The asymmetric sinistral/dextral Markov-Dubins problem

We consider a variation of the classical Markov-Dubins problem dealing with curvature-constrained, shortest paths in the plane with prescribed initial and terminal positions and tangents, when the lower and upper bounds of the curvature are not necessarily equal. The motivation for this problem stems from vehicle navigation applications when the vehicle may be biased in taking turns at a particular direction due to hardware failures or environmental conditions. We employ optimal control to characterize the structure of the shortest path and we resort to geometric techniques to provide sufficient conditions for optimality of the resulting path.

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

[2]  J. Ball.  OPTIMIZATION—THEORY AND APPLICATIONS Problems with Ordinary Differential Equations (Applications of Mathematics, 17) , 1984 .

[3]  Lamberto Cesari,et al.  Optimization-Theory And Applications , 1983 .

[4]  Marilena Vendittelli,et al.  Nonholonomic distance to polygonal obstacles for a car-like robot of polygonal shape , 2006, IEEE Transactions on Robotics.

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

[6]  Marilena Vendittelli,et al.  Obstacle distance for car-like robots , 1999, IEEE Trans. Robotics Autom..

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

[8]  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 .

[9]  Jean-Daniel Boissonnat,et al.  Shortest path synthesis for Dubins non-holonomic robot , 1994, Proceedings of the 1994 IEEE International Conference on Robotics and Automation.

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

[11]  Jean-Daniel Boissonnat,et al.  Accessibility region for a car that only moves forwards along optimal paths , 1993 .

[12]  J. Hedrick,et al.  Optimal Path Planning with a Kinematic Airplane Model , 2007 .

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

[14]  Philippe Souères,et al.  Optimal trajectories for nonholonomic mobile robots , 1998 .

[15]  E. Cockayne,et al.  Plane Motion of a Particle Subject to Curvature Constraints , 1975 .

[16]  H. Sussmann,et al.  The Markov-Dubins problem with angular acceleration control , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[17]  P. Souéres,et al.  Shortest paths synthesis for a car-like robot , 1996, IEEE Trans. Autom. Control..

[18]  Jean-Daniel Boissonnat,et al.  Shortest paths of Bounded Curvature in the Plane , 1991, Geometric Reasoning for Perception and Action.