© 2002 IBM Corporation@ 2005 SJTU Complex Networks & Control Lab
Xiaofan WangShanghai Jiao Tong University
2007.7.19 Guilin
The International Workshop on Complex The International Workshop on Complex
Systems and Networks 2007Systems and Networks 2007
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Collaborators
Mr. Housheng Su (My Ph D student)
Prof. Zongli Lin (Univ. of Virginia)
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Flocking & Synchronization
A large number of agentslimited information and simple rulesorganize into a coordinated motion
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Connection
FlockingFlockingSynchronizationSynchronizationConsensusConsensusRendezvousRendezvousSwarmingSwarming
Distributed coordination Distributed coordination of network of agents:of network of agents:
AgentAgentNetworkNetworkDistributed local controlDistributed local controlGlobal coordinated Global coordinated behavior (behavior (emergentemergent))
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Birds Flocking
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Schools of fish
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Herds of animals
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Mobile robots, UAVs, sensor networks
massive distributed sensing using mobile sensor networks
cooperative unmanned aerial vehicle
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A complex network view of flocking
Node AgentAt any time t, there is an edge between two agents if ||qi(t)-qj(t)||<r
A spatial complex dynamical network with time-varying (switching) topology
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Flocking problems
Suppose that each agent has limited capability. In order to create a coordinated motion
What kind of basic rules should each agent follow?
How to design distributed algorithmsso that these rules hold?
Can we guarantee stability of the coordinated motion
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Boids-Classical Flocking Model
Reynolds , “Flocks, Herd, and Schools: A Distributed Behavioral Model”, Computer Graphics, 21(4),1987.
Three rules:Separation: Steer to avoid collisions with nearby flockmates
Alignment: Steer toward the average heading of local flockmates
Cohesion: Steer to move toward the average position of local flockmates
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Flocking Control AlgorithmDesign distributed control algorithm for each agent
Obstacle avoidance Tracking
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SJTU Flocking Control Algorithm
Goals of Control:
,i i
i i
q pp u==
&
& 1,...,i N=
iq Position
ip Velocity
Separation & CohesionSeparation & Cohesion
AlignmentAlignment
0 || ||ij i j ijd q q e< ≤ − ≤ < ∞
|| || 0i jp p− =
TrackingTracking || || 0ip pγ− =
Challenge: All these goals should be achieved simultaneously
|| || ,i j iq q d j N− ≈ ∀ ∈
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AlignmentAlignment-- Consensus ProtocolConsensus Protocol
i ip u=&
,i i
i i
q pp u==
&
& 1,...,i N=
iq Position
ip Velocity
Goal of Control:AlignmentAlignment || || 0i jp p− =
spatial adjacency matrix bump function
( )( )i
ij j ij N
a q p p∈
= −∑
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TrackingTracking-- Navigational feedbackNavigational feedback
1 2( ) ( )i i iu c q q c p pγ γ= − + −
Virtual leader:
( , )q p
p f q pγ γ
γ γ γ γ
=⎧⎨ =⎩
&
&
1 2, 0c c >
Navigational feedback:
,i i
i i
q pp u==
&
& 1,...,i N=
iq Position
ip Velocity
Goal of Control:
TrackingTracking || || 0ip pγ− =
Do not need any coupling!
What’s the price?
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Separation & CohesionSeparation & Cohesion
|| || ,i j iq q d j N− ≈ ∀ ∈
,i i
i i
q pp u==
&
& 1,...,i N=
iq Position
ip Velocity
Goal of Control:Separation & CohesionSeparation & Cohesion
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Separation & CohesionSeparation & CohesionArtificial Potential Function MethodArtificial Potential Function Method
( )i
i
i q ijj N
u qαψ∈
= − ∇∑{ , , 1,..., }i i jN j q q r j i j N− < ≠ =
( )ijqαψ
,i i
i i
q pp u==
&
& 1,...,i N=
iq Position
ip Velocity
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The Whole Flocking Algorithm
g di i i iu f f f γ= + +
1 22( ) ( )( ) ( ) ( )
1i i
j ii j i ij j i i i
j N j Nj i
q qu q q a q p p c q q c p p
q qα γ γσφ
ε∈ ∈
−= − + − + − + −
− −∑ ∑
Olfati-Saber, Flocking for Multi-Agent Dynamic Systems: Algorithms and Theory, IEEE Trans AC，2006
,i i
i i
q pp u==
&
&
Separation & Cohesion, Alignment, Tracking
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Initial positions are chosen randomly so that the initial net is highly disconnected. No. of edges increases and has a tendency to render the net connected.
