Optimal Gap Navigation for a Disc Robot

  • Rigoberto Lopez-Padilla
  • Rafael Murrieta-Cid
  • Steven M. LaValle
Part of the Springer Tracts in Advanced Robotics book series (STAR, volume 86)


This paper considers the problem of globally optimal navigation with respect to Euclidean distance for disc-shaped, differential-drive robot placed into an unknown, simply connected polygonal region. The robot is unable to build precise geometric maps of the environment. Most of the robot’s information comes from a gap sensor, which indicates depth discontinuities and allows the robot to move toward them. A motion strategy is presented that optimally navigates the robot to any landmark in the region. Optimality is proved and the method is illustrated in simulation.


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  1. 1.
    Canny, J., Reif, J.: New lower bound techniques for robot motion planning problems. In: Proceedings IEEE Symposium on Foundations of Computer Science, pp. 49–60 (1987)Google Scholar
  2. 2.
    Chen, D.Z., Wang, H.: Paths among curved obstacles in the plane. In: Proceedings of Computing Research Repository (2011)Google Scholar
  3. 3.
    Chew, L.P.: Planning the shortest path for a disc in O(n 2 log n) time. In: Proceedings ACM Symposium on Computational Geometry (1985)Google Scholar
  4. 4.
    de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications, 2nd edn. Springer, Berlin (2000)CrossRefGoogle Scholar
  5. 5.
    Durrant-Whyte, H., Bailey, T.: Simultaneous localization and mapping: Part I. IEEE Robotics and Automation Magazine 13(2), 99–110 (2006)CrossRefGoogle Scholar
  6. 6.
    Ghosh, S.K.: Visibility Algorithms in the Plane. Cambridge University Press, Cambridge (2007)CrossRefMATHGoogle Scholar
  7. 7.
    Ghosh, S.K., Mount, D.M.: An output sensitive algorithm for computing visibility graphs. In: Proceedings IEEE Symposium on Foundations of Computer Science, pp. 11–19 (1987)Google Scholar
  8. 8.
    Landa, Y., Tsai, R.: Visibility of point clouds and exploratory path planning in unknown environments. Communications in Mathematical Sciences 6(4), 881–913 (2008)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    LaValle, S.M.: Planning Algorithms. Cambridge University Press, Cambridge (2006), CrossRefMATHGoogle Scholar
  10. 10.
    LaValle, S.M.: Sensing and filtering: A fresh perspective based on preimages and information spaces. Foundations and Trends in Robotics Series. Now Publishers, Delft, The Netherlands (2012)Google Scholar
  11. 11.
    Liu, Y.-H., Arimoto, S.: Finding the shortest path of a disc among polygonal obstacles using a radius-independent graph. IEEE Trans. on Robotics and Automation 11(5), 682–691 (1995)CrossRefGoogle Scholar
  12. 12.
    Lopez-Padilla, R., Murrieta-Cid, R., LaValle, S.M.: Detours for optimal navigation with a disc robot, pp. 1–21 (February 2012),
  13. 13.
    Mitchell, J.S.B.: Shortest paths and networks. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn., pp. 607–641. Chapman and Hall/CRC Press, New York (2004)Google Scholar
  14. 14.
    Mitchell, J.S.B., Papadimitriou, C.H.: The weighted region problem. Journal of the ACM 38, 18–73 (1991)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Murphy, L., Newman, P.: Using incomplete online metric maps for topological exploration with the gap navigation tree. In: Proc. IEEE Int. Conf. on Robotics & Automation (2008)Google Scholar
  16. 16.
    Reif, J.H., Sun, Z.: An efficient approximation algorithm for weighted region shortest path problem. In: Donald, B.R., Lynch, K.M., Rus, D. (eds.) Algorithmic and Computational Robotics: New Directions, pp. 191–203. A.K. Peters, Wellesley (2001)Google Scholar
  17. 17.
    Thrun, S., Burgard, W., Fox, D.: Probabilistic Robotics. MIT Press, Cambridge (2005)MATHGoogle Scholar
  18. 18.
    Tovar, B., Murrieta, R., LaValle, S.M.: Distance-optimal navigation in an unknown environment without sensing distances. IEEE Trans. on Robotics 23(3), 506–518 (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Rigoberto Lopez-Padilla
    • 1
  • Rafael Murrieta-Cid
    • 1
  • Steven M. LaValle
    • 2
  1. 1.Center for Mathematical Research (CIMAT)Guanajuato, Gto.México
  2. 2.University of IllinoisUrbanaUSA

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