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 sampling-based approaches that are appropriate for higher dimensions. We suggest taking samples that are entire low-dimensional manifolds of the configuration space. These samples capture the connectivity of the configuration space much better than isolated point samples. Geometric algorithms then provide powerful primitive operations for complete analysis of the low-dimensional manifolds. We have implemented our framework for the concrete case of a polygonal robot translating and rotating amidst polygonal obstacles. To this end, we have developed a primitive operation for the analysis of an appropriate set of manifolds using arrangements of curves of rational functions. This modular integration of several carefully engineered components has lead to a significant speedup over the PRM sampling-based algorithm, which represents an approach that is prevalent in practice.
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 German-Israeli Foundation (grant no. 969/07), and by the Hermann Minkowski–Minerva Center for Geometry at Tel Aviv University.
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References
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)
Latombe, J.C.: Robot Motion Planning. Kluwer Academic Publishers, Norwell (1991)
LaValle, S.M.: Planning Algorithms. Cambridge University Press, Cambridge (2006)
Reif, J.H.: Complexity of the mover’s problem and generalizations. In: FOCS, pp. 421–427. IEEE Computer Society, Washington, DC, USA (1979)
Lozano-Perez, T.: Spatial planning: A configuration space approach. MIT AI Memo 605 (1980)
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)
Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry. Algorithms and Computation in Mathematics. Springer, Heidelberg (2003)
Canny, J.F.: Complexity of Robot Motion Planning (ACM Doctoral Dissertation Award). MIT Press, Cambridge (1988)
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)
Aronov, B., Sharir, M.: On translational motion planning of a convex polyhedron in 3-space. SIAM J. Comput. 26(6), 1785–1803 (1997)
Avnaim, F., Boissonnat, J., Faverjon, B.: A practical exact motion planning algorithm for polygonal object amidst polygonal obstacles. In: Boissonnat, J.-D., Laumond, J.-P. (eds.) Geometry and Robotics. LNCS, vol. 391, pp. 67–86. Springer, Heidelberg (1989)
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)
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)
Sharir, M.: Algorithmic Motion Planning. In: Handbook of Discrete and Computational Geometry, 2nd edn., CRC Press, Inc., Boca Raton (2004)
Fogel, E., Halperin, D.: Exact and efficient construction of Minkowski sums of convex polyhedra with applications. CAD 39(11), 929–940 (2007)
Hachenberger, P.: Exact Minkowksi sums of polyhedra and exact and efficient decomposition of polyhedra into convex pieces. Algorithmica 55(2), 329–345 (2009)
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)
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)
Kuffner, J.J., Lavalle, S.M.: RRT-Connect: An efficient approach to single-query path planning. In: ICRA, pp. 995–1001. IEEE, Los Alamitos (2000)
Ladd, A.M., Kavraki, L.E.: Generalizing the analysis of PRM. In: ICRA, pp. 2120–2125. IEEE, Los Alamitos (2002)
Hirsch, S., Halperin, D.: Hybrid motion planning: Coordinating two discs moving among polygonal obstacles in the plane. In: WAFR 2002, pp. 225–241 (2002)
Zhang, L., Kim, Y.J., Manocha, D.: A hybrid approach for complete motion planning. In: IROS, pp. 7–14 (2007)
De Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications. Springer, Heidelberg (2008)
Lien, J.M.: Hybrid motion planning using Minkowski sums. In: RSS 2008 (2008)
Yang, J., Sacks, E.: RRT path planner with 3 DOF local planner. In: ICRA, pp. 145–149. IEEE, Los Alamitos (2006)
Salzman, O., Hemmer, M., Raveh, B., Halperin, D.: Motion planning via manifold samples. In: arXiv:1107.0803 (2011)
Siek, J.G., Lee, L.-Q., Lumsdaine, A.: The Boost Graph Library: User Guide and Reference Manual. Addison-Wesley Professional, Reading (2001)
The CGAL Project: CGAL User and Reference Manual. 3.7 edn. CGAL Editorial Board (2010), http://www.cgal.org/
Canny, J., Donald, B., Ressler, E.K.: A rational rotation method for robust geometric algorithms. In: SoCG 1992, pp. 251–260. ACM, New York (1992)
Austern, M.H.: Generic Programming and the STL. Addison-Wesley, Reading (1998)
Berberich, E., Hemmer, M., Kerber, M.: A generic algebraic kernel for non-linear geometric applications. In: SoCG 2011 (2011)
Plaku, E., Bekris, K.E., Kavraki, L.E.: OOPS for motion planning: An online open-source programming system. In: ICRA, pp. 3711–3716. IEEE, Los Alamitos (April 2007)
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)
Berberich, E., Fogel, E., Halperin, D., Mehlhorn, K., Wein, R.: Arrangements on parametric surfaces I: General framework and infrastructure. Mathematics in Computer Science 4(1), 45–66 (2010)
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Salzman, O., Hemmer, M., Raveh, B., Halperin, D. (2011). Motion Planning via Manifold Samples. In: Demetrescu, C., Halldórsson, M.M. (eds) Algorithms – ESA 2011. ESA 2011. Lecture Notes in Computer Science, vol 6942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23719-5_42
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