Motivating Example 1: Omniscient Beings Going Places

1. Specifying Globally, Controlling LocallyERICKLAVINSDepartment of Computer ScienceCalifornia Institute of Technology

2. Motivating Example 1:Omniscient Beings Going Places • Problem Description: Get each robot to its goal with no collisions. • With global knowledge it’s easy. • But it doesn’t scale: • n2 communications • Many solutions (like optimal control) become computationally infeasible as n goes up. Cn(R 2) is connected.

3. Motivating Example 2:Solipsists Going Places(or “The LA Freeway Model”) So what if the robots just treat others like (moving) obstacles? Communication and computation go down. But you can get especially poor performance. A happy medium uses uses just enough communication, sensing and computation to perform the task. But how much is just enough?

4. Decentralized Control • Given that: • There is no leader. • Global state or consensus with more than a constant number of other entities is impractical. • Communication complexity should be linear or better. • What is the right formalism for designing and reasoning about decentralized systems? • Approach: Synthesize local controllers from a global specification.

5. Compiler uses a formal composition to build up factory programs. Example: The Minifactory Product description Theorem 1: The compiler produces live (deadlock free), cyclic distributed programs that respect product flow. Observation: Communication goes up linearly. Throughput is constant. [Klavins, HSCC 2000; Klavins and Koditschek, ICRA 2000] Compiler GOAL: Automated factory assembles copies of product (Pictured is the CMU Minifactory [Hollis, Rizzi, Gowdy ICRA ’97-’99]).

6. Example: Self Assembly Given: A (graph) specification of an assembly. Neighbors should be distance dnbr apart. Non-neighbors should be farther away. Synthesize: Local controllers for each part that have the “emergent” effect of assembling the product. In a simplified model Theorem 1: Only specified product is assembled. Theorem 2: A maximal number of parts are assembled. Observation: Communication is linear since sensing is bounded. Time to p% assembled is independent of n. [Klavins, CCC 2002; Klavins ICRA 2002]

7. 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 -1 Example: Oscillator Networks Equation for Individual Oscillator What graphs are valid specifications? [Klavins, Ghrist and Koditschek, WAFR2000; Klavins, Thesis 2001; Klavins and Koditschek, IJRR 2002,...] Simple locomotion model for stick insect analysis: [Klavins, Komsuoglu, Koditschek & Full, NBR 2000] The system corresponding to this connection graph meets the specification: it has a single, global attracting behavior. Observation: Communication complexity depends on the degree of the connection graph. The same analysis on this system gives multiple stable orbits. The system does not perform the task specified.

8. Toward a Systematic ApproachBased on UNITY [Klavins&Hickey, Submitted to CDC2002] • IDEA: Take a processor view: Specify a decentralized dynamical system as a parallel program. • Each processor (vehicle) owns a set of instructions describing its behavior. • The dynamics of the environment is just another processor (a computationally powerful one!). • As a result, we get: • No continuous/discrete duality. • A formal object amenable to automated reasoning. • No specification/implementation duality.

9. A Sample SpecificationIn which the features of “DRL” are highlighted rule guard dynamics controller spec

10. Non-deterministic Execution Model how g:r transforms the state the kth epoch

11. Specifying Dynamics and Controllers controller specification  is the “eventually” temporal logic relation new temporal logic relation

12. Refinement The road from specification to implementation is paved with refinements. PLAN: Build a toolbox of specification transformations that can be used to systematically refine (global) specifications into (local) implementations.

13. Continuous Communication Refinement means: for some zBk(x). Theorem: Given a partition of the clauses of  that respects variable assignments, CCR() .

14. A Multi-Vehicle Example • Problem: Each vehicle should maintain an estimate of the position of every other vehicle. • Requirement: Nearby vehicles should have better estimates of each other’s position. • What is the least amount of communication necessary to implement com ?

15. f e t time to send time to recv Maintaining Estimates

16. The Refinement Theorem: ’ .

17. Area of nth annulus is 2(2n-1) Choosing the Bound Function Recall that we want to have What is a good ? Assume an infinite number of robots, with density . And suppose that a vehicle communicates with vehicles at distance d every r(d) seconds. Then the total rate for all robots at distance m is Conjecture: Any choice of  results in r(d)=O(d).

18. Charts and Graphs • Choose (d) = kd + 2com • Communication rate goes up as density of vehicles goes up. • Message rate divided by worst case rate stays low as n goes up.

19. The Road Ahead Proof automation Proof of concept with MVWT More natural expression of delays (SWCR doesn’t work) Planning as refinement Communication complexity of tasks