A geometric approach to shortest bounded curvature paths

Consider two elements in the tangent bundle of the Euclidean plane $(x,X),(y,Y)\in T{\mathbb R}^2$. In this work we address the problem of characterizing the paths of bounded curvature and minimal length starting at $x$, finishing at $y$ and having tangents at these points $X$ and $Y$ respectively. This problem was first investigated in the late 50's by Lester Dubins. In this note we present a constructive proof of Dubins' result giving special emphasis on the geometric nature of this problem.

[1]  H. Whitney.  On regular closed curves in the plane , 1937 .

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

[3]  Stephen Smale,et al.  Regular curves on Riemannian manifolds , 1958 .

[4]  L. Dubins.  On plane curves with curvature. , 1961 .

[5]  Harold H. Johnson.  An application of the maximum principle to the geometry of plane curves , 1974 .

[6]  P. Souéres,et al.  The Shortest path synthesis for non-holonomic robots moving forwards , 1994 .

[7]  Richard M. Murray,et al.  A Mathematical Introduction to Robotic Manipulation , 1994 .

[8]  Pankaj K. Agarwal,et al.  Motion planning for a steering-constrained robot through moderate obstacles , 1995, STOC '95.

[9]  H. Sussmann.  Shortest 3-dimensional paths with a prescribed curvature bound , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[10]  F. Monroy-Pérez,et al.  Non-Euclidean Dubins' Problem , 1998 .

[11]  O. C. Durumeric.  Local structure of ideal shapes of knots , 2002 .

[12]  Y. Chitour,et al.  Dubins’ problem on surfaces. I. nonnegative curvature , 2005 .

[13]  Yacine Chitour,et al.  Dubins' Problem on Surfaces II: Nonpositive Curvature , 2006, SIAM J. Control. Optim..

[14]  Nicholas C. Wormald,et al.  Constrained Path Optimisation for Underground Mine Layout , 2007, World Congress on Engineering.

[15]  Ricardo G. Sanfelice,et al.  On the Optimality of Dubins Paths across Heterogeneous Terrain , 2008, HSCC.

[16]  E. Feron,et al.  On the Curvature-Constrained Traveling Salesman Problem , 2008 .

[17]  David G. Kirkpatrick,et al.  Discrete Dubins Paths , 2012, ArXiv.

[18]  Irina S. Dolinskaya,et al.  Time-optimal trajectories with bounded curvature in anisotropic media , 2012, Int. J. Robotics Res..

[19]  J. Hyam Rubinstein,et al.  Curvature-constrained directional-cost paths in the plane , 2012, J. Glob. Optim..

[20]  Stuart R. Phinn,et al.  Coral Reef Remote Sensing: A Guide for Mapping, Monitoring and Management , 2013 .

[21]  Hyam Rubinstein,et al.  The classification of homotopy classes of bounded curvature paths , 2014 .

[23]  José L. Ayala.  On the Topology of the Spaces of Plane Curves , 2014 .

[24]  Jos'e Ayala,et al.  Length minimising bounded curvature paths in homotopy classes , 2014, 1403.4930.

[25]  José L. Ayala,et al.  Non-uniqueness of the homotopy class of bounded curvature paths , 2014, 1403.4911.