Minimum time feedback control of autonomous underwater vehicles

We study the problem of steering a vehicle from its initial position in a 2D, time-varying, ocean flow field to a desired target position in minimum time. In particular, we focus on the case where the magnitude of the flow field sometimes exceeds the speed of the vehicle, and thus controllability is an issue. In order to obtain globally optimal, closed loop trajectories, one solves a dynamic Hamilton Jacobi Bellman equation for the optimal “time-to-go” and associated optimal feedback control law. We do this indirectly via a simple but powerful extremal field algorithm, which allows incremental refinement of the solution and is trivial to parallelize. We characterize solutions and the resulting closed loop optimal trajectories for a time-invariant double gyre flow field and for a numerically-defined, time-varying flow field from a real model of the Adriatic Sea.

[1]  R. Newcomb.  VISCOSITY SOLUTIONS OF HAMILTON-JACOBI EQUATIONS , 2010 .

[2]  Kokichi Sugihara,et al.  Stable marker-particle method for the Voronoi diagram in a flow field , 2007 .

[3]  Naomi Ehrich Leonard,et al.  Collective Motion, Sensor Networks, and Ocean Sampling , 2007, Proceedings of the IEEE.

[4]  Rustam Stolkin,et al.  Optimal AUV path planning for extended missions in complex, fast-flowing estuarine environments , 2007, Proceedings 2007 IEEE International Conference on Robotics and Automation.

[5]  J. Tsitsiklis,et al.  Efficient algorithms for globally optimal trajectories , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[6]  J A Sethian,et al.  A fast marching level set method for monotonically advancing fronts. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[7]  Yan Pailhas,et al.  Path Planning for Autonomous Underwater Vehicles , 2007, IEEE Transactions on Robotics.

[8]  Thomas Holzhüter,et al.  Optimal regulator for the inverted pendulum via Euler-Lagrange backward integration , 2004, Autom..

[9]  J. Sethian,et al.  Ordered upwind methods for static Hamilton–Jacobi equations , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[10]  Alexander Vladimirsky,et al.  Ordered Upwind Methods for Static Hamilton-Jacobi Equations: Theory and Algorithms , 2003, SIAM J. Numer. Anal..

[11]  Matthew Dunbabin,et al.  Go with the flow : optimal AUV path planning in coastal environments , 2009, ICRA 2009.

[12]  Arthur E. Bryson,et al.  Applied Optimal Control , 1969 .