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Discovering a Point Source in Unknown Environments

  • Chapter
Algorithmic Foundation of Robotics VIII

Part of the book series: Springer Tracts in Advanced Robotics ((STAR,volume 57))

Abstract

We consider the inverse problem of discovering the location of a source from very sparse point measurements in a bounded domain that contains impenetrable (and possibly unknown) obstacles. We present an adaptive algorithm for determining themeasurement locations, and ultimately, the source locations. Specifically, we investigate source discovery for the Laplace operator, though the approach can be applied to more general linear partial differential operators. We propose a strategy for the case when the obstacles are unknown and the environment has to be mapped out using a range sensor concurrently with source discovery.

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References

  1. Bardi, M., Capuzzo-Dolcetta, I.: Optimal control and viscosity solutions of Hamilton- Jacobi-Bellman equations. Birkhäuser Boston Inc., Boston (1997); With appendices by Maurizio Falcone and Pierpaolo Soravia

    Book  MATH  Google Scholar 

  2. Choset, H., Lynch, K.M., Hutchinson, S., Kantor, G., Burgard, W., Kavraki, L.E., Thrun, S.: Principles of Robot Motion: Theory, Algorithms, and Implementations. The MIT Press, Cambridge (2005)

    MATH  Google Scholar 

  3. Deng, X., Kameda, T., Papadimitriou, C.: How to learn an unknown environment. i: the rectilinear case. J. ACM 45(2), 215–245 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. El Badia, A., Ha-Duong, T.: On an inverse source problem for the heat equation. Application to a pollution detection problem. J. Inverse Ill-Posed Probl. 10(6), 585–599 (2002)

    MATH  MathSciNet  Google Scholar 

  5. El Badia, A., Ha-Duong, T., Hamdi, A.: Identification of a point source in a linear advectiondispersion- reaction equation: application to a pollution source problem. Inverse Problems 21(3), 1121–1136 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Evans, L.C.: Partial Differential Equations. Amer. Math. Soc. (1998)

    Google Scholar 

  7. Gabriely, Y., Rimon, E.: Competitive complexity of mobile robot on line motion planning problems. In: WAFR, pp. 249–264 (2004)

    Google Scholar 

  8. Gibou, F., Fedkiw, R.P., Cheng, L.-T., Kang, M.: A second-order-accurate symmetric discretization of the Poisson equation on irregular domains. J. Comput. Phys. 176(1), 205–227 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Goodman, J.E., O’Rourke, J. (eds.): Handbook of discrete and computational geometry, 2nd edn. CRC Press LLC, Boca Raton (2004)

    MATH  Google Scholar 

  10. Halperin, D., Kavraki, L.E., Latombe, J.C.: Robot algorithms. In: Atallah, M. (ed.) Handbook of Algorithms and Theory of Computation. CRC Press, Boca Raton (1999)

    Google Scholar 

  11. Halperin, D., Kavraki, L.E., Latombe, J.C.: Robotics. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry. CRC Press, Boca Raton (2004)

    Google Scholar 

  12. Kamon, I., Rivlin, E.: Sensory-based motion planning with global proofs. TRA 13(6), 814–822 (1997)

    Google Scholar 

  13. Kao, M.-Y., Reif, J.H., Tate, S.R.: Searching in an unknown environment: An optimal randomized algorithm for the cow-path problem. In: SODA: ACM-SIAM Symposium on Discrete Algorithms (A Conference on Theoretical and Experimental Analysis of Discrete Algorithms) (1993)

    Google Scholar 

  14. Landa, Y., Galkowski, D., Huang, Y., Joshi, A., Lee, C., Leung, K., Malla, G., Treanor, J., Voroninski, V., Bertozzi, A., Tsai, Y.-H.: Robotic path planning and visibility with limited sensor data. In: American Control Conference, ACC 2007, July 2007, pp. 5425–5430 (2007)

    Google Scholar 

  15. Landa, Y., Tsai, R.: Visibility of point clouds and exploratory path planning in unknown environments. CAM Report 08-28 (Submitted to Communications in Mathematical Sciences) (2008)

    Google Scholar 

  16. Landa, Y., Tsai, R., Cheng, L.-T.: Visibility of point clouds and mapping of unknown environments. Lecture Notes in Computational Science and Engineering, pp. 1014–1025. Springer, Heidelberg (2006)

