The Markov-Dubins problem in the presence of a stochastic drift field

We consider the problem of navigating a small Dubins-type aerial or marine vehicle to a prescribed destination set in minimum expected time and in the presence of a stochastic drift field induced by local winds or currents. First, we present a deterministic control law that is independent of the local winds/currents and their statistics. Next, by employing numerical techniques from stochastic optimal control, we compute an optimal feedback control strategy that incorporates the stochastic variation in the wind when driving the Dubins vehicle to its destination set in minimum expected time. Our analyses and simulations offer a side-by-side comparison of the optimal deterministic and stochastic optimal feedback control laws for this problem, and they illustrate that the deterministic control can, in many cases, capture the salient features of structure of the stochastic optimal feedback control.

[1]  C. Gardiner.  Stochastic Methods: A Handbook for the Natural and Social Sciences , 2009 .

[2]  Efstathios Bakolas,et al.  Optimal steering for kinematic vehicles with applications to spatially distributed agents , 2011 .

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

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

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

[6]  Efstathios Bakolas,et al.  Feedback Navigation in an Uncertain Flowfield and Connections with Pursuit Strategies , 2012 .

[7]  J. Quadrat.  Numerical methods for stochastic control problems in continuous time , 1994 .

[8]  H. Kushner.  Numerical Methods for Stochastic Control Problems in Continuous Time , 2000 .

[9]  T. Pecsvaradi.  Optimal horizontal guidance law for aircraft in the terminal area , 1972 .

[10]  P. Tsiotras,et al.  Time-optimal synthesis for the Zermelo-Markov-Dubins problem: The constant wind case , 2010, Proceedings of the 2010 American Control Conference.

[11]  Phillip R. Chandler,et al.  Tour Planning for an Unmanned Air Vehicle Under Wind Conditions (Preprint) , 2007 .

[12]  V. Patsko,et al.  NUMERICAL STUDY OF THE “HOMICIDAL CHAUFFEUR” DIFFERENTIAL GAME WITH THE REINFORCED PURSUER , 2006 .

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

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

[15]  Efstathios Bakolas,et al.  Optimal Synthesis of the Asymmetric Sinistral/Dextral Markov–Dubins Problem , 2011, J. Optim. Theory Appl..

[16]  Craig A. Woolsey,et al.  Minimum-Time Path Planning for Unmanned Aerial Vehicles in Steady Uniform Winds , 2009 .

[17]  Dejan Milutinovic,et al.  A stochastic approach to Dubins feedback control for target tracking , 2011, 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[18]  Ross P. Anderson,et al.  DUBINS VEHICLE TRACKING OF A TARGET WITH UNPREDICTABLE TRAJECTORY , 2011 .