Validating a HamiltonJacobi Approximation to Hybrid System Reachable Sets

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(No Transcript)
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Motivating applications
(Source Boeing X45-A)
(Source Northrop Grumman-X47A)
(Source NASA Ames)
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Hybrid systems

• Continuous systems controlled by a discrete
  logic embedded systems (autopilot logic)
• Coordinating processes multi-vehicle systems
  interfacing continuous control with coordination
  protocols
• Continuous systems with a phased operation
  (biological cell growth and division)

discrete systems (computer science)
continuous systems (control)
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Verification and Controller Synthesis

• Verification a mathematical proof that the
  system satisfies a property
• Controller synthesis the design of control laws
  to guarantee that the system satisfies the
  property
• Methods give definitive answers, unlike
  simulation
• Often give surprising answers, trajectories which
  one might not think to simulate
• Reduces development time, cost of certification

initial
unsafe
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Verification and Controller Synthesis

• Verification a mathematical proof that the
  system satisfies a property
• Controller synthesis the design of control laws
  to guarantee that the system satisfies the
  property
• Methods give definitive answers, unlike
  simulation
• Often give surprising answers, trajectories which
  one might not think to simulate
• Reduces development time, cost of certification

initial
unsafe
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Verification and Controller Synthesis

• Verification a mathematical proof that the
  system satisfies a property
• Controller synthesis the design of control laws
  to guarantee that the system satisfies the
  property
• Methods give definitive answers, unlike
  simulation
• Often give surprising answers, trajectories which
  one might not think to simulate
• Reduces development time, cost of certification

initial
unsafe
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Verification and Controller Synthesis

• Verification a mathematical proof that the
  system satisfies a property
• Controller synthesis the design of control laws
  to guarantee that the system satisfies the
  property
• Safety Property can be encoded as a condition on
  the systems reachable set of states

initial
unsafe
unsafe
unsafe initialization
safe, under appropriate control
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Example Aircraft Collision Avoidance

• Two identical aircraft at fixed altitude speed

y
v
y
u
x
v
d
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Continuous Reachable Set
Solve
Display
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Collision Avoidance Filter

• Simple demonstration
• Pursuer turn to head toward evader
• Evader turn to head right

Movies
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Blunder Zones for Closely Spaced Approaches
EEM Maneuver 1 accelerate
EEM Maneuver 2 turn 45 deg, accelerate
EEM Maneuver 3 turn 60 deg
evader
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Implementation Display design courtesy of
Chad Jennings, Andy Barrows, David Powell
Blunder Zone is shown by the yellow contour Red
Zone in the green tunnel is the intersection of
the BZ with approach path. The Red Zone
corresponds to an assumed 2 second pilot delay.
The Yellow Zone corresponds to an 8 second pilot
delay
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Map View showing a blunder The BZ calculations
are performed in real time (40Hz) so that the
contour is updated with each video frame.
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Verified Mode Switching in Autopilots
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Use in Cockpit Interface Verification

• Controllable flight envelopes for landing and
  Take Off / Go Around (TOGA) maneuvers may not be
  the same
• Pilots cockpit display may not contain
  sufficient information to distinguish whether
  TOGA can be initiated

existing interface
controllable TOGA envelope
intersection
revised interface
controllable flare envelope
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A More General Problem Structure
Communication Zone
Safety Assurance Zone
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(Decomposed) Centralized Optimization
Neighborhood of ith vehicle
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fixed time horizon
Bargaining start
Fixed time horizon complete global map
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Flight Plans published by aircraft 1
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Another Example
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Flight Plans published by aircraft 1
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moving time horizon
Bargaining start
Receding horizon incomplete global map
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Local Optimization with Constraints

• Constraints embed
• local dynamics coordinated turn and straight
  flight hdi
• input constraints limited turn rate and
  velocity gei
• global coordination constraints minimum safety
  assurance gsij for all j within neighborhood of
  i

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Decomposition I
Centralized Optimization
Decomposed Centralized Optimization
Pareto optimality Nash equilibrium
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Nash Equilibrium for Centralized Problem

• Define Hamiltonian for each subsystem
• is a Nash equilibrium for the centralized
  optimization problem if
• where
• Thus, none of the subsystems can improve its
  solution, with all other subsystems solutions
  remaining fixed.

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Decomposition II
Decomposed Centralized Optimization
Decentralized Optimization
Nash equilibrium Local optimal solutions
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Nash Equilibrium for Decentralized Problem

• Define Hamiltonian for each subsystem
• is a Nash equilibrium for the decentralized
  optimization problem if
• Optimal solutions by each of the
  subsystems

Proposition is a Nash equilibrium of the
centralized problem if and only if it is a Nash
equilibrium of the decentralized problem
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Example Nash Equilibrium at (0,0)
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Using Penalty function methods

• Global contraction function from the local
  optimization structures

• For a particular solution, local optimization of
  the ith vehicle only affects the portion of F
  tied to its own local optimization

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Cooperation Assumptions

• Eliminates cases in which a subsystem is
  artificially acting against a constraint dictated
  by another group

• Eliminates cases in which two subsystems act
  against each other with non-identical constraints

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Nash Bargaining with Multiple Threads

• Multiple solutions, or threads, exist within
  the system

Vehicle 1
Vehicle 2
Vehicle 3
Vehicle 4
Iteration 1
Iteration 2
Iteration 3
Iteration 4
Iteration 4
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Convergence Results

• Global convergence to a (not necessarily
  feasible) Nash solution
• If the gradients of the constraint functions are
  linearly independent (Linear Constraint
  Qualification Condition, LICQ), then global
  convergence to a feasible Nash solution
• Pareto optimality for convex problems

Inalhan, Stipanovic, Tomlin. Decentralized
Optimization, with Application to Multiple
Aircraft Coordination. CDC 2002, Submitted to
JOTA
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4-Vehicle Example
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4-Vehicle Example
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Flight Plans published by aircraft 1
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Applied to other problems of interest

• Decentralized Initialization Procedure Heuristics
• Multiple-Depots (Vehicles), Time-windows for
  access, Priority on objectives and the vehicles
• Iterative selection process carried at each
  vehicle
• Best solution in the fleet is then selected from
  each vehicles solution set

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Spectrum of Approaches
Lack of information ?? Bounded Irrationality
Cooperative incomplete information
Non-cooperative Full information
Cooperative Full information
Non-cooperative No information
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Research Goals

• Design of provably correct and safe decentralized
  control protocols
• Adapt to coordination
• Allow for dynamic reconfiguration
• Treatment of information
• Multi-scale provisioning of data based on inputs
  from various sensing modalities
• Urgency of the need for sensed data
• Available bandwidth
• Verification algorithms
• Used during design phase, to reduce the time
  spent during the validation phase

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Stanford DragonFly UAV
10 ft wingspan
12 ft wingspan
Jang, Teo, Inalhan, and Tomlin, DASC 2001,
Jang and Tomlin, AIAA GNC 2002