On the Power of Manifold Samples in Exploring Configuration Spaces and the Dimensionality of Narrow Passages

Part of the Springer Tracts in Advanced Robotics book series (STAR, volume 86)

Abstract

We extend our study of Motion Planning via Manifold Samples (MMS), a general algorithmic framework that combines geometric methods for the exact and complete analysis of low-dimensional configuration spaces with sampling-based approaches that are appropriate for higher dimensions. The framework explores the configuration space by taking samples that are low-dimensional manifolds of the configuration space capturing its connectivity much better than isolated point samples. The contributions of this paper are as follows: (i) We present a recursive application of MMS in a sixdimensional configuration space, enabling the coordination of two polygonal robots translating and rotating amidst polygonal obstacles. In the adduced experiments for the more demanding test cases MMS clearly outperforms PRM, with over 20-fold speedup in a coordination-tight setting. (ii) A probabilistic completeness proof for the most prevalent case, namely MMS with samples that are affine subspaces. (iii) A closer examination of the test cases reveals that MMS has, in comparison to standard sampling-based algorithms, a significant advantage in scenarios containing high-dimensional narrow passages. This provokes a novel characterization of narrow passages which attempts to capture their dimensionality, an attribute that had been (to a large extent) unattended in previous definitions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Salzman, O., Hemmer, M., Halperin, D.: On the Power of Manifold Samples in Exploring Configuration Spaces and the Dimensionality of Narrow Passages. CoRR abs/1202.5249 (2012), http://arxiv.org/abs/1202.5249
  2. 2.
    Aronov, B., Sharir, M.: On translational motion planning of a convex polyhedron in 3-space. SIAM J. Comput. 26(6), 1785–1803 (1997)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Avnaim, F., Boissonnat, J.D., Faverjon, B.: A practical exact motion planning algorithm for polygonal object amidst polygonal obstacles. In: Proceedings of the Workshop on Geometry and Robotics, pp. 67–86. Springer, London (1989)CrossRefGoogle Scholar
  4. 4.
    Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry, 2nd edn. Algorithms and Computation in Mathematics. Springer (2006)Google Scholar
  5. 5.
    Berberich, E., Fogel, E., Halperin, D., Kerber, M., Setter, O.: Arrangements on parametric surfaces II: Concretizations and applications. MCS 4, 67–91 (2010)MATHMathSciNetGoogle Scholar
  6. 6.
    Berberich, E., Fogel, E., Halperin, D., Mehlhorn, K., Wein, R.: Arrangements on parametric surfaces I: General framework and infrastructure. MCS 4, 45–66 (2010)MATHMathSciNetGoogle Scholar
  7. 7.
    Canny, J.F.: Complexity of Robot Motion Planning (ACM Doctoral Dissertation Award). The MIT Press (1988)Google Scholar
  8. 8.
    Chazelle, B., Edelsbrunner, H., Guibas, L.J., Sharir, M.: A singly exponential stratification scheme for real semi-algebraic varieties and its applications. Theoretical Computer Science 84(1), 77–105 (1991)CrossRefMATHGoogle Scholar
  9. 9.
    Choset, H., Burgard, W., Hutchinson, S., Kantor, G., Kavraki, L.E., Lynch, K., Thrun, S.: Principles of Robot Motion: Theory, Algorithms, and Implementation. MIT Press (2005)Google Scholar
  10. 10.
    Dalibard, S., Laumond, J.P.: Linear dimensionality reduction in random motion planning. I. J. Robotic Res. 30(12), 1461–1476 (2011)CrossRefGoogle Scholar
  11. 11.
    De Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer (2008)Google Scholar
  12. 12.
    Foegl, E., Halperin, D., Wein, R.: CGAL Arrangements and their Applications. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  13. 13.
    Fogel, E., Halperin, D.: Exact and efficient construction of Minkowski sums of convex polyhedra with applications. CAD 39(11), 929–940 (2007)MATHGoogle Scholar
  14. 14.
