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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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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
A
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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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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.