Abstract
We present, implement, and analyze a spectrum of closely-related planners, designed to gain insight into the relationship between classical grid search and probabilistic roadmaps (PRMs). Building on quasi-Monte Carlo sampling literature, we have developed deterministic variants of the PRM that use low-discrepancy and low-dispersion samples, including lattices. Classical grid search is extended using subsampling for collision detection and also the optimal-dispersion Sukharev grid, which can be considered as a kind of lattice-based roadmap to complete the spectrum. Our experimental results show that the deterministic variants of the PRM offer performance advantages in comparison to the original PRM and the recent Lazy PRM. This even includes searching using a grid with subsampled collision checking. Our theoretical analysis shows that all of our deterministic PRM variants are resolution complete and achieve the best possible asymptotic convergence rate, which is shown superior to that obtained by random sampling. Thus, in surprising contrast to recent trends, there is both experimental and theoretical evidence that some forms of grid search are superior to the original PRM.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
N. M. Amato, O. B. Bayazit, L. K. Dale, C. Jones, and D. Vallejo. OBPRM: An obstacle-based PRM for 3D workspaces. In Proceedings of the Workshop on Algorithmic Foundations of Robotics, pages 155–168, 1998.
N. M. Amato and Y. Wu. A randomized roadmap method for path and manipulation planning. In IEEE Int. Conf. Robot. & Autom., pages 113–120, 1996.
J. Barraquand, L. Kavraki, J.-C. Latombe, T.-Y. Li, R. Motwani, and P. Ragha-van. A random sampling scheme for robot path planning. In G. Giralt and G. Hirzinger, editors, Proc. of the 7th International Symposium on Robotics Research, pages 249–264. Springer, New York, NY, 1996.
J. Barraquand and J.-C. Latombe. A Monte-Carlo algorithm for path planning with many degrees of freedom. In IEEE Int. Conf. Robot. & Autom., pages 1712–1717, 1990.
R. Bohlin. Path planning in practice; lazy evaluation on a multi-resolution grid. In IEEE/RS J Int. Conf on Intelligent Robots & Systems, 2001.
R. Bohlin and L. Kavraki. Path planning using Lazy PRM. In IEEE Int. Conf. Robot & Autom., 2000.
R. Bohlin and L. Kavraki. A randomized algorithm for robot path planning based on lazy evalaution. In S. Rajasekaran, P. Pardalos, J. Reif, and J. Rolim, editors, Handbook on Randomized Computation. Kluwer Academic, 2001.
V. Boor, N. H. Overmars, and A. F. van der Stappen. The gaussian sampling strategy for probabilistic roadmap planners. In IEEE Int. Conf. Robot. & Autom., pages 1018–1023, 1999.
M. Branicky, S. M. LaValle, K. Olsen, and L. Yang. Quasi-randomized path planning. In Proc. IEEE InVl Conf. on Robotics and Automation, pages 1481–1487, 2001.
B. R. Donald. A search algorithm for motion planning with six degrees of freedom. Artif IntelL, 31: 295–353, 1987.
B. R. Donald, P. G. Xavier, J. Canny, and J. Reif. Kinodynamic planning. Journal of the ACM, 40: 1048–66, November 1993.
B. Faverjon. Obstacle avoidance using an octree in the configuration space of a manipulator. In IEEE Int. Conf. Robot. & Autom., pages 504–512, 1984.
B. Faverjon and P. Tournassoud. A local based method for path planning of manipulators with a high number of degrees of freedom. In IEEE Int. Conf. Robot & Autom., pages 1152–1159, 1987.
C. Holleman and L. E. Kavraki. A framework for using the workspace medial axis in PRM planners. In IEEE Int. Conf Robot & Autom., pages 1408–1413, 2000.
D. Hsu, L. E. Kavraki, J.-C. Latombe, R. Motwani, and S. Sorkin. On finding narrow passages with probabilistic roadmap planners. In et al. P. Agarwal, editor, Robotics: The Algorithmic Perspective, pages 141–154. A.K. Peters, Wellesley, MA, 1998.
D. Hsu, J.-C. Latombe, and R. Motwani. Path planning in expansive configuration spaces. Int. J. Comput. Geom. & Appl, 4: 495–512, 1999.
