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Research at Research at IntelIntel
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
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2Research at Research at IntelIntel
Large-Scale Modular Robots
PolyBot, PARC
Atron, SDU
tens ofmodules
Claytronics
thousands of modules
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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
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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.)
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Probabilistic approachConceptually easy:find locations/orientations that best match observations among
modules
Observation model
Goal: maximize likelihood
the most likely locationof module i
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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
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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
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Suppose add evidence in different order
1 2
3
tightly connectedcomponents first
weak region later(few observations)
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connectivity graph / MRF
Algorithm Overview
… … … …
Hierarchically partitionconnectivity graph
Incorporate evidence betweencomponents bottom-up
1 2
rigid body alignment
partition merge
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Technical Challenges
How do we identify “weak” regions?1
Is the algorithm scalable?2
3 Can the algorithm be distributed?
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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
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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’
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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
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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
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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
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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
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estimate
estimate afterrefinement
Selected Results (sparse test case)
groundtruth
(all same)
incrementalsolution
Robust SDP[Biswas et al., 2006]
our solution
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Accuracy
Classical MDS
Regularized SDP
Incremental
Our solution
RMS error[module radii]
better
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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
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Distributed 3D Results
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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%
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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