1. Decision Making Under Uncertainty PI Meeting - June 20, 2001 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. Distributed Control of Multiple Vehicle Systems Motivation

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

4. Distributed Control of Multiple Vehicle Systems Background: Multiple Aircraft Maneuvers Safe if …

5. Distributed Control of Multiple Vehicle Systems A Simple Protocol Case 1: Case 2: Case 3: Case 4: Case 5: Case 6:

6. Distributed Control of Multiple Vehicle Systems 3 aircraft collision avoidance

7. Distributed Control of Multiple Vehicle Systems 10 aircraft collision avoidance

8. Distributed Control of Multiple Vehicle Systems Robust to Uncertainties in Position However, current protocol is centralized, not robust to communication uncertainty

9. Distributed Control of Multiple Vehicle Systems Game Theoretic Approach

10. Distributed Control of Multiple Vehicle Systems Analytic Computation of Blunder Zone

11. Distributed Control of Multiple Vehicle Systems Sample Trajectories Segment 1 Segment 2 Segment 3

12. Distributed Control of Multiple Vehicle Systems Application to Formation Flight possible for a two aircraft system what about multiple (>2) aircraft?

13. Distributed Control of Multiple Vehicle Systems Directed Graph Example of FMS BSEP(A=F)P(A=T) FFF0.990.01 TFF0.30.7 FTF0.20.8 FFT0.150.85 TTF0.10.9 TTT0.010.99 TFT0.050.95 FTT0.030.97 Continuous behavior?

14. Distributed Control of Multiple Vehicle Systems Aircraft motion is presented with hybrid modes Provides a basis for embedding discrete decisions, finite dimensional optimization, discrete state propagation Reachability algorithms Hybrid Model of Aircraft V_cruise63 m/sec V_minimum123 m/sec V_maximum90.6 m/sec W_maximum1.2 deg/sec

15. Distributed Control of Multiple Vehicle Systems Hybrid Model of Aircraft Continuous dynamics – planar kinematic model Our examples: hybrid model with five flight modes i th vehicle k th step optimal variable

16. Distributed Control of Multiple Vehicle Systems Example (continued) Motion of Vehicle 1 Motion of Vehicle 2

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

18. Distributed Control of Multiple Vehicle Systems 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 Cost (from desired) vs. Mode Selection

19. Distributed Control of Multiple Vehicle Systems Red SAFE Blue UNSAFE Matrix Game Structure for Hybrid Modes

20. Distributed Control of Multiple Vehicle Systems No safe mode for vehicle #1 for every mode selection of vehicle #2 No safe mode for vehicle #2 for every mode selection of vehicle #1 Coordination is needed

21. Distributed Control of Multiple Vehicle Systems Dynamic Coordination Problem

22. Distributed Control of Multiple Vehicle Systems Local to the i th Vehicle Local optimization by i th vehicle based on global information set D i Group optimization by k th vehicle based on information set S i

23. Distributed Control of Multiple Vehicle Systems Decentralized Optimization

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

25. Distributed Control of Multiple Vehicle Systems LOCAL HAMILTONIAN LOCAL DECENTRALIZED OPTIMAL Perspective of the i th vehicle

26. Distributed Control of Multiple Vehicle Systems 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. Distributed Control of Multiple Vehicle Systems GLOBAL COORDINATION PROBLEM GLOBAL LAGRANGIAN CONDITION FOR CENTRALIZED GLOBAL OPTIMALITY Global Perspective

28. Distributed Control of Multiple Vehicle Systems 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 Nash Equilibrium

29. Distributed Control of Multiple Vehicle Systems C 1 =0.7C 2 =0.8C 3 =0.6C 4 =0.9 Example: 4 Vehicle Coordination

30. Distributed Control of Multiple Vehicle Systems 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) Local optimization given the constraint “information set”: {x j,y j,u j } i Example: 4 Vehicle Coordination

31. Distributed Control of Multiple Vehicle Systems Approximate Penalty Function: Exact Penalty Function: Penalty Methods

32. Distributed Control of Multiple Vehicle Systems State propagation and safety constraints are naturally embedded in the cost function Global Optimization

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

34. Distributed Control of Multiple Vehicle Systems Example 1

35. Distributed Control of Multiple Vehicle Systems Example 1

36. Distributed Control of Multiple Vehicle Systems Example 2

37. Distributed Control of Multiple Vehicle Systems Example 2

38. Distributed Control of Multiple Vehicle Systems Iteration Results

39. Distributed Control of Multiple Vehicle Systems Dynamic Horizon Pointwise optimal control law is easily outperformed Global decreasing trend for –total coordination cost –constraint violation

40. Distributed Control of Multiple Vehicle Systems Example: Multiple Vehicle Mission Design

41. Distributed Control of Multiple Vehicle Systems 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 Multiple Vehicle Mission Design

42. Distributed Control of Multiple Vehicle Systems 3 Dimensional Perspective –The tubes represent 2.5 km radius safety zones –X[km] * Y[km] * Time[min] Higher Dimensions

43. Distributed Control of Multiple Vehicle Systems Testbed #2: Stanford DragonFly Test Platform Aircraft DragonFly AircraftNew Airframe

44. Distributed Control of Multiple Vehicle Systems DragonFly Avionics Single-board Computer GPS board IMU Air-Data Probe TsTs TsTs TsTs TcTc Control Command Actuator Control Computer Servo Control Vehicle Control Navigation Path Planning Path Planning Data Logging Data Logging Communication Communication … … …

45. Distributed Control of Multiple Vehicle Systems Software Architecture

46. Distributed Control of Multiple Vehicle Systems 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