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Distributed Localization of Modular Robot Ensembles

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

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Distributed Localization of Modular Robot Ensembles

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

  2. Claytronics thousands of modules Large-Scale Modular Robots PolyBot, PARC Atron, SDU tens ofmodules 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 estimateof relative location intensity of reading Global estimate for all modules Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008

  4. Challenges 1 Dense, irregular structurehard to apply sparse approximations Modular robot structure: dense SLAM problem, sparse (courtesy E. Brunskill et al.) 2 Massively parallel system Limited processing 8MHz CPU 4kB RAM,128kB ROM ¼10 nodes ¼10,000 nodes 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 the most likely location of module i Goal: maximize likelihood Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008

  6. convergence greedy initialization hypothesizedoptimum Try 1: Optimize Likelihood initialize greedily with a subset of observations then optimize likelihood with local iterative method greedy initialization With bad initialization, convergence very slow; may get stuck in local optima 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 convergence partial solution Number of iterations weak region:few observations step Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008

  8. 3 Suppose add evidence in different order tightly connectedcomponents first 1 2 weak region later(few observations) Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008

  9. merge … … … … rigid body alignment Algorithm Overview Hierarchically partitionconnectivity graph Incorporate evidence betweencomponents bottom-up 1 2 connectivity graph / MRF partition Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008

  10. 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 Can the algorithm be distributed? 3 Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008

  12. Ordering as a graph cut problem few edges / observationsbetween the components A B connectivity graph many edges / observationswithin the component Objective optimized in normalized cut [Shi, Malik, 2000] Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008

  13. greedy abstraction Abstraction: G’ Scaling up Bad news: • normalized cut relatively slow: O(N1.5) • requires entire connectivity graph In practice, not so bad:compute normcut on an abstraction of connectivity graph Original connectivity: G cut in G’ Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008

  14. Putting it all together return to level k-1 recurse to level k+1 local optimization(1st order+precond.) closed-form[Umeyama, 1991] greedy spectral 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 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 1st and 2nd order statistics of (pi,qi) leader {pi} {qi} Leverage aggregation + problem structure for global coordination Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008

  17. g Experimental Setup 2D: Placed modules in gravitationalfield, let them settle 3D: Rasterized realistic models,randomized orientations • 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

  18. Selected Results (sparse test case) groundtruth (all same) incrementalsolution Robust SDP [Biswas et al., 2006] our solution estimate estimate afterrefinement Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008

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

  20. better Scalability Total numberof updates £ 106 4 Number of iterations increases very slowly with size of ensemble 3 gradientthreshold 0.1 2 gradientthreshold 1 1 0 0 2000 5000 10000 Number of modules Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008

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

  22. Communication Complexity (number of messages / module) 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 Distributed Localization of Modular Robot Ensembles – Robotics: Science and Systems 2008

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