1. Distributed Cooperative Control of Multiple Vehicle Formations Using Structural Potential Functions Reza Olfati-Saber Postdoctoral Scholar Control and Dynamical Systems California Institute of Technology Olfati@cds.caltech.edu UCLA, March 2nd, 2002

2. Outline Introduction Multi-vehicle Formations Past Research Coordinated Tasks –Stabilization/Tracking –Rejoin/Split/Reconfiguration Maneuvers Why Distributed Control? Formation Graphs –Rigidity/Foldability of Graphs Potential Functions Distributed Control Laws Simulation Results Conclusions

3. Introduction Definition: Multi-agent Systems are systems that consist of multiple agents or vehicles with several sensors/actuators and the capability to communicate with one another to perform coordinated tasks. Applications: – Automated highways – Air traffic control – Satellite formations – Search and rescue operations – Robots capable of playing games (e.g. soccer/capture the flag) – Formation flight of UAV’s (Unmanned Aerial Vehicles)

4. Multi-Vehicle Formations A group of vehicles with a specific set of inter-vehicle distances is called a Multi-Vehicle Formation. Formation Stabilization Dynamics:

5. Past Research Robotics: navigation using artificial potential functions (Rimon and Koditschek, 1992) Multi-vehicle Systems: –Coordinated control of groups using artificial potentials (Leonard and Fiorelli, 2001) –Information flow on graphs associated with multi-vehicle systems (Fax and Murray, 2001)

6. Why Distributed Control? No vehicle knows the state/control of all other vehicles No vehicle knows its relative configuration/velocity w.r.t. all other vehicles unless n = 2,3 The control law for each vehicle must be distributed so that the overall computational complexity of the problem is acceptable for large number of vehicles A system controlled via a centeralized computer does not function if that computer breaks.

7. What is a Formation?

8. Formation Representation

9. Coordinated Tasks attitude Tracking Trajectory Rejoin Split Reconfiguration Diamond Formation Delta Formation Two Formations One Formation Switching

10. Split/Rejoin Maneuvers

11. Operational Graph

12. Formation Graphs 1 2 3 4 ii) measures iii) knows its desired distance to must be an Edge means i) is a neighbor of Formation Graph: Connectivity Matrix Distance Matrix Set of Vertices

13. Rigidity 23 4 1 Definition: A planar formation graph with n nodes and 2n-3 critical links is called a rigid formation graph. Definition: A critical link is a link that eliminates a mobility degree of freedom of a multi-body system. aa bb c d Remark: c (or d) is called a single mobility degree of freedom of the formation graph. c

14. Foldability 23 4 1 Definition: A rigid formation graph is foldable iff the set of structural constraints associated with the formation graph does not have a unique solution. Definition: The following non-redundant set of equations are called structural constraints of a formation graph. aa bb c d c 4 Deviation Variable:

15. Node Orientation 1 2 3 1 2 3

16. Unambiguous FG’s Definition: A formation graph is called unambiguous if it is both rigid and unfoldable. 1 2 3 4 5 6 1 2 3 4 5 6 7

17. Potential Functions Potential Function: Force:

18. Distributed Control Laws Potential Function : Hamiltonian : indices of the neighbors of Theorem(ROS-RMM-IFAC’02): The following state feedback is a gradient-based bounded and distributed control law that achieves collision-free local asymptotic stabilization of any unambiguous desired formation graph.

19. Operational Graph

20. Split Maneuvers

21. Rejoin Maneuver

22. Reconfiguration I

23. Reconfiguration II

24. Tracking

25. Conclusions Introducing a framework for formal specification of unambiguous formation graphs of multi-vehicle systems that is compatible with formation control. Providing a Lyapunov function and a bounded and distributed state feedback that performs coordinated tasks such as formation stabilization/tracking, split/rejoin, and reconfiguration maneuvers. Introducing a Hybrid System that represents split, rejoin, and reconfiguration maneuvers in a unified framework as a discrete-state transition where each discrete-state is an unambiguous formation graph.