Y. K. Hwang and N. Ahuja. A potential field approach to path planning. IEEE Trans. Robot & Autom., 8 (l): 23–32, February 1992.
L. E. Kavraki. Computation of configuration-space obstacles using the Fast Fourier Transform. IEEE Trans. Robot & Autom., 11 (3): 408–413 1995.
L. E. Kavraki, P. Svestka, J.-C. Latombe, and M. H. Overmars. Probabilistic roadmaps for path planning in high-dimensional configuration spaces. IEEE Trans. Robot. & Autom., 12 (4): 566–580, June 1996.
F. Lamiraux and J.-P. Laumond. On the expected complexity of random path planning. In IEEE Int. Conf. Robot & Autom., pages 3306–3311, 1996.
J.-C. Latombe. Robot Motion Planning. Kluwer Academic Publishers, Boston, MA, 1991.
S. M. LaValle and J. J. Kuffner. Rapidly-exploring random trees: Progress and prospects. In B. R. Donald, K. M. Lynch, and D. Rus, editors, Algorithmic and Computational Robotics: New Directions, pages 293–308. A K Peters, Wellesley, MA, 2001.
P. Leven and S. Hutchinson. Real-time motion planning in changing environments. In Proc. International Symposium on Robotics Research, 2000.
S. R. Lindemann and S. M. LaValle. Incremental low-discrepancy lattice methods for motion planning. 2003. Submitted to IEEE International Conference on Robotics and Automation.
J. Matousek. Geometric Discrepancy. Springer-Verlag, Berlin, 1999.
E. Mazer, J. M. Ahuactzin, ana P. Bessière. The Ariadne’s clew algorithm. J. Artificial Intell. Res., 9: 295–316, November 1998.
E. Mazer, G. Talbi, J. M. Ahuactzin, and P. Bessière. The Ariadne’s clew algorithm. In Proc. Int. Conf. of Society of Adaptive Behavior, Honolulu, 1992.
H. Niederreiter. Random Number Generation and Quasi-Monte-Carlo Methods. Society for Industrial and Applied Mathematics, Philadelphia, USA, 1992.
H. Niederreiter and C. P. Xing. Nets, (t,s)-sequences, and algebraic geometry. In P. Hellekalek and G. Larcher, editors, Random and Quasi-Random Point Sets, Lecture Notes in Statistics, Vol. 138, pages 267–302. Springer-Verlag, Berlin, 1998.
C. Pisula, K. Hoff, M. Lin, and D. Manoch. Randomized path planning for a rigid body based on hardware accelerated Voronoi sampling. In Proc. Workshop on Algorithmic Foundation of Robotics, 2000.
G. Sanchez and J.-C. Latombe. A single-query bi-directional probabilistic roadmap planner with lazy collision checking. In Int. Symp. Robotics Research, 2001.
T. Simeon, J.-P. Laumond., and C. Nissoux. Visibility based probabilistic roadmaps for motion planning. Advanced Robotics Journal, 14 (6), 2000.
Thierry Simeon, 2002. Personal communication.
A. G. Sukharev. Optimal strategies of the search for an extremum. U.S.S.R. Computational Mathematics and Mathematical Physics, 11 (4), 1971. Translated from Russian, Zh. Vuchisl. Mat i Mat Fiz11, 4, 910–924, 1971.
X. Wang and F. J. Hickernell. Randomized halton sequences. Math. Comp. Modelling, 32: 887–899, 2000.
S. A. Wilmarth, N. M. Amato, and P. F. Stiller. MAPRM: A probabilistic roadmap planner with sampling on the medial axis of the free space. In IEEE Int. Conf. Robot. & Autom., pages 1024–1031, 1999.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
LaValle, S.M., Branicky, M.S. (2004). On the Relationship between Classical Grid Search and Probabilistic Roadmaps. In: Boissonnat, JD., Burdick, J., Goldberg, K., Hutchinson, S. (eds) Algorithmic Foundations of Robotics V. Springer Tracts in Advanced Robotics, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45058-0_5
Download citation
DOI: https://doi.org/10.1007/978-3-540-45058-0_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-07341-0
Online ISBN: 978-3-540-45058-0
eBook Packages: Springer Book Archive