS XX XX vivi Y ij ij For any two agents iand j : SWARMS 33 Theorem [ Moshtagh, Jadbabaie and Daniilidis, CDC05]: Consider the system of N equations If the proximity graph of the agents is fixed and connected, then the control laws result in flocking. Furthermore the consensus state is locally asymptotically stable. A similar result holds in the case of switching graphs, if the union graph is connected in time. YwYw ZwZw vjvj TiSTiS XX XX vivi Y ij XwXw ij SWARMS 34 Lyapunov-based proof Lyapunov function: measure of discrepancy between velocity vectors Using LaSalles invariance principle, all trajectories converge to the largest invariant set within the set: velocity vectors will synchronize. In 2-d this is v T L v It Can be shown that For 0