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Origin of our ideaFlocking with minority of informed agents
Assumption in previous algorithm: Each agent is an informed agent
In contrast with some phenomena in the nature May be difficult to implement in engineering applications
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Two Nature Examples
A few informed individuals within a fish school are known to be able to influence the ability of the school to navigate towards a target
Only about 5% of the bees within a honeybee swarm can guide the group to a new nest site
lifts off to fly to a new nest site, only the scouts know in what direction the swarm must fly
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1 22( ) ( ) ( ) ( )
1i i
j ii j i j i i i
j N j Nj i
q qu q q p p c q q c p p
q qα γ γσφ
ε∈ ∈
−= − + − + − + −
− −∑ ∑
Informed agent
Flocking with minority of informed agents
2( ) ( )
1i i
j ii j i j i
j N j Nj i
q qu q q p p
q qα σφ
ε∈ ∈
−= − + −
− −∑ ∑Uninformed agent
( , )q pγ γ
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Our interests: behavior of the group when only a small fraction of agents are informed agents.
Flocking with minority of informed agents
Challenges:
An informed agent not only is influenced by the virtual leader but might also be influenced by some uninformed agents.
It’s not obvious that an informed agent will track the virtual leader, not to mention those uninformed agents.
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Our interests: behavior of the group when only a small fraction of agents are informed agents.
Flocking with minority of informed agents
Our contributions:
Theory: Not only all informed agents but also some uninformed agents will DO track the virtual leader.
Simulation: majority of uninformed agents will INDEED track the virtual leader.
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Suppose that the initial energy is finite. 0Q
(1) The distance between any informed agent and the virtual leader is not larger than for all 0 12 /Q c 0t ≥
Cohesive of Informed Agents
1 22( ) ( ) ( ) ( )
1i i
j ii j i j i i i
j N j Nj i
q qu q q p p c q q c p p
q qα γ γσφ
ε∈ ∈
−= − + − + − + −
− −∑ ∑
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(2) All informed agents asymptotically move with the desired velocity . pγ
Velocity Matching of Informed Agents
1 22( ) ( ) ( ) ( )
1i i
j ii j i j i i i
j N j Nj i
q qu q q p p c q q c p p
q qα γ γσφ
ε∈ ∈
−= − + − + − + −
− −∑ ∑
How about the uniformed agents?
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Assumption: For an uninformed agent in the group, assume that there always exist a path of finite length between the uninformed agent and one informed agent.
2( ) ( )
1i i
j ii j i j i
j N j Nj i
q qu q q p p
q qα σφ
ε∈ ∈
−= − + −
− −∑ ∑
Cohesive & Velocity Matching of Uninformed Agents
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0 1 02 / ( )Q c N M r+ −
pγ
(3) The distance between the uninformed agent and the virtual leader is not larger than
(4) The uninformed agent asymptotically moves with the desired velocity
2( ) ( )
1i i
j ii j i j i
j N j Nj i
q qu q q p p
q qα σφ
ε∈ ∈
−= − + −
− −∑ ∑
Cohesive & Velocity Matching of Uninformed Agents
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Simulation Results: N=100, M0=10
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10−2
10−1
100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
δ
P
N=100N=300N=500N=1000
the fraction of agents that eventually move with the desired velocity
the fraction of randomly chosen informed agentsAll estimates are the results of averaging over 50 realizations.For any given group size N, P is an increasing function of δ.The larger the group, the smaller the δ is needed to guide the group with a given P. Example: In order for 80% of the agents to move with the same desired velocity.For sufficiently large groups, only a very small fraction of informed agents will guide most agents in the group.
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Conclusion
A very small group of informed agents can cause most of the agents to move with the desired velocity.
Help to understand flocking behaviors in the nature
Provide a framework for guiding the design of engineering multi-agent systems.
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