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Distributed Control of Multiple Vehicle Systems

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Title: 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 Stanfor
d 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 (gt2) aircraft?

13
Directed Graph Example of FMS
B S E P(AF) P(AT)
F F F 0.99 0.01
T F F 0.3 0.7
F T F 0.2 0.8
F F T 0.15 0.85
T T F 0.1 0.9
T T T 0.01 0.99
T F T 0.05 0.95
F T T 0.03 0.97
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

V_cruise 63 m/sec
V_minimum 123 m/sec
V_maximum 90.6 m/sec
W_maximum 1.2 deg/sec
15
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
t0.2 min.
t0.4 min.
t0.6 min.
t0.8 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
C10.7 C20.8 C30.6 C40.9
30
Example 4 Vehicle Coordination
Local optimization given the constraint
information set xj,yj,uji
  • 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
Dynamic Horizon
  • Global decreasing trend for
  • total coordination cost
  • constraint violation
  • Pointwise optimal control law is easily
    outperformed

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 vehicles
    solution set

42
Higher Dimensions
  • 3 Dimensional Perspective
  • The tubes represent 2.5 km radius safety zones
  • Xkm Ykm Timemin

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
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