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
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
Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
3Research at Research at IntelIntel
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
Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
4Research at Research at IntelIntel
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.)
Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
5Research at Research at IntelIntel
Probabilistic approachConceptually easy:find locations/orientations that best match observations among
modules
Observation model
Goal: maximize likelihood
the most likely locationof module i
Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
6Research at Research at IntelIntel
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
Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
7Research at Research at IntelIntel
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
Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
8Research at Research at IntelIntel
Suppose add evidence in different order
1 2
3
tightly connectedcomponents first
weak region later(few observations)
Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
9Research at Research at IntelIntel
connectivity graph / MRF
Algorithm Overview
… … … …
Hierarchically partitionconnectivity graph
Incorporate evidence betweencomponents bottom-up
1 2
rigid body alignment
partition merge
Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
10Research at Research at IntelIntel
Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
11Research at Research at IntelIntel
Technical Challenges
How do we identify “weak” regions?1
Is the algorithm scalable?2
3 Can the algorithm be distributed?
Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
12Research at Research at IntelIntel
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
Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
13Research at Research at IntelIntel
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’
Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
14Research at Research at IntelIntel
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 Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
15Research at Research at IntelIntel
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
Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
16Research at Research at IntelIntel
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
Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
17Research at Research at IntelIntel
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
Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
18Research at Research at IntelIntel
estimate
estimate afterrefinement
Selected Results (sparse test case)
groundtruth
(all same)
incrementalsolution
Robust SDP[Biswas et al., 2006]
our solution
Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
19Research at Research at IntelIntel
Accuracy
Classical MDS
Regularized SDP
Incremental
Our solution
RMS error[module radii]
better
Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
20Research at Research at IntelIntel
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 Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
21Research at Research at IntelIntel
Distributed 3D Results
Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
22Research at Research at IntelIntel
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%
Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008
23Research at Research at IntelIntel
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
Top Related