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Persistence Acquisition and Maintenance for

Autonomous Formations

Brad C. YUNational ICT Australia Limited

The Australian National University

With Baris Fidan & Brian D.O. Anderson

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Aim

• To provide basic concepts about rigid formation control

what’s a rigid formation?How to keep it rigid?

• To stimulate the interest of applying graph theory in control

systems modeled by graphs….

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Outline

• Introduction to Rigid Formation Control• Rigid Persistent (Acquiring Persistence)

• Maintaining Persistent Formation• Conclusion

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Introduction to Rigid Formation Control

• Many Control Tasks exist for Multiagent Systems

In particular, we looked at Preserving Rigid Formation (the shape)

during a continuous move

Tools:Graph Theory

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Control Scenarios

• Goal: To maintain a formation shape during a continuous move (i.e. To preserve all the inter-agent distances)

• Method: maintaining certain inter-agent distances

• Distance between agent X and Y may be maintained

– Jointly by X and Y: modelled by undirected graphs, rigid graph theory applicable.

– Unilaterally by X : modelled by directed graphs. Need to validate or modify all rigidity type questions and theories.

Motivated us to develop Persistence Framework

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Rigidity and Minimal Rigidity

Fully connected, Rigid

Minimally rigid = Keep formation rigid with minimal number of edges

remove 3 edges

Not rigid

remove 3 edges

Rigid

There are ways of checking 2D rigidity graphically or using linear

algebra

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Rigidity notion is insufficient in directed case

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4NOT RIGID

So, need to take direction constraints into account in addition to distance constraints

But, if 3 moves, 4 is unable to react

??

Rigidity insufficient because

•Essentially undirected notion

Directed distance constraints

X

B is rigid.

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Persistence

• Rigidity: “All constraints satisfied structure preserved”

• Constraint Consistence: “Every agent tries to satisfy all its constraints all the constraints are satisfied”

• Persistence: “Every agent tries to satisfy all its constraints structure preserved”

Persistence ||

Rigidity + C. Consistence

Rig. NO C.C. YES

Rig. YESC.C. NO

Rig. YESC.C. YES

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Characterization of persistence

A persistent graph in D dimensions (D = 2 or 3) remains persistent after deletion of an edge leaving a vertex with out-degree > D

Examples (D=2) :

Graph remains persistent

Obtained graph not rigid not persistent

Initial graph was not persistent

Persistence Test: A graph is persistent iff all subgraphs obtained by removing edges leaving vertices with d+ > D until all vertices have d+ <= D are rigid

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From rigidity to persistence

• Rigid formations Persistent formations

Why?• This exercise reduces control complexity by notably

half.• Simpler communication protocol (one-way sensing)

Question:• What are the rules of assigning directions

(asymmetric control structure) to establish persistence from rigidity?

– No solution for general graphs.– We consider several special classes of graphs

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Acquiring Persistence for Wheel Formations

NOT Persistent

Persistent

The red agents are overloaded with 3 constraints, apply persistence test by removing edges, resulting in a non-rigid graph

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Acquiring Persistence Circle Formations, C

Sensing Radius of one agent doubled,

Two new edges established

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Acquiring Persistence for Circle Formations(C2)

For all agents of C, let sensing radius

be doubled, one obtains C2 graph

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Maintaining Persistent Formation

• DOF , denoted as in the following, is an abstraction of agent’s autonomy in its movement

• An agent’s DOF defines its “role” in the formation

• Consider this 3D formation,

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Transfer of DOF

•Change of agents’ “role”(esp. leadership) of a formation may be required as part of mission plan, new agent carrying new mission maybe added as leader

•Transfer of DOF can be made via a general technique we developed for formation in arbitrary dimension (s)

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Ok!

Join us

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*** 3D

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Future Work

Practical

1> Obtain actual control laws to keep distance effectively constant

2> Relax the (highly) abstracted Point-Agent to one with orientation and/or dimensionality and/or shape

Theoretical

3> Find solutions to direction assignment for general graphs

4> Characterize formation robustness

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Link Loss and/or Agent Loss

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• On behalf of co-authors, I would like to acknowledge the contribution of J.M. Hendrickx and V.D. Blondel to the persistent framework.

Thank You

• I would like to thank the ISSNIP2005 committee for the Student Grant.