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 email@example.com. IASTED CONTROL AND APPLICATIONS
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
IASTED CONTROL AND APPLICATIONS
May 24-26, 2006, Montreal, Quebec, Canada
Flocking of Birds
Schooling of Fish
More Examples from Nature
A Swarm of Locusts
Synchronously Flashing Fireflies
Autonomous Formation Flying and UAV Networks
Mobile Robot Networks
Crowd Dynamics and Building Egress
Multi-Robot Remote Manipulation
In order to analyze such systems and design coordination strategies, several questions must be addressed:
In this talk we assume:
We present a unifying approach to:
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
More generally, an N-DOF Lagrangian system
where H is the total energy. Therefore, the system is passive
from input to output
Directed – Not Balanced
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.
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.
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.
We can also prove output synchronization when the coupling between agents is nonlinear,
where is a (passive) nonlinearity of the form
 Nikhil Chopra and Mark W. Spong, “Output Synchronization of Networked Passive Systems,” IEEE Transactions on Automatic Control, submitted, December, 2005
 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.
Since each agent is assumed to be passive, let
be the storage functions for the N agents
and define the Lyapunov-Kraskovski functional
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
Connectivity of the graph interconnection then implies output synchronization.
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
Consider a system of four point masses with second-order dynamics
connected in a ring topology
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
coupling the passive outputs leads to
and the agents synchronize as shown below
Consider two coupled pendula with dynamics
is the phase of the i-th oscillator,
Kuramoto Oscillators are systems of the form
is the natural frequency and is the coupling strength.
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
Consider a network of N Lagrangian systems
As before, define the input torque as
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.