Algorithmica

, Volume 67, Issue 4, pp 547–565 | Cite as

Motion Planning via Manifold Samples

  • Oren Salzman
  • Michael Hemmer
  • Barak Raveh
  • Dan Halperin
Article

Abstract

We present a general and modular algorithmic framework for path planning of robots. Our framework combines geometric methods for exact and complete analysis of low-dimensional configuration spaces, together with practical, considerably simpler sampling-based approaches that are appropriate for higher dimensions. In order to facilitate the transfer of advanced geometric algorithms into practical use, we suggest taking samples that are entire low-dimensional manifolds of the configuration space that capture the connectivity of the configuration space much better than isolated point samples. Geometric algorithms for analysis of low-dimensional manifolds then provide powerful primitive operations. The modular design of the framework enables independent optimization of each modular component. Indeed, we have developed, implemented and optimized a primitive operation for complete and exact combinatorial analysis of a certain set of manifolds, using arrangements of curves of rational functions and concepts of generic programming. This in turn enabled us to implement our framework for the concrete case of a polygonal robot translating and rotating amidst polygonal obstacles. We show that this instance of the framework is probabilistically complete. Moreover, we demonstrate that the integration of several carefully engineered components leads to significant speedup over the popular PRM sampling-based algorithm, which represents the more simplistic approach that is prevalent in practice.

