Distributed Control of Multiple Vehicle Systems

1. Distributed Control of Multiple Vehicle Systems Claire Tomlin and Gokhan Inalhan with Inseok Hwang Rodney Teo and Jung Soon Jang Department of Aeronautics and Astronautics Stanford University

2. Motivation

3. Application Areas • Aviation surveillance / imaging • Search / Rescue / Disaster relief • Precision Agriculture • Environmental Control & Monitoring • UCAV Fleets • Communication Relays • Remote sensing / distributed data acquisition

4. Background: Multiple Aircraft Maneuvers Safe if …

5. A Simple Protocol Case 1: Case 2: Case 3: Case 4: Case 5: Case 6:

6. 3 aircraft collision avoidance

7. 10 aircraft collision avoidance

8. Robust to Uncertainties in Position • However, current protocol is centralized, not robust to communication uncertainty

9. Game Theoretic Approach

10. Analytic Computation of Blunder Zone

11. Sample Trajectories Segment 2 Segment 3 Segment 1

12. Application to Formation Flight • possible for a two aircraft system • what about multiple (>2) aircraft?

13. Directed Graph Example of FMS Continuous behavior?

14. Hybrid Model of Aircraft • Aircraft motion is presented with hybrid modes • Provides a basis for embedding discrete decisions, finite dimensional optimization, discrete state propagation • Reachability algorithms

15. ith vehicle optimal kth step variable Hybrid Model of Aircraft • Continuous dynamics – planar kinematic model • Our examples: hybrid model with five flight modes

16. Example (continued) Motion of Vehicle 1 Motion of Vehicle 2

17. Trees of Possible Locations for each Vehicle Vehicle 1 Vehicle 2 t=0.2 min. t=0.4 min. t=0.6 min. t=0.8 min.

18. Cost (from desired) vs. Mode Selection • Mode Sequence 245 (base ten) : 1-4-4-0 (base five) Vmax for 0.1 min; Left Turn for 0.1 min; Left Turn for 0.1 min; Vcruise for 0.1min

19. Matrix Game Structure for Hybrid Modes Blue UNSAFE Red SAFE

20. Coordination is needed No safe mode for vehicle #2 for every mode selection of vehicle #1 No safe mode for vehicle #1 for every mode selection of vehicle #2

21. Dynamic Coordination Problem

22. Local to the ith Vehicle Local optimization by ith vehicle based on global information set Di Group optimization by kth vehicle based on information set Si

23. Decentralized Optimization

24. Local Optimization by each Vehicle LOCAL COORDINATION PROBLEM local cost function inter-vehicular constraints Individual state propagation inter-vehicular constraints Local vehicle constraints local information set (neighborhood)

25. Perspective of the ith vehicle LOCAL HAMILTONIAN LOCAL DECENTRALIZED OPTIMAL

26. Result • Our iterative algorithm based on local decentralized optimization converges to a global decentralized optimal solution thus at each iteration As L is bounded below by zero, convergence is guaranteed

27. Global Perspective GLOBAL COORDINATION PROBLEM GLOBAL LAGRANGIAN CONDITION FOR CENTRALIZED GLOBAL OPTIMALITY

28. Nash Equilibrium • The global decentralized optimal solution corresponds to a Nash Equilibria of the centralized optimization problem for an M-player game with each player cost function corresponding to • and the constraints to

29. Example: 4 Vehicle Coordination

30. Example: 4 Vehicle Coordination Local optimization given the constraint “information set”: {xj,yj,uj}i • Each aircraft penalizes its own deviation from its desired flight path subject to • Minimum safety constraints (penalty functions) • Aircraft dynamics and flight modes (state propagation)

31. Penalty Methods • Approximate Penalty Function: • Exact Penalty Function:

32. Global Optimization • State propagation and safety constraints are naturally embedded in the cost function

33. Testbed #1: Networked Simulation Local Control Process Aircraft # 1 Aircraft # 3 RBNB Matlink Client/Server Layer TCP-IP TCP-IP TCP-IP TCP-IP Aircraft # 4 Aircraft # 2 RBNB Server TCP-IP TCP-IP WORLD MODEL

34. Example 1

35. Example 1

36. Example 2

37. Example 2

38. Iteration Results

39. Pointwise optimal control law is easily outperformed Global decreasing trend for total coordination cost constraint violation Dynamic Horizon

40. Example: Multiple Vehicle Mission Design

41. Multiple Vehicle Mission Design • Decentralized Initialization Procedure Heuristics • Multiple-Depots(Vehicles), Time-windows for access, Priority on objectives and the vehicles • Iterative selection process carried via each vehicle • Best solution then selected from each vehicle’s solution set

42. Higher Dimensions • 3 Dimensional Perspective • The tubes represent 2.5 km radius safety zones • X[km] * Y[km] * Time[min]

43. Testbed #2: Stanford DragonFly Test Platform DragonFly Aircraft New Airframe

44. DragonFly Avionics Actuator Control Computer Single-board Computer GPS board Tc Ts Control Command IMU Servo Control Ts • Vehicle Control • Navigation • Path Planning • Data Logging • Communication • … Air-Data Probe Ts …

45. Software Architecture

46. Directions • Application of algorithm directly to probabilistic hybrid models (Koller) • Numerical implementation issues (Saunders) • Evolution of the algorithm in a dynamic environment (connect operator) • Dynamic visitation problems