1. 1 Persistence Acquisition and Maintenance for Autonomous Formations Brad C. YU National ICT Australia Limited The Australian National University With Baris Fidan & Brian D.O. Anderson

2. 2 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 ….

3. 3 Outline Introduction to Rigid Formation Control Rigid  Persistent (Acquiring Persistence) Maintaining Persistent Formation Conclusion

4. 4 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

5. 5 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

6. 6 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

7. 7 Rigidity notion is insufficient in directed case 1 3 2 4 A B C 1 3 2 4 1 3 2 4 NOT 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.

8. 8 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. YES C.C. NO Rig. YES C.C. YES A B C 1 3 2 41 3 2 41 3 2 4

9. 9 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

10. 10 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

11. 11 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

12. 12 Acquiring Persistence Circle Formations, C Sensing Radius of one agent doubled, Two new edges established

13. 13 Acquiring Persistence for Circle Formations(C 2 ) For all agents of C, let sensing radius be doubled, one obtains C 2 graph

14. 14 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, * *** ** *

15. 15 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) 2 3 1 4 5 Ok! Join us ** * *** 3D

16. 16 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

17. 17 Link Loss and/or Agent Loss

18. 18 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.