Minimum-time trajectories for kinematic mobile robots and other planar rigid bodies with finite control sets

This paper presents first attempts at a method for searching for time-optimal trajectories for a general model of mobile robots that includes Dubins and Reeds-Shepp cars, differential-drive robots, and omnidirectional robots as special cases. The paper takes as a starting point recent results by the authors that describe necessary conditions on the trajectories, based on Pontryagin's Maximum Principle. These necessary conditions reduce the problem of finding an optimal trajectory between start and goal to a few one-dimensional search problems. This search is not formally guaranteed to find a near-optimal trajectory if the sampling of the search space is not fine enough, but comparison to existing analytical results for specific systems, and a complete numerical search over trajectories with only a few control switches, demonstrates effectiveness of the method.

[1]  M. L. Chambers.  The Mathematical Theory of Optimal Processes , 1965 .

[2]  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 .

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

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

[5]  Subhash Suri,et al.  Curvature-constrained shortest paths in a convex polygon (extended abstract) , 1998, SCG '98.

[6]  Monique Chyba,et al.  Designing Efficient Trajectories for Underwater Vehicles Using Geometric Control Theory , 2005 .

[7]  Devin J. Balkcom,et al.  Generalizing Dubins Curves: Minimum-time Sequences of Body-fixed Rotations and Translations in the Plane , 2010, Int. J. Robotics Res..

[8]  Subhash Suri,et al.  Curvature-Constrained Shortest Paths in a Convex Polygon , 2002, SIAM J. Comput..

[9]  Andrew D. Lewis,et al.  Optimal control for a simplified hovercraft model , 2000 .

[10]  François G. Pin,et al.  Time-Optimal Trajectories for Mobile Robots With Two Independently Driven Wheels , 1994, Int. J. Robotics Res..

[11]  Antonio Bicchi,et al.  Optimal paths in a constrained image plane for purely image-based parking , 2008, 2008 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[12]  Devin J. Balkcom,et al.  The Minimum-Time Trajectories for an Omni-Directional Vehicle , 2006, WAFR.

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

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

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

[16]  Kevin M. Lynch,et al.  Stable Pushing: Mechanics, Controllability, and Planning , 1995, Int. J. Robotics Res..

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

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

[19]  Rafael Murrieta-Cid,et al.  A Motion Planner for Maintaining Landmark Visibility with a Differential Drive Robot , 2008, WAFR.

[20]  Jean-Yves Fourquet,et al.  Minimum time motion of a mobile robot with two independent, acceleration-driven wheels , 1997, Proceedings of International Conference on Robotics and Automation.

[21]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[22]  Kenneth Y. Goldberg,et al.  Motion Planning Under Uncertainty for Image-guided Medical Needle Steering , 2008, Int. J. Robotics Res..

[23]  Raffaello D'Andrea,et al.  Near-optimal dynamic trajectory generation and control of an omnidirectional vehicle , 2004, Robotics Auton. Syst..

[24]  Andrei Furtuna.  Minimum time kinematic trajectories for self-propelled rigid bodies in the unobstructed plane , 2011 .

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

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

[27]  Devin J. Balkcom,et al.  Minimum Wheel-Rotation Paths for Differential-Drive Mobile Robots , 2006, Proceedings 2006 IEEE International Conference on Robotics and Automation, 2006. ICRA 2006..