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.
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), and by the Hermann Minkowski-Minerva Center for Geometry at Tel Aviv University.
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References
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
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.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)
Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry, 2nd edn. Algorithms and Computation in Mathematics. Springer (2006)
Berberich, E., Fogel, E., Halperin, D., Kerber, M., Setter, O.: Arrangements on parametric surfaces II: Concretizations and applications. MCS 4, 67–91 (2010)
Berberich, E., Fogel, E., Halperin, D., Mehlhorn, K., Wein, R.: Arrangements on parametric surfaces I: General framework and infrastructure. MCS 4, 45–66 (2010)
Canny, J.F.: Complexity of Robot Motion Planning (ACM Doctoral Dissertation Award). The MIT Press (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)
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)
Dalibard, S., Laumond, J.P.: Linear dimensionality reduction in random motion planning. I. J. Robotic Res. 30(12), 1461–1476 (2011)
De Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer (2008)
Foegl, E., Halperin, D., Wein, R.: CGAL Arrangements and their Applications. Springer, Heidelberg (2012)
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)
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)
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)
Hsu, D., Latombe, J.C., Motwani, R.: Path planning in expansive configuration spaces. Int. J. Comp. Geo. & App. 4, 495–512 (1999)
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)
Kavraki, L.E., Latombe, J.C., Motwani, R., Raghavan, P.: Randomized query processing in robot path planning. JCSS 57(1), 50–60 (1998)
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)
Kuffner, J.J., Lavalle, S.M.: RRT-Connect: An efficient approach to single-query path planning. In: ICRA, pp. 995–1001 (2000)
Ladd, A.M., Kavraki, L.E.: Generalizing the analysis of PRM. In: ICRA, pp. 2120–2125. IEEE Press (2002)
Latombe, J.C.: Robot Motion Planning. Kluwer Academic Publishers, Norwell (1991)
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)
LaValle, S.M.: Planning Algorithms. Cambridge University Press, Cambridge (2006)
Lien, J.M.: Hybrid motion planning using Minkowski sums. In: RSS 2008 (2008)
Lozano-Perez, T.: Spatial planning: A configuration space approach. MIT AI Memo 605 (1980)
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)
Reif, J.H.: Complexity of the mover’s problem and generalizations. In: FOCS, pp. 421–427. IEEE Computer Society, Washington, DC (1979)
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)
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)
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)
Sharir, M.: Algorithmic Motion Planning, Handbook of Discrete and Computational Geometry, 2nd edn. CRC Press, Inc., Boca Raton (2004)
Siek, J.G., Lee, L.Q., Lumsdaine, A.: The Boost Graph Library: User Guide and Reference Manual. Addison-Wesley Professional (2001)
The CGAL Project: CGAL User and Reference Manual, 3.7 edn. CGAL Editorial Board (2010), http://www.cgal.org/
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)
Yang, J., Sacks, E.: RRT path planner with 3 DOF local planner. In: ICRA, pp. 145–149 (2006)
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)
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Salzman, O., Hemmer, M., Halperin, D. (2013). On the Power of Manifold Samples in Exploring Configuration Spaces and the Dimensionality of Narrow Passages. In: Frazzoli, E., Lozano-Perez, T., Roy, N., Rus, D. (eds) Algorithmic Foundations of Robotics X. Springer Tracts in Advanced Robotics, vol 86. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36279-8_19
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