Transcript Slide 1
Optimization
Based
Approaches to
Autonomy
SAE Aerospace Control and Guidance Systems
Committee (ACGSC) Meeting
Harvey’s Resort, Lake Tahoe, Nevada
March 3, 2005
Cedric Ma
Northrop Grumman Corporation
Outline
Introduction
Level of Autonomy
Optimization and Autonomy
Autonomy Hierarchy and Applications
Path Planning with Mixed Integer Linear Programming
Optimal Trajectory Generation with
Nonlinear Programming
Summary and Conclusions
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Autonomy in Vehicle Applications
PACK LEVEL COORDINATION
TEAM
TACTICS
NAVIGATION
LANDING
FORMATION FLYING
RENDEZVOUS & REFUELING
COOPERATIVE SEARCH
OBSTACLE AVOIDANCE
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Autonomy: Boyd’s OODA “Loop”
Observe
Orient
Implicit
Guidance
& Control
Unfolding
Circumstances
Observations
Feed
Forward
Genetic
Heritage
Unfolding
Interaction
With
Environment
Act
Implicit
Guidance
& Control
Cultural
Traditions
Analyses &
Synthesis
New
Information
Outside
Information
Decide
Feed
Forward
Decision
(Hypothesis)
Previous
Experience
Feedback
Feed
Forward
Action
(Test)
Unfolding
Interaction
With
Environment
Feedback
Note how orientation shapes observation, shapes decision, shapes action, and in turn is shaped by the feedback and
other phenomena coming into our sensing or observing window.
Also note how the entire “loop” (not just orientation) is an ongoing many-sided implicit cross-referencing process
of projection, empathy, correlation, and rejection.
From “The Essence of Winning and Losing,” John R. Boyd, January 1996.
Defense and the National Interest, http://www.d-n-i.net, 2001
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Level of Autonomy
Ground Operation
Activities performed off-line
Tele-Operation
Awareness of sensor / actuator interfaces
Executes commands uploaded from the
ground
1 Ground operation Reactive Control
Awareness of the present situation
2 Tele-operation
Simple reflexes, i.e. no planning required
3 Reactive Control
A condition triggers an associated action
4 Responsive Control
5 Deliberative Control Responsive Control
Awareness of past actions
Remembers previous actions
Remembers features of the environment
Remembers goals
Goal of Optimization
Deliberative Control
Based Autonomy
Awareness of future possibilities
Reasons about future consequences
Chooses optimal paths / plans
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Optimization and Autonomy
Observe
Orient
Implicit
Guidance
& Control
Unfolding
Circumstances
Observations
Feed
Forward
Genetic
Heritage
Unfolding
Interaction
With
Environment
Act
Implicit
Guidance
& Control
Cultural
Traditions
New
Information
Outside
Information
Decide
Analyses &
Synthesis
Feed
Forward
Decision
(Hypothesis)
Previous
Experience
Feedback
Feedback
Formulation of problem
shapes the “Orient” mechanism
Feed
Forward
Action
(Test)
Unfolding
Interaction
With
Environment
Objective/Reward Function
Constraints/Rules
(i.e. Dynamics/Goal)
Vehicle
State
Optimizer
Optimal
Control/Decision
Determines best course of action based on
current objective, while meeting constraints
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Autonomy Hierarchy
Mission
Planning
Planning & Scheduling, Resource Allocation & Sequencing
Task Sequencing, Auto Routing
Time Scale: ~1 hr
Cooperative
Control
Multi-Agent Coordination, Pack Level Organization
Formation Flying, Cooperative Search & Electronic Warfare
Conflict Resolution, Task Negotiation, Team Tactics
Time Scale: ~1 min
“Navigation,” Motion Planning
Obstacle/Collision/Threat Avoidance
Time Scale: ~10s
Path
Planning
“Guidance,” Contingency Handling
Trajectory
Landing, Rendezvous, Refueling
Generation
Time Scale: ~1s
“Control,” Disturbance Rejection
Trajectory
Applications: Stabilization, Adaptive
Following/
Reconfigurable Control, FDIR
Inner Loop
Time Scale: ~0.1s
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Path Planning with
Mixed-Integer
Linear Programming
(MILP)
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Overview: Path Planning
Path Planning bridges the gap between Mission
Planner/AutoRouter and Individual Vehicle Guidance
Acts on an “intermediate” time scale between that of
mission planner (minutes) and guidance (<seconds)
Short reaction time
Mission waypoints
Nap-of-the-Earth Flight
Multi-vehicle Coordination
Terrain Navigation
Obstacle Avoidance
Collision Avoidance
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Path-Planning with MILP
Mixed-Integer Linear Programming
Linear Programs (LP) with integer variables
COTS MILP solver: ILOG CPLEX
Vehicle dynamics as linear constraints:
Limit velocity, acceleration, climb/turn rate
Resulting path is given to 4-D guidance
Integer variables can model:
Obstacle collision constraints (binary)
