On the Relationship Between Classical Grid Search and Probabilistic Roadmaps

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 the 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 dispersion-optimal 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, multiple-query PRM and the single-query, Lazy PRM. Surprisingly, even some forms of grid search yield performance that is comparable to the original PRM. 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 to be superior to that obtained by random sampling. Thus, in surprising contrast to recent trends, there is both experimental and theoretical evidence that the randomization used in the original PRM is not advantageous.

[1]  Steven M. LaValle,et al.  Incremental low-discrepancy lattice methods for motion planning , 2003, 2003 IEEE International Conference on Robotics and Automation (Cat. No.03CH37422).

[2]  Robert Bohlin,et al.  Path planning in practice; lazy evaluation on a multi-resolution grid , 2001, Proceedings 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems. Expanding the Societal Role of Robotics in the the Next Millennium (Cat. No.01CH37180).

[3]  Steven M. LaValle,et al.  Quasi-randomized path planning , 2001, Proceedings 2001 ICRA. IEEE International Conference on Robotics and Automation (Cat. No.01CH37164).

[4]  Fred J. Hickernell,et al.  Randomized Halton sequences , 2000 .

[5]  Lydia E. Kavraki,et al.  A framework for using the workspace medial axis in PRM planners , 2000, Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065).

[6]  Lydia E. Kavraki,et al.  Path planning using lazy PRM , 2000, Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065).

[7]  Fred J. Hickernell,et al.  Extensible Lattice Sequences for Quasi-Monte Carlo Quadrature , 2000, SIAM J. Sci. Comput..

[8]  Thierry Siméon,et al.  Visibility-based probabilistic roadmaps for motion planning , 2000, Adv. Robotics.

[9]  Mark H. Overmars,et al.  The Gaussian sampling strategy for probabilistic roadmap planners , 1999, Proceedings 1999 IEEE International Conference on Robotics and Automation (Cat. No.99CH36288C).

[10]  Lydia E. Kavraki,et al.  On finding narrow passages with probabilistic roadmap planners , 1998 .

[11]  Daniel Vallejo,et al.  OBPRM: an obstacle-based PRM for 3D workspaces , 1998 .

[12]  Rajeev Motwani,et al.  Path planning in expansive configuration spaces , 1997, Proceedings of International Conference on Robotics and Automation.

[13]  Lydia E. Kavraki,et al.  Probabilistic roadmaps for path planning in high-dimensional configuration spaces , 1996, IEEE Trans. Robotics Autom..

[14]  Florent Lamiraux,et al.  On the expected complexity of random path planning , 1996, Proceedings of IEEE International Conference on Robotics and Automation.

[15]  Nancy M. Amato,et al.  A randomized roadmap method for path and manipulation planning , 1996, Proceedings of IEEE International Conference on Robotics and Automation.

[16]  I. Sloan.  Lattice Methods for Multiple Integration , 1994 .

[17]  Pierre Bessière,et al.  The Ariadne's Clew Algorithm , 1993, J. Artif. Intell. Res..

[18]  Lydia E. Kavraki,et al.  Computation of configuration-space obstacles using the fast Fourier transform , 1993, [1993] Proceedings IEEE International Conference on Robotics and Automation.

[19]  Narendra Ahuja,et al.  A potential field approach to path planning , 1992, IEEE Trans. Robotics Autom..

[20]  John F. Canny,et al.  Computing Roadmaps of General Semi-Algebraic Sets , 1991, Comput. J..

[21]  Koichi Kondo,et al.  Motion planning with six degrees of freedom by multistrategic bidirectional heuristic free-space enumeration , 1991, IEEE Trans. Robotics Autom..

[22]  Bruce Randall Donald,et al.  Real-time robot motion planning using rasterizing computer graphics hardware , 1990, SIGGRAPH.

[23]  Jean-Claude Latombe,et al.  A Monte-Carlo algorithm for path planning with many degrees of freedom , 1990, Proceedings., IEEE International Conference on Robotics and Automation.

[24]  Bernhard Glavina,et al.  Solving findpath by combination of goal-directed and randomized search , 1990, Proceedings., IEEE International Conference on Robotics and Automation.

[25]  W. Fischer,et al.  Sphere Packings, Lattices and Groups , 1990 .

[26]  Michael A. Erdmann,et al.  On probabilistic strategies for robot tasks , 1989 .

[27]  John Canny,et al.  The complexity of robot motion planning , 1988 .

[28]  Bruce Randall Donald,et al.  A Search Algorithm for Motion Planning with Six Degrees of Freedom , 1987, Artif. Intell..

[29]  Larry S. Davis,et al.  Multiresolution path planning for mobile robots , 1986, IEEE J. Robotics Autom..

[30]  Chee-Keng Yap,et al.  A "Retraction" Method for Planning the Motion of a Disc , 1985, J. Algorithms.

[31]  Bernard Faverjon,et al.  Obstacle avoidance using an octree in the configuration space of a manipulator , 1984, ICRA.

[32]  J. Schwartz,et al.  On the “piano movers” problem. II. General techniques for computing topological properties of real algebraic manifolds , 1983 .

[33]  J. Schwartz,et al.  On the Piano Movers' Problem: III. Coordinating the Motion of Several Independent Bodies: The Special Case of Circular Bodies Moving Amidst Polygonal Barriers , 1983 .

[34]  Tomás Lozano-Pérez,et al.  Spatial Planning: A Configuration Space Approach , 1983, IEEE Transactions on Computers.

[35]  J. Halton.  On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals , 1960 .

[36]  J. Hammersley.  MONTE CARLO METHODS FOR SOLVING MULTIVARIABLE PROBLEMS , 1960 .

[37]  H. Weyl.  Über die Gleichverteilung von Zahlen mod. Eins , 1916 .

[38]  M. Pavone,et al.  Kinodynamic Planning , 2021, Encyclopedia of Robotics.

[39]  G. Swaminathan.  Robot Motion Planning , 2006 .

[40]  S. Tezuka.  Quasi-Monte Carlo — Discrepancy between Theory and Practice , 2002 .

[41]  Mark H. Overmars,et al.  A Comparative Study of Probabilistic Roadmap Planners , 2002, WAFR.

[42]  Y. Wang,et al.  An Historical Overview of Lattice Point Sets , 2002 .

[43]  Jean-Claude Latombe,et al.  A Single-Query Bi-Directional Probabilistic Roadmap Planner with Lazy Collision Checking , 2001, ISRR.

[44]  Steven M. LaValle,et al.  Rapidly-Exploring Random Trees: Progress and Prospects , 2000 .

[45]  Nancy M. Amato,et al.  MAPRM: a probabilistic roadmap planner with sampling on the medial axis of the free space , 1999, Proceedings 1999 IEEE International Conference on Robotics and Automation (Cat. No.99CH36288C).

[46]  Dinesh Manocha,et al.  Randomized Path Planning for a Rigid Body Based on Hardware Accelerated Voronoi Sampling , 1999 .

[47]  H. Niederreiter,et al.  Nets, ( t, s )-Sequences, and Algebraic Geometry , 1998 .

[48]  F. J. Hickernell.  Lattice rules: how well do they measure up? in random and quasi-random point sets , 1998 .

[49]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[50]  A. G. Sukharev.  Optimal strategies of the search for an extremum , 1971 .