    Google Scholar 

  17. Latombe, J.-C.: Robot motion planning. Kluwer Academic Publishers, Dordrecht (1991)

    Google Scholar 

  18. Laubach, S.L., Burdick, J.W.: An autonomous sensor-based path-planning for planetary microrovers. In: ICRA (1999)

    Google Scholar 

  19. LaValle, S.M.: Planning Algorithms. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  20. Ling, L., Hon, Y.C., Yamamoto, M.: Inverse source identification for Poisson equation. Inverse Problems in Science and Engineering 13(4), 433–447 (2005)

    Article  MathSciNet  Google Scholar 

  21. Lumelsky, V.J., Stepanov, A.A.: Path planning strategies for a point mobile automaton moving amidst unknown obstacles of arbitrary shape. Algorithmica 2, 403–430 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  22. Osher, S., Fedkiw, R.P.: Level set methods: an overview and some recent results. J. Comput. Phys. 169(2), 463–502 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  23. Papadimitriou, C.H., Yannakakis, M.: Shortest paths without a map. Theoretical Comput. Sci. 84, 127–150 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  24. Sethian, J.A.: Level set methods and fast marching methods, 2nd edn. Cambridge University Press, Cambridge (1999); Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science

    MATH  Google Scholar 

  25. Tovar, B., Guilamo, L., LaValle, S.M.: Gap navigation trees: A minimal representation for visibility-basedtasks. In: Erdmann, M., Hsu, D., Overmars, M., van der Stappen, A.F. (eds.) Algorithmic Foundations of Robotics VI. Springer, Heidelberg (2005)

    Google Scholar 

  26. Tovar, B., Murrieta-Cid, R., LaValle, S.M.: Distance-optimal navigation in an unknown environment withoutsensing distances. IEEE Trans. Robotics 23(3), 506–518 (2007)

    Article  Google Scholar 

  27. Tsai, R., Kao, C.-Y.: A level set formulation for visibility and its discretization. Journal of Scientific Computing (to appear)

    Google Scholar 

  28. Tsai, Y.-H.R., Cheng, L.-T., Osher, S., Burchard, P., Sapiro, G.: Visibility and its dynamics in a PDE based implicit framework. J. Comput. Phys. 199(1), 260–290 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  29. Tsitsiklis, J.: Efficient algorithms for globally optimal trajectories. IEEE Transactions on Automatic Control 40(9), 1528–1538 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  30. Urrutia, J.: Art gallery and illumination problems. In: Sack, J.R., Urrutia, J. (eds.) Handbook of Computational Geometry. Elsevier Science, Amsterdam (2000)

    Google Scholar 

  31. Wang, Y., Chirikjian, G.: A new potential field method for robot path planning. In: Proceedings of the IEEE International Robotics and Automation Conference, San Francisco, CA, pp. 977–982 (2000)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institute for Computational and Applied Mathematics, Westfaelische Wilhelms Universitaet Muenster,  

    Martin Burger

  2. University of California, Los Angeles

    Yanina Landa

  3. University of Texas at Austin,  

    Nicolay M. Tanushev & Richard Tsai

Authors
  1. Martin Burger
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  2. Yanina Landa
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  3. Nicolay M. Tanushev
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  4. Richard Tsai
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Editor information

Editors and Affiliations

  1. Department of Mechanical Engineering, The Johns Hopkins University, 21218, Baltimore, Maryland, USA

    Gregory S. Chirikjian

  2. Carnegie Mellon University, 5000 Forbes Ave., Newell-Simon Hall 3211, 15213, Pittsburgh, PA, USA

    Howie Choset

  3. Sistemas Digitales, Instituto Tecnológico, Autónomo de México, Rio Hondo 1, Progreso Tizapán, 01080, México, D.F., México

    Marco Morales

  4. Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, 60208, Evanston, IL, USA

    Todd Murphey

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© 2009 Springer-Verlag Berlin Heidelberg

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Burger, M., Landa, Y., Tanushev, N.M., Tsai, R. (2009). Discovering a Point Source in Unknown Environments. In: Chirikjian, G.S., Choset, H., Morales, M., Murphey, T. (eds) Algorithmic Foundation of Robotics VIII. Springer Tracts in Advanced Robotics, vol 57. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00312-7_41

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  • DOI: https://doi.org/10.1007/978-3-642-00312-7_41

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00311-0

  • Online ISBN: 978-3-642-00312-7

  • eBook Packages: EngineeringEngineering (R0)

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