Control Modes, Threat Exposure
Nonlinear Functions: RCS, Dynamics
Min. Time, Acceleration, Altitude, Threat
Objective function includes terms for:
Acceleration, Non-Arrival, Terminal, Altitude,
Threat Exposure
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Basic Obstacle Avoidance Problem
Vehicle Dynamic Constraints
Double Integrator dynamics
Max acceleration
Max velocity
Objective Function (summed over each time step)
Acceleration (1-norm) in x, y, z
Distance to destination (1-norm)
Altitude (if applicable)
Obstacle Constraints (integer)
x – M b 1 ≤ x1
x + M b 2 ≥ x2
One set per obstacle per time step
y – M b3 ≤ y 1
No cost associated with obstacles
y+Mb ≥y
4
2
b1 + b2 + b3 + b4 ≤ 3
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Receding Horizon MILP Path-Planning
Path is computed periodically,
with most current information
Planning horizon, replan period
chosen based on problem type,
computational requirements, &
environment
Only subset of current plan is
executed before replanning
RH reduces computation time
Shorter planning horizon
Does not plan to destination
RH introduces robustness to
path planning
Pop-up obstacles
Unexpected obstacle movement
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Obstacle Avoidance
Nap of the Earth Flight
Treetop level
Urban Low Altitude
Operations
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Collision Avoidance
Problem is formulated identically
as Obstacle Avoidance in MILP
Air vehicles are moving obstacles
Path calculation based on
expected future trajectory of other
vehicles
Dealing with Uncertainty
Vehicles of uncertain intent can be
enlarged with time
Receding Horizon
Frequent replanning
Change in planned path (blue)
in response to changes in
intruder movement
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Coordinated Conflict Resolution
3-D Multi-Vehicle PathPlanning problem
Centralized version
“Decentralized Cooperative
Trajectory Planning of
Multiple Aircraft with Hard
Safety Guarantees” by MIT
Loiter maneuvers can be used
to produce provably safe
trajectories
Minimum separation distance
is specified in problem
formulation
No limit to number of vehicles
Non-cooperative vehicles are
treated as moving obstacles
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Threat Avoidance
Purpose: To avoid detection
by known threats by planning
trajectory behind opaque
obstacles
Shadow-like “Safe Zones”
One per threat/obstacle pair
Well defined for convex
obstacles
Nice topological properties
Vehicle hiding
behind building
On-time arrival
at destination
Patent Pending: Docket No.
000535-030
Threat
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Summary
MILP Path Planning
MILP: Fast Global Optimization
No suboptimal local minima
Branch & Bound provides fast
tree-search
Commercial solver on RTOS
Tractability Trade-off:
Time Discretization
Constraints active only at
discrete points in time
Time Scale Refinement
Linear dynamics/constraints
Formulation should properly
capture nonlinearity of solution
space
True global minimum is in a
neighborhood of MILP optimal
solution
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Optimal Trajectory
Generation with
Nonlinear
Programming
(NLP)
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Problems & Goal of Trajectory Generation
Currently, the primary method is pre-generated waypoint routes with
little/no adaptation or reaction to threats or condition changes
Even the latest vehicles have low autonomy levels and are doing exactly
what they are told, largely indifferent to the world around them
What are the potential gains of Near Real Time Trajectory Generation?
Improved Effectiveness
Reduced operator workload – force multiplier
Mission planning / re-planning
Account for range and time delays
Improved Survivability?
UAV trades success/risk
Limp-home capability
Autonomous threat mitigation
(RCS, SAM, Small Arms, AA Fire)
Air/Air Engagement
Accurate release of cheap ‘dumb’ ordinance
GOAL DRIVEN AUTONOMY
Command ‘What’ not ‘How’
How best can we mimic (improve?) on human skill and speed at
trajectory generation in complex environments?
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Classical Trajectory Optimization Problem
Cost:
Constraints:
Issues:
• Becomes the traditional two
point constrained boundary value
problem
• Computationally expensive due
to equality constraints from the
system, environment and
actuation dynamics
• Currently intractable in required
time for effective control
Hope?
• Perhaps our systems contain a structure which allows all solutions of the
system, (trajectories) to be smoothly mapped, from a set of free trajectories
in a reduced dimensional space. Algebraic solutions in this reduced space
would implicitly satisfying the dynamic constraints of the original system.
