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Distributed Localization ofModular Robot Ensembles

Robotics: Science and Systems25 June 2008

Stanislav Funiak, Michael Ashley-RollmanSeth Copen Goldstein

Carnegie Mellon University

Padmanabhan Pillai, Jason Campbell

Intel Research Pittsburgh

Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008

2Research at Research at IntelIntel

Large-Scale Modular Robots

PolyBot, PARC

Atron, SDU

tens ofmodules

Claytronics

thousands of modules



Internal Localization

Goal: recover the location of all modules from local observations

(in 2D or 3D)Neighboring modules(uncertain observations)

Local estimateof relative location

Global estimatefor all modules

intensity of reading



ChallengesDense, irregular structure hard to apply sparse approximations

1

Modular robot structure: dense SLAM problem, sparse

2 Massively parallel system

¼ 10,000 nodes ¼ 10 nodes

Limited processing8MHz CPU4kB RAM,128kB ROM

(courtesy E. Brunskill et al.)



Probabilistic approachConceptually easy:find locations/orientations that best match observations among

modules

Observation model

Goal: maximize likelihood

the most likely locationof module i



Try 1: Optimize Likelihood

initialize greedily with a subset of observationsthen optimize likelihood with local iterative method

With bad initialization, convergence very slow; may get stuck in local optima

greedy initialization convergence

hypothesizedoptimum

greedy initialization



Try 2: Incremental Optimization

maximize for progressively larger set of modules

loop closing

partial solution

convergence

Nu

mb

er o

f it

erat

ion

sstepweak region:

few observations



Suppose add evidence in different order

1 2

3

tightly connectedcomponents first

weak region later(few observations)



connectivity graph / MRF

Algorithm Overview

… … … …

Hierarchically partitionconnectivity graph

Incorporate evidence betweencomponents bottom-up

1 2

rigid body alignment

partition merge



Technical Challenges

How do we identify “weak” regions?1

Is the algorithm scalable?2

3 Can the algorithm be distributed?



Ordering as a graph cut problem

Objective optimized in normalized cut [Shi, Malik, 2000]

connectivity graph

A B

few edges / observationsbetween the components

many edges / observationswithin the component



Scaling up

Bad news:• normalized cut relatively slow: O(N1.5)• requires entire connectivity graph

Original connectivity: G

greedyabstraction

cut in G’

In practice, not so bad:compute normcut on an abstraction of connectivity graph

Abstraction: G’



Putting it all together

greedy spectral closed-form[Umeyama, 1991]

local optimization(1st order+precond.)

recurse to level k+1

return to level k-1



Distributed Implementation

Algorithmic challenges• carry out the phases (abstraction, cut,

alignment)in a distributed setting

• robustness to failures, changes in topology

Implementation challenges• many phases, pass information from one to

another• inherently asynchronous system• message-passing programming tedious

Declarative programming language Meld

complete implementation in < 500 lines



Example: Rigid body alignment

Want to find best rigid transformation t,

Solution: aggregate 1st and 2nd order statistics of (pi, qi)

{pi} {qi}

leader

Leverage aggregation + problem structure for global coordination



Experimental Setup

2D: Placed modules in gravitationalfield, let them settle

3D: Rasterized realistic models,randomized orientations

g

DPRSim simulator: http://www.pittsburgh.intel-research.net/dprweb/• physical interaction among modules• sensing• communication

Centralized and distributed experiments



estimate

estimate afterrefinement

Selected Results (sparse test case)

groundtruth

(all same)

incrementalsolution

Robust SDP[Biswas et al., 2006]

our solution



Accuracy

Classical MDS

Regularized SDP

Incremental

Our solution

RMS error[module radii]

better



Scalability

0 2000 5000 100000

1

4

3

£ 106

Number of modules

Total numberof updates

better

2

gradientthreshold 1

gradientthreshold 0.1

Number of iterations increases very slowly with size of ensemble



Distributed 3D Results



Communication Complexity

Procedure / Test case

5 £ 5 £ 5 10 £ 10 £ 10

Neighbor detection 5 0.5% 5 0.3%

Graph abstraction 80 7.7% 124 7.3%

Normalized cut – agg. – dissemination

38 3.7%27 2.7%

63 3.7% 48 2.8%

Rigid alignment – agg.– dissemination

73 7.0%27 2.7%

114 6.7% 48 2.8%

Gradient descent 783 75.8% 1294 76.3%

(number of messages / module)

Gradient descent 783 75.8% 1294 76.3%



Conclusions

• Presented approach for localization in modular robots– Order of evidence affects approximation

– Normalized cut provides an effective heuristic

– Lends itself to a distributed implementation

• The approach yields an effective algorithm– Outperforms Euclidean embedding, simpler heuristics

– Scalable

– Low communication complexity

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