Specifying Globally, Controlling Locally E RIC K LAVINS Department of Computer Science California Institute of Technology

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: n 2 communications Many solutions (like optimal control) become computationally infeasible as n goes up. C n ( R 2 ) is connected.

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?

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.

Example: The Minifactory 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 uses a formal composition to build up factory programs. Product description Automated factory assembles copies of product (Pictured is the CMU Minifactory [Hollis, Rizzi, Gowdy ICRA ’97-’99]). Compiler GOAL:

Example: Self Assembly Given: A (graph) specification of an assembly. Neighbors should be distance d nbr 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]

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

Toward a Systematic Approach Based 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.

A Sample Specification In which the features of “DRL” are highlighted guard rule dynamics controller spec

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

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

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.

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

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 ?

Maintaining Estimates t time to recv time to send e f

The Refinement Theorem:  ’  .

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 Area of nth annulus is 2  (2n-1) Conjecture: Any choice of  results in r(d)=O(d).

Charts and Graphs vehiclesvehicles/m 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.

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