Keywords

Motion planning Computational geometry Manifolds 

References

  1. 1.
    Amato, N.M., Wu, Y.: A randomized roadmap method for path and manipulation planning. In: IEEE International Conference on Robotics and Automation, pp. 113–120 (1996) CrossRefGoogle Scholar
  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) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Avnaim, F., Boissonnat, J., Faverjon, B.: A practical exact motion planning algorithm for polygonal objects amidst polygonal obstacles. In: Geometry and Robotics. Lecture Notes in Computer Science, vol. 391, pp. 67–86. Springer, Berlin (1989) CrossRefGoogle Scholar
  4. 4.
    Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry, 2nd edn. Springer, Secaucus (2006) MATHGoogle Scholar
  5. 5.
    Berberich, E., Fogel, E., Halperin, D., Mehlhorn, K., Wein, R.: Arrangements on parametric surfaces I: General framework and infrastructure. Math. Comput. Sci. 4(1), 45–66 (2010) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Berberich, E., Hemmer, M., Kerber, M.: A generic algebraic kernel for non-linear geometric applications. In: Symposium on Computational Geometry, pp. 179–186 (2011) Google Scholar
  7. 7.
    Berenson, D., Srinivasa, S.S., Ferguson, D., Kuffner, J.J.: Manipulation planning on constraint manifolds. In: International Conference on Robotics and Automation, pp. 625–632 (2009) Google Scholar
  8. 8.
    Canny, J., Donald, B., Ressler, E.K.: A rational rotation method for robust geometric algorithms. In: Symposium on Computational Geometry, pp. 251–260 (1992) Google Scholar
  9. 9.
    Canny, J.F.: Complexity of Robot Motion Planning (ACM Doctoral Dissertation Award). MIT Press, Cambridge (1988) Google Scholar
  10. 10.
    Chazelle, B., Edelsbrunner, H., Guibas, L.J., Sharir, M.: A singly exponential stratification scheme for real semi-algebraic varieties and its applications. Theor. Comput. Sci. 84(1), 77–105 (1991) CrossRefMATHGoogle Scholar
  11. 11.
    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, Cambridge (2005) Google Scholar
  12. 12.
    Şucan, I.A., Moll, M., Kavraki, L.E.: The open motion planning library. IEEE Robotics & Automation Magazine (2012). http://ompl.kavrakilab.org
  13. 13.
    De Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Berlin (2008) Google Scholar
  14. 14.
    Dobrowolski, P.: An algorithm for computing the exact configuration space of a rotating object in 3-space. Int. J. Comput. Sci. 39(4), 363–376 (2012) Google Scholar
  15. 15.
    Fogel, E., Halperin, D.: Exact and efficient construction of Minkowski sums of convex polyhedra with applications. Comput. Aided Des. 39(11), 929–940 (2007) CrossRefMATHGoogle Scholar
  16. 16.
    Fogel, E., Halperin, D., Wein, R.: CGAL Arrangements and Their Applications—A Step-by-Step Guide, Geometry and Computing vol. 7. Springer, Berlin (2012) CrossRefGoogle Scholar
  17. 17.
    Hachenberger, P.: Exact Minkowski sums of polyhedra and exact and efficient decomposition of polyhedra into convex pieces. Algorithmica 55(2), 329–345 (2009) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Halperin, D., Sharir, M.: A near-quadratic algorithm for planning the motion of a polygon in a polygonal environment. Discrete Comput. Geom. 16(2), 121–134 (1996) MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hirsch, S., Halperin, D.: Hybrid motion planning: Coordinating two discs moving among polygonal obstacles in the plane. In: Workshop on the Algorithmic Foundations of Robotics, pp. 225–241 (2002) Google Scholar
  20. 20.
    Hsu, D., Latombe, J.C., Motwani, R.: Path planning in expansive configuration spaces. Int. J. Comput. Geom. Appl. 9(4/5), 495–512 (1999) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kavraki, L.E., Kolountzakis, M.N., Latombe, J.C.: Analysis of probabilistic roadmaps for path planning. IEEE Trans. Robot. Autom. 14(1), 166–171 (1998) CrossRefGoogle Scholar
  22. 22.
    Kavraki, L.E., Svestka, P., Latombe, J.C., Overmars, M.: Probabilistic roadmaps for path planning in high dimensional configuration spaces. IEEE Trans. Robot. Autom. 12(4), 566–580 (1996) CrossRefGoogle Scholar
  23. 23.
    Kuffner, J.J., Lavalle, S.M.: RRT-Connect: An efficient approach to single-query path planning. In: IEEE International Conference on Robotics and Automation, pp. 995–1001 (2000) Google Scholar
  24. 24.
    Ladd, A.M., Kavraki, L.E.: Generalizing the analysis of PRM. In: IEEE International Conference on Robotics and Automation, pp. 2120–2125 (2002) Google Scholar
  25. 25.
    Latombe, J.C.: Robot Motion Planning. Kluwer Academic, Norwell (1991) CrossRefGoogle Scholar
  26. 26.
    Lavalle, S.M.: Rapidly-exploring random trees: A new tool for path planning. Technical Report 98-11, Computer Science Department, Iowa State University (1998) Google Scholar
  27. 27.
    LaValle, S.M.: Planning Algorithms. Cambridge University Press, Cambridge (2006) CrossRefMATHGoogle Scholar
  28. 28.
    Lien, J.M.: Hybrid motion planning using Minkowski sums. In: Robotics: Science and Systems (2008) Google Scholar
  29. 29.
    Lien, J.M.: A simple method for computing Minkowski sum boundary in 3D using collision detection. In: Workshop on the Algorithmic Foundations of Robotics, pp. 401–415 (2008) Google Scholar
  30. 30.
    Lozano-Perez, T.: Spatial Planning: A Configuration Space Approach. MIT Press, Cambridge (1980). AI Memo 605 Google Scholar
  31. 31.
    Mayer, N., Fogel, E., Halperin, D.: Fast and robust retrieval of Minkowski sums of rotating convex polyhedra in 3-space. In: SPM, pp. 1–10 (2010) Google Scholar
  32. 32.
    Porta, J.M., Jaillet, L., Bohigas, O.: Randomized path planning on manifolds based on higher-dimensional continuation. Int. J. Robot. Res. 31(2), 201–215 (2012) CrossRefGoogle Scholar
  33. 33.
    Reif, J.H.: Complexity of the mover’s problem and generalizations. In: Symposium on Foundations of Computer Science, pp. 421–427. IEEE Comput. Soc., Washington (1979) Google Scholar
  34. 34.
    Salzman, O., Hemmer, M., Halperin, D.: On the power of manifold samples in exploring configuration spaces and the dimensionality of narrow passages. In: Workshop on the Algorithmic Foundations of Robotics (2012, to appear). arXiv:1202.5249
  35. 35.
    Salzman, O., Hemmer, M., Raveh, B., Halperin, D.: Motion planning via manifold samples. In: European Symposium on Algorithms, pp. 493–505 (2011) Google Scholar
  36. 36.
    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) MathSciNetCrossRefGoogle Scholar
  37. 37.
    Schwartz, J.T., Sharir, M.: On the “piano movers” problem: II. General techniques for computing topological properties of real algebraic manifolds. Adv. Appl. Math. 4(3), 298–351 (1983) MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Sharir, M.: Algorithmic Motion Planning, Handbook of Discrete and Computational Geometry, 2nd edn. CRC Press, Boca Raton (2004) Google Scholar
  39. 39.
    Siek, J.G., Lee, L.Q., Lumsdaine, A.: The Boost Graph Library: User Guide and Reference Manual. Addison-Wesley, Reading (2001) Google Scholar
  40. 40.
    The CGAL Project: CGAL User and Reference Manual, 3.7 edn. CGAL Editorial Board (2010). http://www.cgal.org/
  41. 41.
    Varadhan, G., Manocha, D.: Accurate Minkowski sum approximation of polyhedral models. Graph. Models 68(4), 343–355 (2006) CrossRefMATHGoogle Scholar
  42. 42.
    Wein, R.: Exact and efficient construction of planar Minkowski sums using the convolution method. In: European Symposium on Algorithms, pp. 829–840 (2006) Google Scholar
  43. 43.
    Yang, J., Sacks, E.: RRT path planner with 3 DOF local planner. In: IEEE International Conference on Robotics and Automation, pp. 145–149 (2006) Google Scholar
  44. 44.
    Zhang, L., Kim, Y.J., Manocha, D.: A hybrid approach for complete motion planning. In: International Conference on Intelligent Robots and Systems, pp. 7–14 (2007) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Oren Salzman
    • 1
  • Michael Hemmer
    • 1
  • Barak Raveh
    • 1
    • 2
  • Dan Halperin
    • 1
  1. 1.Tel-Aviv UniversityTel-AvivIsrael
  2. 2.The Hebrew UniversityJerusalemIsrael

Personalised recommendations