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Trajectory Generation: Current Methods
Brute force numerical method solution
of the dynamic and constraint ODE’s
Iteration 1:
e
Solution Method
1) Guess control e(t)
2) Propagate dynamics from beginning to
end (simulate)
3) Propagate constraints from beginning
to end (simulate)
4) Check for constraint violation
5) Modify guess e(t)
6) Repeat until feasible/optimal solution
obtained. (optimize)
e
Vast complexity and extremely long
solution times are addressed by
either/both:
Very simple control curves
All calculations performed offline
(selected/looked-up online)
Much of previous work in subject
devoted to improving ‘wisdom’ of next
guess
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Iteration 2:
Differential Systems Suggest an Elegant
Solution
Perhaps our systems contain a structure which allows all solutions of the
system, (trajectories) to be smoothly mapped by a set of free trajectories in a
reduced dimensional space. Algebraic solutions in this reduced space would
implicitly satisfying the dynamic constraints of the original system dynamics
and constraint ODE’s
Constraints are mapped into the flat
space as well and also become time
independent
Direct Solutions! We are modifying
the same curve we are optimizing!
Local Support: Every solution is only
affected by the trajectory near it
Basically a curve fit problem
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Differential Flatness
Definition:
A system is said to be differentially
flat if there exists variables z1,…,zm
of the form
such that (x,u) can be expressed in
terms of z and its derivatives by an
equation of the form
Example:
(Point-to-Point):
Differential Constraints are
reduced to algebraic equations in
the Flat space!
Note: Dynamic Feedback Linearization via endogenous feedback
is equivalent to differential flatness.
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Instinct Autonomy: Now Using Flatness
Simply find any curve that
satisfies the constraints in the
flat space
Solution Method
1) Map system to flat space using
‘w-1’
2) Guess trajectory of flat output zn
3) Compare against constraints (in
flat space)
4) Optimize over control points
When completed apply ‘w’
function to convert back to
normal space
Much simpler control space, no
simulation required:
Very simple to manipulate
curves
All calculations performed online on the vehicle
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z1
z2
e
Too Good to be True? What did we Lose?
It seems reasonable that such a reduction in complexity
would result in some sort of approximation
Many systems lose nothing at all!
Linear models that are controllable (including nonminimum phase)
Fully-flat nonlinear models
Some systems make reasonable assumptions
Conventional A/C make identical assumptions as dynamic
inversion
Some systems are very much less obvious and more
complicated
This is one of the hardest questions of Differential
Flatness – identifying the flat output can be very difficult
Modern configurations are very challenging!
After one stabilization loop, most systems become
differentially flat (or very close to it)
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SEC Autonomous Trajectory Generation
GO TO WP_D
GO TO WP_A
GO TO Rnwy_3
GO TO WP_C
GO TO WP_A
GO TO WP_S
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Summary
MILP Path Planning
MILP: Fast Global Optimization
No suboptimal local minima
Branch & Bound provides fast
tree-search
Commercial solver on RTOS
Tractability Trade-off:
Time Discretization
Constraints active only at
discrete points in time
Time Scale Refinement
Linear dynamics/constraints
Formulation should properly
capture nonlinearity of solution
space
True global minimum is in a
neighborhood of MILP optimal
solution
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Optimal Trajectory Generation
OTG: Fast Nonlinear Optimization
Optimal control for full nonlinear
systems
Differential Flatness property
allows problem to be mapped to
lower dimensional space for
NLP solver
Absence of dynamics in new
space speeds optimization
Easier constraint propagation
Problem setup should focus on
right “basin of attraction”
NLP solver seeks locally optimal
solutions via SQP methods
Good initial guess
Use in conjunction with global
methods, i.e. MILP
Conclusions
Optimization based approaches help achieve a higher level of
autonomy by enabling autonomous decision making
Cast autonomy applications into standard optimization problems,
to be solved using existing optimization tools and framework
Benefits: no need to build custom solver, existing body of theory,
continued improvement in solver technology
Future: broad range of complex autonomy applications are
enabled by a wide, continuous spectrum of powerful optimization
engines and approaches
Challenge: advanced development of V&V, sensing, & fusion
technology, leading to widespread certification and adoption
Thanks/Credits:
NTG/OTG Approach: Mark Milam/NGST, Prof. R. Murray/Caltech
MILP Approach: Prof. Jonathan How/MIT
Autonomy Slides: Jonathan Mead/NGST
OTG Slides: Travis Vetter/NGIS
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