1. Research at Intel Distributed Localization of Modular Robot Ensembles Robotics: Science and Systems 25 June 2008 Stanislav Funiak, Michael Ashley-Rollman Seth Copen Goldstein Carnegie Mellon University Padmanabhan Pillai, Jason Campbell Intel Research Pittsburgh

2. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 2 Large-Scale Modular Robots PolyBot, PARC Atron, SDU tens of modules Claytronics thousands of modules

3. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 3 Internal Localization Goal: recover the location of all modules from local observations (in 2D or 3D) Neighboring modules (uncertain observations) Local estimate of relative location Global estimate for all modules intensity of reading

4. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 4 Challenges Dense, irregular structure hard to apply sparse approximations 1 Modular robot structure: denseSLAM problem, sparse 2 Massively parallel system ¼ 10,000 nodes ¼ 10 nodes Limited processing 8MHz CPU 4kB RAM, 128kB ROM (courtesy E. Brunskill et al.)

5. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 5 Probabilistic approach Conceptually easy: find locations/orientations that best match observations among modules Observation model Goal: maximize likelihood the most likely location of module i

6. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 6 Try 1: Optimize Likelihood initialize greedily with a subset of observations then optimize likelihood with local iterative method With bad initialization, convergence very slow; may get stuck in local optima greedy initialization convergence hypothesized optimum greedy initialization

7. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 7 Try 2: Incremental Optimization maximize for progressively larger set of modules loop closing partial solution convergence Number of iterations step weak region: few observations

8. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 8 Suppose add evidence in different order 12 3 tightly connected components first weak region later (few observations)

9. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 9 connectivity graph / MRF Algorithm Overview ………… Hierarchically partition connectivity graph Incorporate evidence between components bottom-up 12 rigid body alignment partitionmerge

10. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 10

11. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 11 Technical Challenges How do we identify “weak” regions? 1 Is the algorithm scalable? 2 3 Can the algorithm be distributed?

12. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 12 Ordering as a graph cut problem Objective optimized in normalized cut [Shi, Malik, 2000] connectivity graph AB few edges / observations between the components many edges / observations within the component

13. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 13 Scaling up Bad news: normalized cut relatively slow: O(N 1.5 ) requires entire connectivity graph Original connectivity: G greedy abstraction cut in G’ In practice, not so bad: compute normcut on an abstraction of connectivity graph Abstraction: G’

14. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 14 Putting it all together greedyspectral closed-form [Umeyama, 1991] local optimization (1 st order+precond.) recurse to level k+1 return to level k-1

15. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 15 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

16. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 16 Example: Rigid body alignment Want to find best rigid transformation t, Solution: aggregate 1 st and 2 nd order statistics of ( p i, q i ) {pi}{pi}{qi}{qi} leader Leverage aggregation + problem structure for global coordination

17. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 17 Experimental Setup 2D:Placed modules in gravitational field, 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

18. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 18 estimate estimate after refinement Selected Results (sparse test case) ground truth (all same) incremental solution Robust SDP [Biswas et al., 2006] our solution

19. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 19 Accuracy Classical MDS Regularized SDP Incremental Our solution RMS error [module radii] better

20. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 20 Scalability 02000 500010000 0 1 4 3 £ 10 6 Number of modules Total number of updates better 2 gradient threshold 1 gradient threshold 0.1 Number of iterations increases very slowly with size of ensemble

21. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 21 Distributed 3D Results

22. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 22 Communication Complexity Procedure / Test case5 £ 5 £ 510 £ 10 £ 10 Neighbor detection5 0.5% 5 0.3% Graph abstraction80 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 descent783 75.8%1294 76.3% (number of messages / module) Gradient descent783 75.8%1294 76.3%

23. Research at Intel Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008 23 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