1. 1 Analysis of Autonomous Multiagent Formations From Rigidity to Persistence DSTO-NICTA Meeting 22 July 2005 Brad C. YU National ICT Australia Limited

2. 2 Outline Problem Description Recap on Rigidity Persistence Structural Persistence Summary and Future Work

3. 3 Problem What’s the right information and control structure and the right analysis model to maintain formation?

4. 4 Outline Problem Description Recap on Rigidity Persistence Structural Persistence Summary and Future Work

5. 5 Rigid Formations A formation is a collection of agents (point agents initially) in two or three dimensional space A formation is rigid if the distance between each pair of agents does not change over time Steady motion of a formation normally needs rigidity In a rigid formation, normally only some distances are explicitly maintained, with the rest being consequentially maintained. The distances ab,bc,cd,ad and ac are explicitly maintained and the distance bd is maintained as a result of the topology. a b c d Rigidity is an Undirected Notion!

6. 6 Scenarios of Control Undirected notion of rigidity requires peer-wise control, it’s a joint effort of both agents How do we deal with scenarios with leader-follower structure? LeaderFollower

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 B is rigid. But, if 3 moves, 4 is unable to react ??  Rigidity insufficient because Essentially undirected notion (although definition OK for directed graphs) Directed distance constraints

8. 8 Outline Problem Description Recap on Rigidity Persistence Structural Persistence Summary and Future Work

9. 9 Constraint Consistence (C. C.) This leads to new notion of “persistence” Intuitively, an agent CANNOT have too many directed distance constraints, i.e., constraints on the distances to be maintained from other agents. An agent is fitting if it satisfies all constraints in all cases A formation is C. C. if every agent is fitting Example: p(1) p(3) p(2) p(4) p’ fitting w.r.t. 1,2,3 But 4 is non-fitting  p not C.C. (although p rigid) p’(3) =p’(2) p’(4)= =p’(1)

10. 10 Rigidity to 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 1 3 2 4 1 3 2 4 1 3 2 4 Rig. NO C.C. YES Rig. YES C.C. NO Rig. YES C.C. YES A B C

11. 11 Half-way summary Rigidity has been studied extensively with aid of undirected graphs (distance constraints) Persistence is a recently developed notion, principally due to ourselves and collaborators, for analysis of formations with leader-follower type of control structure (directed distance constraints) We have developed systematic ways of checking persistence which involves repetitive tests for rigidity. What if we look at the formation structure as a whole??? (is it structurally persistent? ---- the new question)

12. 12 Outline Problem Description Recap on Rigidity Persistence Structural Persistence Summary and Future Work

13. 13 Two leaders ’ problem in 3D formation A persistent formation is NOT structurally persistent if it has two leaders. 2D persistent formation is always structurally persistent. 1 2 3 4 5 ?? A leader is an agent that has NO constraints on its movement. 1 and 2 are the leaders in this 3D formation

14. 14 Collections of autonomous vehicles (Point agents) Rigid Persistent, but two leader agents may create problem Persistent, and only one leader agent Persistence to Structural Persistence Rigid PersistentStruc Pers. Structural Persistence

15. 15 Analysis of Structural Persistence Our studies show that in contrast to the fact that Rigidity and persistence only deal with individual agent’s constraints collectively Structural persistence deals with a formation as a whole. It puts constraints on the allocation of “degrees of freedom”

16. 16 Allocation of DOF DOF is an abstraction of agent’s autonomy in its movement An agent’s DOF defines its “role” in the formation In 3D formation, no-constraint = 3 DOF = move freely in 3D = a leader one constraint = 2 DOF = move in sphere two constraints = 1 DOF = move in circle three constraints = no DOF = fixed (w.r.t formation) four or more constraints ??

17. 17 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 technique we developed for formation in arbitrary dimension (s) 2 3 1 4 5 Ok! Join us Sorry, I have too many guys to follow! Alright, I can give one DOF to you. Same here! Let’s lead together Let me follow 5 then

18. 18 Outline Problem Description Recap on Rigidity Persistence Structural Persistence Summary and Future Work

19. 19 A summary of results NotionHandlesWorks in RigidityDistance Constraints2D, 3D PersistenceDirected Distance Const.2D Stru. Pers.(as above)2D, 3D How to construct a structurally persistent formation? Tip: Henneberg Sequence What about the robustness of the formation?? Tip: Redundancy

20. 20 Future Work (cont’d) Practical 1> Formation health and quality measurement/assurance auto discovering, decentralized detection algorithms/procedures 2> Formation robustness, self-curing (closing ranks), splitting, merging formations, 3> Implementation/deployment related control issues (e.g., cycles in the graph = a loop of L-F, who is the leader?) Theoretical 4> Link graphical results to that of Linear Algebra 5> Extend/generalize results to arbitrary dimensions 6> Still many open problems around

21. 21 Future work The Grand Challenge: A rigorous analysis framework based on Graph Theory and Linear Algebra (Matroid) that works for both 2D and 3D formations in practical applications. With comprehensive results based on the framework in formation construction, manipulation, closing-ranks with guaranteed level of robustness, and at any time, the health and quality of such formation can be measured and determined at low computational complexity.

22. 22 Main References: [1]J. M. Hendrickx, B.D.O. Anderson, V. Blondel & J. –C. Delvenne, “ Directed graphs for the analysis of rigidity and persistence in autonomous agent systems”, submitted to International Journal of Robust Nonlinear Control, 2005 [2]J. M. Hendrickx, B. Fidan, C. YU, B.D.O. Anderson & V. Blondel, “ Rigidity and Persistence of Three and Higher Dimensional Directed Formations”, to appear in MARS05, Barcelona, Sep 2005 [3]C. YU, J. M. Hendrickx, B. Fidan & B.D.O Anderson, “Structural Persistence of Three Dimensional Autonomous Formations”, to appear in MARS05, Barcelona, Sep 2005 [4]C. YU, B. Fidan & B.D.O. Anderson, “Persistence Acquisition and Maintenance for Autonomous Formations”, Submitted to the 2nd International Conference on Intelligent Sensors, Sensor Networks and Information Processing, Melbourne, Dec 2005. Thank you