Coordination of Multi-Agent Systems Mark W. Spong Donald Biggar Willett Professor Department of Electrical and Computer Engineering and The Coordinated

Coordination of Multi-Agent Systems

Mark W. SpongDonald Biggar Willett Professor

Department of Electrical and Computer Engineeringand The Coordinated Science Laboratory

University of Illinois at Urbana-Champaign, [email protected]

IASTED CONTROL AND APPLICATIONS

May 24-26, 2006, Montreal, Quebec, Canada

IASTED CONTROL AND APPLICATIONSMay 24-26, 2006, Montreal, Quebec, Canada

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


Synchronously Flashing Fireflies

A Swarm of Locusts

More Examples from Nature


Autonomous Formation Flying and UAV Networks

Examples from Engineering


Crowd Dynamics and Building Egress

Mobile Robot Networks

Examples from Social Dynamics and Engineering Systems


Example from Bilateral Teleoperation

Multi-Robot Remote Manipulation


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


Synchronization of Metronomes

Example:


Fundamental Questions

In order to analyze such systems and design coordination strategies, several questions must be addressed:

1. What are the dynamics of the individual agents?

2. How do the agents exchange information?

3. How do we couple the available outputs to achieve synchronization?


Fundamental Assumptions

In this talk we assume:

1. that the agents are governed by passive dynamics.

2. that the information exchange among agents is described by a balanced graph, possibly with switching topology and time delays in communication.


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


Definition of A Passive System


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


Passivity of Lagrangian SystemsMore generally, an N-DOF Lagrangian system

satisfies

where H is the total energy. Therefore, the system is passivefrom input to output


Graph Theory

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1 4


32

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3

2

1 5 32

1 4

Examples of Communication Graphs

All-to-All Coupling

(Balanced -Undirected)Directed – Not Balanced

Balanced-Directed


Synchronization of Multi-Agents


First Results

Suppose the systems are coupled by the control law

Theorem: If the communication graph is weakly connected and balanced, then the system is globally stable and the agents output synchronize.

where K is a positive gain and is the set of agents communicating with agent i.


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.


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.


where is a (passive) nonlinearity of the form

Further Extensions

We can also prove output synchronization when the coupling between agents is nonlinear,


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.


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


Nonlinear Positive-Real Lemma


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


Barbalat’s Lemma can be used to show that

and, therefore,

Connectivity of the graph interconnection then implies output synchronization.


Some Examples32

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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


The closed loop system is therefore

and the outputs (states) synchronize as shown


Second-Order Example

Consider a system of four point masses with second-order dynamics

connected in a ring topology

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where which is passive from

The key here is to define ``the right’’ passive output. Define a preliminary feedback

so that the dynamic equations become

to


coupling the passive outputs leads to

and the agents synchronize as shown below


Simulation Results


Example: Coupled Pendula

Consider two coupled pendula with dynamics

and suppose



is the natural frequency and is the coupling strength.

Kuramoto Oscillators

Kuramoto Oscillators are systems of the form

where is the phase of the i-th oscillator,


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


Multi-Robot Coordination With Delay

Consider a network of N Lagrangian systems

As before, define the input torque as

which yields

where


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.


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.


THANK YOU!

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Coordination of Multi-Agent Systems Mark W. Spong Donald Biggar Willett Professor Department of Electrical and Computer Engineering and The Coordinated