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Caging Polygons with Two and Three Fingers

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Algorithmic Foundation of Robotics VII

Part of the book series: Springer Tracts in Advanced Robotics ((STAR,volume 47))

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Abstract

We present algorithms for computing all placements of two and three fingers that cage a given polygonal object with n edges in the plane. A polygon is caged when it is impossible to take the polygon to infinity without penetrating one of the fingers. Using a classification into squeezing and stretching cagings, we provide an algorithm that reports all caging placements of two disc fingers in O(n 2logn) time. Our result extends and improves a recent solution for point fingers. In addition, we construct a data structure requiring O(n 2) storage that can answer in O(logn) whether two fingers in a query placement cage the polygon. We also study caging with three point fingers. Given the placements of two so-called base fingers, we report all placements of the third finger so that the three fingers jointly cage the polygon. Using the fact that the boundary of the set of placements for the third finger consists of equilibrium grasps, we present an algorithm that reports all placements of the third finger in O(n 6log2 n) deterministic time and O(n 6logn(loglogn)3) expected time. Our results extend previous solutions that only apply to convex polygons.

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References

  1. Bicchi, A., Kumar, V.: Robotic grasping and contact: A review. In: ICRA, pp. 348–353. IEEE, Los Alamitos (2000)

    Google Scholar 

  2. Davidson, C., Blake, A.: Caging planar objects with a three-finger one-parameter gripper. In: ICRA, pp. 2722–2727. IEEE, Los Alamitos (1998)

    Google Scholar 

  3. de Berg, M., van Kreveld, M., Overmars, M., Schwartzkopf, O.: Computational Geometry: Algorithms and Applications. Springer, Heidelberg (1997)

    MATH  Google Scholar 

  4. Erickson, J., Thite, S., Rothganger, F., Ponce, J.: Capturing a convex object with three discs. In: ICRA, pp. 2242–2247. IEEE, Los Alamitos (2003)

    Google Scholar 

  5. Henzinger, M.R., King, V.: Randomized fully dynamic graph algorithms with polylogarithmic time per operation. J. ACM 46(4), 502–516 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Holm, J., de Lichtenberg, K., Thorup, M.: Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. In: STOC 1998: Proceedings of the 30th annual ACM symposium on Theory of computing, pp. 79–89. ACM Press, New York (1998)

    Chapter  Google Scholar 

  7. Kuperberg, W.: Problems on polytopes and convex sets. In: DIMACS Workshop on Polytopes, pp. 584–589 (1990)

    Google Scholar 

  8. Markenscoff, X., Ni, L., Papadimitriou, C.H.: The geometry of grasping. Int. J. Robotics Res. 9(1), 61–74 (1990)

    Article  Google Scholar 

  9. Mason, M.: Mechanics of Robotic Manipulation. Intelligent Robotics and Autonomous Agents Series. MIT Press, Cambridge (2001)

    Google Scholar 

  10. Pipattanasomporn, P., Sudsang, A.: Two-finger caging of concave polygon. In: ICRA, pp. 2137–2142. IEEE, Los Alamitos (2006)

    Google Scholar 

  11. Rimon, E., Blake, A.: Caging 2d bodies by 1-parameter two-fingered gripping systems. In: ICRA, vol. 2, pp. 75–91. IEEE, Los Alamitos (1995)

    Google Scholar 

  12. Rimon, E., Burdick, J.W.: Mobility of bodies in contact—part i: A 2nd–order mobility index for multiple–finger grasps. IEEE Tr. on Robotics and Automation 14(5), 696–717 (1998)

    Article  Google Scholar 

  13. Sudsang, A.: A sufficient condition for capturing an object in the plane with disc-shaped robots. In: ICRA, pp. 682–687. IEEE, Los Alamitos (2002)

    Google Scholar 

  14. Sudsang, A., Luewirawong, T.: Capturing a concave polygon with two disc-shaped fingers. In: ICRA, pp. 1121–1126. IEEE, Los Alamitos (2003)

    Google Scholar 

  15. Sudsang, A., Ponce, J.: A new approach to motion planning for disc-shaped robots manipulating a polygonal object in the plane. In: ICRA, pp. 1068–1075. IEEE, Los Alamitos (2000)

    Google Scholar 

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Authors and Affiliations

  1. Department of Information and Computing Sciences, Utrecht University,  

    Mostafa Vahedi & A. Frank van der Stappen

Authors
  1. Mostafa Vahedi
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  2. A. Frank van der Stappen
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Srinivas Akella Nancy M. Amato Wesley H. Huang Bud Mishra

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Vahedi, M., van der Stappen, A.F. (2008). Caging Polygons with Two and Three Fingers. In: Akella, S., Amato, N.M., Huang, W.H., Mishra, B. (eds) Algorithmic Foundation of Robotics VII. Springer Tracts in Advanced Robotics, vol 47. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68405-3_5

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  • DOI: https://doi.org/10.1007/978-3-540-68405-3_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-68404-6

  • Online ISBN: 978-3-540-68405-3

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