Specifying Globally, Controlling Locally ERIC KLAVINS Department of Computer Science California Institute of Technology

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Specifying Globally, Controlling LocallyERIC
KLAVINSDepartment of Computer
ScienceCalifornia Institute of Technology
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Motivating Example 1Omniscient Beings Going
Places

• Problem Description Get each robot to its goal
  with no collisions.
• With global knowledge its easy.
• But it doesnt scale
• n2 communications
• Many solutions (like optimal control) become
  computationally infeasible as n goes up.

Cn(R 2) is connected.
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Motivating Example 2Solipsists 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?
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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.

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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).
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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
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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.
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Toward a Systematic ApproachBased on UNITY
KlavinsHickey, 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.

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A Sample SpecificationIn which the features of
DRL are highlighted
rule
guard
dynamics
controller spec
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Non-deterministic Execution Model
how gr transforms the state
the kth epoch
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Specifying Dynamics and Controllers
controller specification
? is the eventually temporal logic relation
new temporal logic relation
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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.
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Continuous Communication Refinement
means for some z ?Bk(x).
Theorem Given a partition of the clauses of ?
that respects variable assignments, CCR(?) ? ?.
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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 others position.
• What is the least amount of communication
  necessary to implement ?com ?

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Maintaining Estimates
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The Refinement
Theorem ? ? ?.
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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).
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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.

vehicles vehicles/m
10 0.20
20 0.41
30 0.61
40 0.82
50 1.02
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The Road Ahead
Proof automation Proof of concept with
MVWT More natural expression of delays (SWCR
doesnt work) Planning as refinement Communicati
on complexity of tasks