Distributed Control of Multiple Vehicle Systems

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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
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Motivation
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Application Areas

• Aviation surveillance / imaging
• Search / Rescue / Disaster relief
• Precision Agriculture
• Environmental Control Monitoring

• UCAV Fleets
• Communication Relays
• Remote sensing / distributed data acquisition

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Background Multiple Aircraft Maneuvers
Safe if
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A Simple Protocol
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
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3 aircraft collision avoidance
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10 aircraft collision avoidance
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Robust to Uncertainties in Position

• However, current protocol is centralized, not
  robust to communication uncertainty

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Game Theoretic Approach
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Analytic Computation of Blunder Zone
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Sample Trajectories
Segment 2
Segment 3
Segment 1
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Application to Formation Flight

• possible for a two aircraft system
• what about multiple (gt2) aircraft?

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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?
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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
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Hybrid Model of Aircraft

• Continuous dynamics planar kinematic model

• Our examples hybrid model with five flight modes

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Example (continued)
Motion of Vehicle 1
Motion of Vehicle 2
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Trees of Possible Locations for each Vehicle
Vehicle 1
Vehicle 2
t0.2 min.
t0.4 min.
t0.6 min.
t0.8 min.
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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

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Matrix Game Structure for Hybrid Modes
Blue UNSAFE
Red SAFE
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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
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Dynamic Coordination Problem
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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
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Decentralized Optimization
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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)
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Perspective of the ith vehicle
LOCAL HAMILTONIAN
LOCAL DECENTRALIZED OPTIMAL
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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

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

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Example 4 Vehicle Coordination
C10.7 C20.8 C30.6 C40.9
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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)

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Penalty Methods

• Approximate Penalty Function

• Exact Penalty Function

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Global Optimization

• State propagation and safety constraints are
  naturally embedded in the cost function

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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
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Example 1
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Example 1
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Example 2
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Example 2
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Iteration Results
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Dynamic Horizon

• Global decreasing trend for
• total coordination cost
• constraint violation

• Pointwise optimal control law is easily
  outperformed

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Example Multiple Vehicle Mission Design
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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

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Higher Dimensions

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

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Testbed 2 Stanford DragonFly Test Platform
DragonFly Aircraft
New Airframe
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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

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Software Architecture
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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