Persistence Acquisition and Maintenance for Autonomous Formations

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Persistence Acquisition and Maintenance
forAutonomous Formations

• Brad C. YU
• National 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
• whats 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
remove 3 edges
Not rigid
Minimally rigid Keep formation rigid with
minimal number of edges
remove 3 edges
There are ways of checking 2D rigidity
graphically or using linear algebra
Rigid
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Rigidity notion is insufficient in directed case
But, if 3 moves, 4 is unable to react
B is rigid.
A
NOT RIGID
? Rigidity insufficient because

• Essentially undirected notion

B
X
??
So, need to take direction constraints into
account in addition to distance constraints
C
Directed distance constraints
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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

Rig. NO C.C. YES
A
Rig. YES C.C. NO
B
Persistence Rigidity C. Consistence
Rig. YES C.C. YES
C
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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 gt D
Obtained graph not rigid ? not persistent
Examples (D2)
Graph remains persistent
Initial graph was not persistent
Persistence Test A graph is persistent iff all
subgraphs obtained by removing edges leaving
vertices with d gt D until all vertices have d
lt 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 agents autonomy in its movement
• An agents 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
2



3D
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Future Work

• Practical
• 1gt Obtain actual control laws to keep distance
  effectively constant
• 2gt Relax the (highly) abstracted Point-Agent to
  one with orientation and/or dimensionality and/or
  shape
• Theoretical
• 3gt Find solutions to direction assignment for
  general graphs
• 4gt Characterize formation robustness

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Link Loss and/or Agent Loss
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Thank You

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

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