    Hachenberger, P.: Exact Minkowksi sums of polyhedra and exact and efficient decomposition of polyhedra into convex pieces. Algorithmica 55(2), 329–345 (2009)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Halperin, D., Sharir, M.: A near-quadratic algorithm for planning the motion of a polygon in a polygonal environment. Disc. Comput. Geom. 16(2), 121–134 (1996)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Hirsch, S., Halperin, D.: Hybrid Motion Planning: Coordinating Two Discs Moving Among Polygonal Obstacles in the Plane. In: Boissonat, J.-D., Burdick, J., Goldberg, K., Hutchinson, S. (eds.) Algorithmic Foundation Robotics. STAR, vol. 7, pp. 239–255. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  17. 17.
    Hsu, D., Latombe, J.C., Motwani, R.: Path planning in expansive configuration spaces. Int. J. Comp. Geo. & App. 4, 495–512 (1999)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Kavraki, L.E., Kolountzakis, M.N., Latombe, J.C.: Analysis of probabilistic roadmaps for path planning. IEEE Trans. Robot. Automat. 14(1), 166–171 (1998)CrossRefGoogle Scholar
  19. 19.
    Kavraki, L.E., Latombe, J.C., Motwani, R., Raghavan, P.: Randomized query processing in robot path planning. JCSS 57(1), 50–60 (1998)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Kavraki, L.E., Svestka, P., Latombe, J.C., Overmars, M.: Probabilistic roadmaps for path planning in high dimensional configuration spaces. IEEE Transactions on Robotics and Automation 12(4), 566–580 (1996)CrossRefGoogle Scholar
  21. 21.
    Kuffner, J.J., Lavalle, S.M.: RRT-Connect: An efficient approach to single-query path planning. In: ICRA, pp. 995–1001 (2000)Google Scholar
  22. 22.
    Ladd, A.M., Kavraki, L.E.: Generalizing the analysis of PRM. In: ICRA, pp. 2120–2125. IEEE Press (2002)Google Scholar
  23. 23.
    Latombe, J.C.: Robot Motion Planning. Kluwer Academic Publishers, Norwell (1991)CrossRefGoogle Scholar
  24. 24.
    Lavalle, S.M.: Rapidly-exploring random trees: A new tool for path planning. In Computer Science Dept., Iowa State University Tech. Rep., pp. 98–11 (1998)Google Scholar
  25. 25.
    LaValle, S.M.: Planning Algorithms. Cambridge University Press, Cambridge (2006)CrossRefMATHGoogle Scholar
  26. 26.
    Lien, J.M.: Hybrid motion planning using Minkowski sums. In: RSS 2008 (2008)Google Scholar
  27. 27.
    Lozano-Perez, T.: Spatial planning: A configuration space approach. MIT AI Memo 605 (1980)Google Scholar
  28. 28.
    Plaku, E., Bekris, K.E., Kavraki, L.E.: OOPS for motion planning: An online open-source programming system. In: ICRA, pp. 3711–3716. IEEE (2007)Google Scholar
  29. 29.
    Reif, J.H.: Complexity of the mover’s problem and generalizations. In: FOCS, pp. 421–427. IEEE Computer Society, Washington, DC (1979)Google Scholar
  30. 30.
    Salzman, O., Hemmer, M., Raveh, B., Halperin, D.: Motion Planning via Manifold Samples. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 493–505. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  31. 31.
    Schwartz, J.T., Sharir, M.: On the ”piano movers” problem: I. The case of a two-dimensional rigid polygonal body moving amidst polygonal barriers. Commun. Pure Appl. Math. 35, 345–398 (1983)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Schwartz, J.T., Sharir, M.: On the ”piano movers” problem: II. General techniques for computing topological properties of real algebraic manifolds. Advances in Applied Mathematics 4(3), 298–351 (1983)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Sharir, M.: Algorithmic Motion Planning, Handbook of Discrete and Computational Geometry, 2nd edn. CRC Press, Inc., Boca Raton (2004)Google Scholar
  34. 34.
    Siek, J.G., Lee, L.Q., Lumsdaine, A.: The Boost Graph Library: User Guide and Reference Manual. Addison-Wesley Professional (2001)Google Scholar
  35. 35.
    The CGAL Project: CGAL User and Reference Manual, 3.7 edn. CGAL Editorial Board (2010), http://www.cgal.org/
  36. 36.
    Wein, R.: Exact and Efficient Construction of Planar Minkowski Sums Using the Convolution Method. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 829–840. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  37. 37.
    Yang, J., Sacks, E.: RRT path planner with 3 DOF local planner. In: ICRA, pp. 145–149 (2006)Google Scholar
  38. 38.
    Zhang, L., Huang, X., Kim, Y.J., Manocha, D.: D-plan: Efficient collision-free path computation for part removal and disassembly. Journal of Computer-Aided Design and Applications (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Tel-Aviv UniversityTel AvivIsrael

Personalised recommendations