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 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.

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

[2]  Lydia E. Kavraki.  Computation of configuration-space obstacles using the fast Fourier transform , 1995, IEEE Trans. Robotics Autom..

[3]  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.

[4]  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).

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

[6]  El-Ghazali Talbi,et al.  The Ariadne's clew algorithm; From Animal to Animats , 1992 .

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

[8]  B. Faverjon,et al.  A local based approach for path planning of manipulators with a high number of degrees of freedom , 1987, Proceedings. 1987 IEEE International Conference on Robotics and Automation.

[9]  Bernard Chazelle,et al.  The Discrepancy Method , 1998, ISAAC.

[10]  Bruce Randall Donald,et al.  Kinodynamic motion planning , 1993, JACM.

[11]  Peter Shirley,et al.  Discrepancy as a Quality Measure for Sample Distributions , 1991, Eurographics.

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

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

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

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

[16]  N. J. A. Sloane,et al.  Sphere Packings, Lattices and Groups , 1987, Grundlehren der mathematischen Wissenschaften.

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

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

[19]  Steven M. LaValle,et al.  Steps toward derandomizing RRTs , 2004, Proceedings of the Fourth International Workshop on Robot Motion and Control (IEEE Cat. No.04EX891).

[20]  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).

[21]  Steven M. LaValle,et al.  On the Relationship between Classical Grid Search and Probabilistic Roadmaps , 2004, Int. J. Robotics Res..

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

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

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

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

[26]  Fred J. Hickernell,et al.  Monte Carlo and Quasi-Monte Carlo Methods 2000 , 2002 .

[27]  V. Rich.  Personal communication , 1989, Nature.

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

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

[30]  Jean-Claude Latombe,et al.  Robot motion planning , 1970, The Kluwer international series in engineering and computer science.

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

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

[33]  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).

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

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

[36]  Seth Hutchinson,et al.  A Framework for Real-time Path Planning in Changing Environments , 2002, Int. J. Robotics Res..

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

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

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

[40]  S. Tezuka.  Uniform Random Numbers: Theory and Practice , 1995 .

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

[42]  S. LaValle,et al.  Randomized Kinodynamic Planning , 2001 .

[43]  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).

[44]  Michael O. Rabin,et al.  Transaction Protection by Beacons , 1983, J. Comput. Syst. Sci..

[45]  P. Hellekalek,et al.  Random and Quasi-Random Point Sets , 1998 .

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

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

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

[49]  Blake Hannaford,et al.  Resolution-First Scanning of Multidimensional Spaces , 1993, CVGIP Graph. Model. Image Process..

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

[51]  B. Donald,et al.  Kinodynamic Motion Planning 1 Kinodynamic Motion Planning 5 , 1993 .

[52]  R. A. Mitchell.  Error estimates arising from certain pseudorandom sequences in a quasirandom search method , 1990 .

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

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

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

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

[57]  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).

[58]  J. Schwartz,et al.  On the “piano movers'” problem I. The case of a two‐dimensional rigid polygonal body moving amidst polygonal barriers , 1983 .

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

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

[61]  Rajeev Motwani,et al.  Path Planning in Expansive Configuration Spaces , 1999, Int. J. Comput. Geom. Appl..

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

[63]  Bruce Randall Donald,et al.  Algorithmic and Computational Robotics: New Directions , 2001 .

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

[65]  B. Faverjon,et al.  Probabilistic Roadmaps for Path Planning in High-Dimensional Con(cid:12)guration Spaces , 1996 .

[66]  Jean-Claude Latombe,et al.  Proceedings of the workshop on Algorithmic foundations of robotics , 1995 .

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

[68]  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).

[69]  J. T. Shwartz,et al.  On the Piano Movers' Problem : III , 1983 .