Modeling and Planning with Robust Hybrid Automata

Cooperative Control of Distributed Autonomous Vehicles in Adversarial Environments2001 MURI: UCLA, CalTech, Cornell, MIT

Dahleh/Feron/Williams

May 14, 2001

UCLA

Brief update on MIT status

Investigators• Dahleh• Feron• Massaquoi • Williams

Students• Z.-H. Mao (PhD)• G. Kotsalis (PhD)• K. Santarelli (PhD)• T. Schouwenaars (PhD)• M. Valenti (PhD)• A. Walcott (PhD)

Outline• Robust Hybrid Automaton concepts

• Model-Based Programming of autonomous explorers

• Game-theoretic concepts

Problem Formulation• Basic problem for autonomous vehicles/robots:

• Generate and execute a (sub)-optimal motion plan, satisfying given boundary conditions, flight envelope and obstacle avoidance constraints, in a dynamic and uncertain environment– Nonlinear control

• Steering of underactuated, non-holonomic systems

• Stabilization/tracking for nonlinear systems

• Flight envelope protection

– Robotics/Artificial Intelligence• Path planning (obstacle avoidance) for non-holonomic dynamical

systems

– Computer science/Software Engineering• Hard real-time constraints

Research supported by AFOSR, Draper, ONR

Hierarchical decomposition• Need to introduce a hierarchical structure to achieve

computational tractability, e.g. (Stengel, 93):– “Strategic layer”: Task scheduling, goal planning– “Tactical layer”: Guidance, navigation– “Reflexive layer”: Tracking, control, estimation

• General hierarchical systems, derived from arbitrary decompositions, can be extremely hard to analyze and verify

• Design a hierarchical system such that it offers safety and performance guarantees by construction– Analysis and verification: robustness analysis problem

• Consistent hierarchical system

System Quantization • Quantization of feasible trajectories into trajectory

primitives– formalization of the concept of “maneuver”

– Consistent abstraction of the system dynamics

• Hierarchical decomposition of the control tasks:– Maneuver sequencing (guidance, trajectory planning)

– Maneuver execution (control, trajectory tracking)

• Control synthesis:– Build a “maneuver library” (with feedback control)

– Behavioral programming: Solve a mixed-integer program on a “small” space

– Hybrid control system with performance and safety guarantees by design.

Maneuver Automaton• Two classes of trajectory primitives ( trim trajectories + maneuvers )

• Construct a “Maneuver Library”, with a finite number of primitives

• Generate trajectories by sequencing such primitives– All generated trajectories are solutions of the system’s diff. equations

– All generated trajectories satisfy the flight envelope constraints (assuming F(x,u)=F(hx,u))

HoverForward flight

Steady left turn

Steady right turn

Example of planning in a free environment

0 5 10 15 20 25 30 35 40-300

-200

-100

0

100

200

300

400

actual positionactual velocitycommanded position"maneuver switch"

Model-based Autonomy

• How do we program explorers that reason quickly and extensively from commonsense models?

• How do we coordinate heterogeneous teams of robots -- in space, air and land -- to perform complex exploration?

• How do we couple reasoning, adaptivity and learning to create robust agents?

• How do we incorporate model-based autonomy into every day, ubiquitous computing devices?

Programmers generate breadth of functions from commonsense models in light of mission goals.

Model-based Autonomy

• Model-based Reactive Programming• Programmer guides state evolution at strategic levels.

• Commonsense Modeling • Programmer specifies commonsense, compositional

models of spacecraft behavior.

• Model-based Execution Kernel• Reason through system interactions on the fly,

performing significant search & deduction

within the reactive control loop.

Model-based Programming ofCooperating Explorers

Programmers and operators must reason through system-wide interactions to :

• select deadlinesselect deadlines• select timing select timing

constraintsconstraints• allocate resourcesallocate resources

Managing Interactions for Cooperation

• select among select among redundant redundant proceduresprocedures

• Evaluate outcomesEvaluate outcomes• Plan contingenciesPlan contingencies

Model-based Cooperative Programming

c If c next A Unless c next A A, B Always A

Choose reward

A in time [t-,t+]

Decision-theoreticTemporal Planner

• Model-based Programs• Specify team behaviors as concurrent programs.

• Specify options using decision theoretic choice.

• Specify timing constraints between activities.

• Model-based Execution

• Achieves correctness and economy

Pre-plans threads of execution that are optimal and temporally consistent.

• Responds at reactive timescales

Perform planning as graph search

Enroute

Mission Scenario

HOMEHOME

RENDEZVOUSRENDEZVOUS RESCUERESCUE AREAAREA

Diverge

RESCUE LOCATIONRESCUE LOCATION

MEETING POINTMEETING POINT

Station: ABC

Station: XYZ

ONETWO

Enroute Activity:

RendezvousRendezvous Rescue AreaRescue Area

Corridor 2

Corridor 1

Corridor 3

Enroute

3

1

4 5

8

9 10

13

2

6 7 11 12

425

440

30

1

0

0

0

0 0

0

0

0

0

[450,540]

price = 425

price = 425

price = 440

price = 0

price = 0

price = 0

price = 30 price = 0

price = 1 price = 0

price = 0

price = 0

0price = 425

Path P = 1 3 4 5 8

9 10

11 12

13 2

Extend Path

Enroute Activity:

• Least cost threads of execution generated by extended auction algorithm

Start Node : 1End Node: 2

Temporal planning is combined with randomized path planning to find a collision free corridor

4 5

xinit

Path 1

xgoal

Xobs

Game-theoretic concepts(Feron and DeMot)

Problem:

•Navigation of a number of vehicles to a target

•Target located at a position that is known with respect to the vehicles or in a known region with a certain known probability distribution

•Vehicles have visual information about a local part of the environment

•Adversarial, unknown environment

Issues:• Many cooperating vehicles vs. single vehicle missions

•Continuously updating available information

Approach:•Game theory

Illustrative Example

Obstacle

TargetAdversary

Agents

Two-agent gameOne agent gets to target fastPure strategy

Agent

Single-agent gameGet to target fastRequires mixed strategy

?

Initial Observations

• Multiple vehicles yield pure strategies whereas for single vehicles a mixed strategy is optimal

• Continuously information updates? Applicability of certainty equivalence principles (eg Basar & Bernhardt, Birkhauser, 1991)• More general setting: nature chooses the position

of an arbitrary amount of obstacles in the unexplored areas - Need for well-defined models