Coordination of Multi-Agent Systems

1. Coordination of Multi-Agent Systems Mark W. Spong Donald Biggar Willett Professor Department of Electrical and Computer Engineering and The Coordinated Science Laboratory University of Illinois at Urbana-Champaign, USA mspong@uiuc.edu IASTED CONTROL AND APPLICATIONS May 24-26, 2006, Montreal, Quebec, Canada

2. Introduction • The problem of coordination of multiple agents arises in numerous applications, both in natural and in man-made systems. • Examples from nature include: Flocking of Birds Schooling of Fish

3. More Examples from Nature A Swarm of Locusts Synchronously Flashing Fireflies

4. Examples from Engineering Autonomous Formation Flying and UAV Networks

5. Examples from Social Dynamics and Engineering Systems Mobile Robot Networks Crowd Dynamics and Building Egress

6. Example from Bilateral Teleoperation Multi-Robot Remote Manipulation

7. Other Examples • Other Examples: • circadian rhythm • contraction of coronary pacemaker cells • firing of memory neurons in the brain • Superconducting Josephson junction arrays • Design of oscillator circuits • Sensor networks

8. Synchronization of Metronomes Example:

9. Fundamental Questions In order to analyze such systems and design coordination strategies, several questions must be addressed: • What are the dynamics of the individual agents? • How do the agents exchange information? • How do we couple the available outputs to achieve synchronization?

10. Fundamental Assumptions In this talk we assume: • that the agents are governed by passive dynamics. • that the information exchange among agents is described by a balanced graph, possibly with switching topology and time delays in communication.

11. Outline of Results We present a unifying approach to: • Output Synchronization of Passive Systems • Coordination of Multiple Lagrangian Systems • Bilateral Teleoperation with Time Delay • Synchronization of Kuramoto Oscillators

12. Definition of A Passive System

13. Examples of Passive Systems In much of the literature on multi-agent systems, the agents are modeled as first-order integrators This is a passive system with storage function since

14. Passivity of Lagrangian Systems More generally, an N-DOF Lagrangian system satisfies where H is the total energy. Therefore, the system is passive from input to output

15. 3 2 1 4 Graph Theory

16. 3 2 3 2 4 1 5 1 4 3 2 1 4 Examples of Communication Graphs Balanced-Directed All-to-All Coupling (Balanced -Undirected) Directed – Not Balanced

17. Synchronization of Multi-Agents

18. First Results Suppose the systems are coupled by the control law where K is a positive gain and is the set of agents communicating with agent i. Theorem: If the communication graph is weakly connected and balanced, then the system is globally stable and the agents output synchronize.

19. Some Corollaries 1) If the agents are governed by identical linear dynamics then, if (C,A) is observable, output synchronization implies state synchronization 2) In nonlinear systems without drift, the outputs converge to a common constant value.

20. Some Extensions We can also prove output synchronization for systems with delay and dynamically changing graph topologies, i.e. provide the graph is weakly connected pointwise in time and there is a unique path between nodes i and j.

21. Further Extensions We can also prove output synchronization when the coupling between agents is nonlinear, where is a (passive) nonlinearity of the form

22. Technical Details • The proofs of these results rely on methods from Lyapunov stability theory, Lyapunov-Krasovski theory and passivity-based control together with graph theoretic properties of the communication topology. • References: [1] Nikhil Chopra and Mark W. Spong, “Output Synchronization of Networked Passive Systems,” IEEE Transactions on Automatic Control, submitted, December, 2005 [2] Nikhil Chopra and Mark W. Spong, “Passivity-Based Control of Multi-Agent Systems,” in Advances in Robot Control: From Everyday Physics to Human-Like Movement, Springer-Verlag, to appear in 2006.

23. Technical Details Since each agent is assumed to be passive, let ,…, be the storage functions for the N agents and define the Lyapunov-Kraskovski functional

24. Nonlinear Positive-Real Lemma

25. Now, after some lengthy calculations, using Moylan’s theorem and assuming that the interconnection graph is balanced, one can show that

26. Barbalat’s Lemma can be used to show that and, therefore, Connectivity of the graph interconnection then implies output synchronization.

27. 3 2 1 4 Some Examples Consider four agents coupled in a ring topology with dynamics Suppose there is a constant delay T in communication and let the control input be

28. The closed loop system is therefore and the outputs (states) synchronize as shown

29. 3 2 1 4 Second-Order Example Consider a system of four point masses with second-order dynamics connected in a ring topology

30. The key here is to define ``the right’’ passive output. Define a preliminary feedback so that the dynamic equations become where which is passive from to

31. coupling the passive outputs leads to and the agents synchronize as shown below

32. Simulation Results

33. Example: Coupled Pendula Consider two coupled pendula with dynamics and suppose

35. is the phase of the i-th oscillator, where Kuramoto Oscillators Kuramoto Oscillators are systems of the form is the natural frequency and is the coupling strength.

36. Suppose that the oscillators all have the same natural frequency and define Then we can write the system as and our results immediately imply synchronization

37. Multi-Robot Coordination With Delay Consider a network of N Lagrangian systems As before, define the input torque as which yields where

38. Coupling the passive outputs yields and one can show asymptotic state synchronization. This gives new results in bilateral teleoperation without the need for scattering or wave variables, as well as new results on multi-robot coordination.

39. Conclusions • The concept of Passivity allows a number of results from the literature on multi-agent coordination, flocking, consensus, bilateral teleoperation, and Kuramoto oscillators to be treated in a unified fashion.

40. THANK YOU! QUESTIONS?