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Distributed Cooperative Control of Multiple Vehicle Formations Using Structural Potential Functions Reza Olfati-Saber Postdoctoral 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 2 nd, 2002

Outline • • • Introduction Multi-vehicle Formations Past Research Coordinated Tasks – Stabilization/Tracking – 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

Introduction Definition: Multi-agent Systems are systems that consist of multiple agents or vehicles with 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)

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

Past Research • Robotics: navigation using artificial potential functions (Rimon and Koditschek, 1992) • 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)

Why Distributed Control? • No vehicle knows the state/control of all other vehicles • 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.

What is a Formation? What is a Formation?

Formation Representation Formation Representation

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

Split/Rejoin Maneuvers Split/Rejoin Maneuvers

Operational Graph Operational Graph

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

Rigidity Remark: c (or d) is called a single mobility degree of freedom of Rigidity Remark: c (or d) is called a single mobility degree of freedom of the formation graph. Definition: A planar formation graph with n nodes and 2 n-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. 1 a d 2 a 3 c b b 4

Foldability 1 Definition: The following non-redundant a a d set of equations are called Foldability 1 Definition: The following non-redundant a a d set of equations are called structural constraints of a formation graph. 2 c b b Deviation Variable: 4 Definition: A rigid formation graph is foldable iff the set of structural constraints associated with the formation graph does not have a unique solution. 4 3

Node Orientation 3 1 2 3 Node Orientation 3 1 2 3

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

Potential Functions Potential Function: Force: Potential Functions Potential Function: Force:

Distributed Control Laws Hamiltonian : Potential Function : indices of the neighbors of Theorem(ROS-RMM-IFAC’ Distributed Control Laws Hamiltonian : Potential Function : 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.

Operational Graph Operational Graph

Split Maneuvers Split Maneuvers

Rejoin Maneuver Rejoin Maneuver

Reconfiguration I Reconfiguration I

Reconfiguration II Reconfiguration II

Tracking Tracking

Conclusions • Introducing a framework formal specification of unambiguous formation graphs of multi-vehicle systems Conclusions • Introducing a framework 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.