HEADER / FOOTER INFORMATION (SUCH AS PRIVATE / CONFIDENTIAL)

Optimization Based Approaches to Autonomy

March 3, 2005

Cedric MaNorthrop Grumman Corporation

SAE Aerospace Control and Guidance Systems Committee (ACGSC) MeetingHarvey’s Resort, Lake Tahoe, Nevada

2

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

3

Autonomy in Vehicle Applications

FORMATION FLYING

PACK LEVEL COORDINATION

RENDEZVOUS & REFUELING

OBSTACLE AVOIDANCE

COOPERATIVE SEARCH

NAVIGATION

TEAMTACTICS

LANDING

4

Autonomy: Boyd’s OODA “Loop”

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.

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.

FeedForward

Observations Decision(Hypothesis)

Action(Test)

CulturalTraditions

GeneticHeritage

NewInformation Previous

Experience

Analyses &Synthesis

FeedForward

FeedForward

ImplicitGuidance& Control

ImplicitGuidance& Control

UnfoldingInteraction

WithEnvironmentUnfolding

InteractionWith

Environment Feedback

Feedback

OutsideInformation

UnfoldingCircumstances

Observe Orient Decide Act

Defense and the National Interest, http://www.d-n-i.net, 2001

5

Level of Autonomy Ground Operation

Activities performed off-line Tele-Operation

Awareness of sensor / actuator interfaces Executes commands uploaded from the

ground Reactive Control

Awareness of the present situation Simple reflexes, i.e. no planning required A condition triggers an associated action

Responsive Control Awareness of past actions Remembers previous actions Remembers features of the environment Remembers goals

Deliberative Control Awareness of future possibilities Reasons about future consequences Chooses optimal paths / plans

12345

Ground operationTele-operationReactive ControlResponsive ControlDeliberative Control

Goal of OptimizationBased Autonomy

6

OptimalControl/Decision

Objective/Reward FunctionConstraints/Rules(i.e. Dynamics/Goal)

FeedForward

Observations Decision(Hypothesis)

Action(Test)

CulturalTraditions

GeneticHeritage

NewInformation Previous

Experience

Analyses &Synthesis

FeedForward

FeedForward

ImplicitGuidance& Control

ImplicitGuidance& Control

UnfoldingInteraction

WithEnvironmentUnfolding

InteractionWith

Environment Feedback

Feedback

OutsideInformation

UnfoldingCircumstances

Observe Orient Decide Act

Optimization and Autonomy

Optimizer

VehicleState

Determines best course of action based on current objective, while meeting constraints

Formulation of problemshapes the “Orient” mechanism

7

Autonomy Hierarchy

CooperativeControl

MissionPlanning

PathPlanning

TrajectoryGeneration

TrajectoryFollowing/Inner Loop

Planning & Scheduling, Resource Allocation & SequencingTask Sequencing, Auto Routing

Time Scale: ~1 hr

Multi-Agent Coordination, Pack Level OrganizationFormation Flying, Cooperative Search & Electronic Warfare

Conflict Resolution, Task Negotiation, Team TacticsTime Scale: ~1 min

“Navigation,” Motion PlanningObstacle/Collision/Threat AvoidanceTime Scale: ~10s

“Guidance,” Contingency HandlingLanding, Rendezvous, RefuelingTime Scale: ~1s

“Control,” Disturbance RejectionApplications: Stabilization, AdaptiveReconfigurable Control, FDIRTime Scale: ~0.1s

8

HEADER / FOOTER INFORMATION (SUCH AS PRIVATE / CONFIDENTIAL)

Path Planning with Mixed-IntegerLinear Programming (MILP)

Path Planning with Mixed-IntegerLinear Programming (MILP)

9

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

Collision Avoidance

Obstacle AvoidanceTerrain Navigation

Multi-vehicle Coordination

Nap-of-the-Earth Flight

10

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

11

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) One set per obstacle per time step No cost associated with obstacles

x – M b1 ≤ x1

x + M b2 ≥ x2

y – M b3 ≤ y1

y + M b4 ≥ y2

b1 + b2 + b3 + b4 ≤ 3

12

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

13

Obstacle Avoidance

Treetop level

Nap of the Earth Flight

Urban Low AltitudeOperations

14

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

15

Coordinated Conflict Resolution

3-D Multi-Vehicle Path-Planning 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

16

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

Patent Pending: Docket No. 000535-030

Threat

Vehicle hiding behind building

On-time arrivalat destination

17

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

Summary

18

HEADER / FOOTER INFORMATION (SUCH AS PRIVATE / CONFIDENTIAL)

Optimal Trajectory Generation withNonlinear Programming(NLP)

Optimal Trajectory Generation withNonlinear Programming(NLP)

19

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?

20

Classical Trajectory Optimization Problem

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.

Cost:

Constraints:

21

Trajectory Generation: Current Methods Brute force numerical method solution

of the dynamic and constraint ODE’s

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 violation5) Modify guess e(t) 6) Repeat until feasible/optimal solution

obtained. (optimize) 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

e

e

Iteration 1:

Iteration 2:

22

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

23

Definition: A system is said to be differentially flat if there exists variables z1,…,zm of the formsuch that (x,u) can be expressed in terms of z and its derivatives by an equation of the form

Note: Dynamic Feedback Linearization via endogenous feedback is equivalent to differential flatness.

Example: (Point-to-Point):

Differential Constraints are reduced to algebraic equations in the Flat space!

Differential Flatness

24

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 on-

line on the vehicle

z2

e

z1

25

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 non-minimum 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)

26

SEC Autonomous Trajectory Generation

GO TO WP_C

GO TO Rnwy_3

GO TO WP_A

GO TO WP_AGO TO WP_D

GO TO WP_S

27

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

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

Summary

28

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