Modeling and Planning with Robust Hybrid Automata

Cooperative Control of Distributed Autonomous Vehicles in Adversarial Environments

2001 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”: “Reflexive layer”:

Guidance, navigation 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

space : Solve a mixed-integer program on a “small” – 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( Y h x,u)) Steady left turn Hover Forward flight Steady right turn

0 -100 -200 -300 0

Example of planning in a free environment

400 300 200 actual position actual velocity commanded position "maneuver switch" 100 5 10 15 20 25 30 35 40

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?

Model-based Autonomy

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

• 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 of Cooperating Explorers

Managing Interactions for Cooperation

Programmers and operators must reason through system-wide interactions to : • select among redundant procedures • Evaluate outcomes • Plan contingencies • select deadlines • select timing constraints • allocate resources

Model-based Cooperative

•

Programming

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

  • Specify options using decision theoretic choice.

   • Specify timing constraints between activities.

  c If c next A Unless c next A A, B Always A Choose reward A in time [t ,t + ] • 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

Decision-theoretic Temporal Planner

HOME Station: ABC Mission Scenario

TWO

RENDEZVOUS

ONE Enroute

RESCUE AREA Station: XYZ Diverge RESCUE LOCATION MEETING POINT

Enroute Activity:

Enroute

Corridor 2 Rendezvous Corridor 1 Corridor 3 Rescue Area

Enroute Activity: • Least cost threads of execution generated by extended auction algorithm price = 425 [450,540] 1

0 0

3 price = 425 0 4 price = 425 6 price = 440

425

440 5 price = 0

0 0

8 0 price = 0

0

7 price = 0 9 price = 30 11 price = 1

30 1

price = 0 2

Extend Path 0

10 price = 0

0

13

0

price = 0 12 price = 0

Path P = 1



3



4



5



8

Start Node : 1 End Node: 2

9



10 11



12 13



2

x init

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

Path 1

X obs

x goal

5

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

Adversary Target Two-agent game Requires mixed strategy Obstacle ?

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