Skip to main content

Motion Planning via Manifold Samples


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.

This is a preview of subscription content, access via your institution.

Fig. 1
Algorithm 1
Algorithm 2
Algorithm 3
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8


  1. 1.

    A subdivision of the plane into zero-dimensional, one-dimensional and two-dimensional cells, called vertices, edges and faces, respectively induced by the curves.

  2. 2.

    Considering vertical lines and not vertical slabs as discussed in Sect. 3 is done for simplifying the proof. This is obviously a special case of the same family of manifolds thus the presented proof applies for the case of vertical slabs.


  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)

    Chapter  Google 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)

    MathSciNet  Article  MATH  Google 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)

    Chapter  Google Scholar 

  4. 4.

    Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry, 2nd edn. Springer, Secaucus (2006)

    MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    Article  MATH  Google 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).

  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)

    Article  MATH  Google 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)

    Book  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Book  Google 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)

  27. 27.

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

    Book  MATH  Google 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)

    Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  MATH  Google 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).

  41. 41.

    Varadhan, G., Manocha, D.: Accurate Minkowski sum approximation of polyhedral models. Graph. Models 68(4), 343–355 (2006)

    Article  MATH  Google 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 

Download references

Author information



Corresponding author

Correspondence to Oren Salzman.

Additional information

This work has been supported in part by the 7th Framework Programme for Research of the European Commission, under FET-Open grant number 255827 (CGL—Computational Geometry Learning), by the Israel Science Foundation (grant no. 1102/11), by the German-Israeli Foundation (grant no. 969/07), and by the Hermann Minkowski–Minerva Center for Geometry at Tel Aviv University.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Salzman, O., Hemmer, M., Raveh, B. et al. Motion Planning via Manifold Samples. Algorithmica 67, 547–565 (2013).

Download citation


  • Motion planning
  • Computational geometry
  • Manifolds