Metric cells: Towards complete search for optimal trajectories

This paper presents a definition of convexity useful for describing local optimality in configuration spaces, proves that finding convex regions is relatively easy, and presents an algorithm for approximating the free configuration space using a set of such convex regions. The paper examines simple but interesting systems: serial planar arms with revolute joints, and a Reeds-Shepp car. The paper experimentally explores an approach for finding good (although not necessarily optimal) trajectories using the derived data structure.

[1]  A. Bellaïche.  The tangent space in sub-riemannian geometry , 1994 .

[2]  Vijay Kumar,et al.  Invariants for homology classes with application to optimal search and planning problem in robotics , 2012, Annals of Mathematics and Artificial Intelligence.

[3]  Gerardo Lafferriere,et al.  A Differential Geometric Approach to Motion Planning , 1993 .

[4]  P. Soueres,et al.  Shortest paths to obstacles for a polygonal car-like robot , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[5]  Devin J. Balkcom,et al.  Analytical time-optimal trajectories for an omni-directional vehicle , 2012, 2012 IEEE International Conference on Robotics and Automation.

[6]  Guy Desaulniers.  On shortest paths for a car-like robot maneuvering around obstacles , 1996, Robotics Auton. Syst..

[7]  Matthew T. Mason,et al.  Mechanics of Robotic Manipulation , 2001 .

[8]  Nancy M. Amato,et al.  Toggle PRM: A Coordinated Mapping of C-Free and C-Obstacle in Arbitrary Dimension , 2012, WAFR.

[9]  Steven M. LaValle,et al.  Planning algorithms , 2006 .

[10]  F. Jean,et al.  Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning , 2014 .

[11]  L. Dubins.  On Curves of Minimal Length with a Constraint on Average Curvature, and with Prescribed Initial and Terminal Positions and Tangents , 1957 .

[12]  Nancy M. Amato,et al.  UOBPRM: A uniformly distributed obstacle-based PRM , 2012, 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[13]  Emilio Frazzoli,et al.  Efficient Collision Checking in Sampling-Based Motion Planning , 2012, WAFR.

[14]  Timothy Bretl,et al.  Proving path non-existence using sampling and alpha shapes , 2012, 2012 IEEE International Conference on Robotics and Automation.

[15]  Jean-Claude Latombe,et al.  Nonholonomic multibody mobile robots: Controllability and motion planning in the presence of obstacles , 2005, Algorithmica.

[16]  Andrew Chi-Chih Yao,et al.  An Almost Optimal Algorithm for Unbounded Searching , 1976, Inf. Process. Lett..

[17]  Devin J. Balkcom,et al.  Time Optimal Trajectories for Bounded Velocity Differential Drive Vehicles , 2002, Int. J. Robotics Res..

[18]  Cong Wang,et al.  On soft predicates in subdivision motion planning , 2013, SoCG '13.

[19]  L. Shepp,et al.  OPTIMAL PATHS FOR A CAR THAT GOES BOTH FORWARDS AND BACKWARDS , 1990 .

[20]  Emilio Frazzoli,et al.  Sampling-based algorithms for optimal motion planning , 2011, Int. J. Robotics Res..

[21]  Emilio Frazzoli,et al.  Algorithmic Foundations of Robotics X - Proceedings of the Tenth Workshop on the Algorithmic Foundations of Robotics, WAFR 2012, MIT, Cambridge, Massachusetts, USA, June 13-15 2012 , 2013, WAFR.

[22]  Marilena Vendittelli,et al.  Shortest Paths to Obstacles for a Polygonal Dubins Car , 2009, IEEE Transactions on Robotics.

[23]  J. Sussmann,et al.  SHORTEST PATHS FOR THE REEDS-SHEPP CAR: A WORKED OUT EXAMPLE OF THE USE OF GEOMETRIC TECHNIQUES IN NONLINEAR OPTIMAL CONTROL. 1 , 1991 .

[24]  Bruce Randall Donald,et al.  Provably good approximation algorithms for optimal kinodynamic planning: Robots with decoupled dynamics bounds , 1995, Algorithmica.

[25]  Jean-Paul Laumond,et al.  Robot Motion Planning and Control , 1998 .

[26]  Kostas E. Bekris,et al.  Sparse Methods for Efficient Asymptotically Optimal Kinodynamic Planning , 2014, WAFR.

[27]  Robin Deits,et al.  Computing Large Convex Regions of Obstacle-Free Space Through Semidefinite Programming , 2014, WAFR.

[28]  J. Whitehead,et al.  CONVEX REGIONS IN THE GEOMETRY OF PATHS , 1932 .