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Maximin Approach to the Ship Collision Avoidance Problem via Multiple-Subarc Sequential Gradient-Restoration Algorithm

Author

Listed:
  • A. Miele

    (Rice University)

  • T. Wang

    (Rice University)

Abstract

The ideal strategy for ship collision avoidance under emergency conditions is to maximize wrt the controls the timewise minimum distance between a host ship and an intruder ship. This is a maximin problem or Chebyshev problem of optimal control in which the performance index being maximinimized is the distance between the two ships. Based on the multiple-subarc sequential gradient-restoration algorithm, a new method for solving the maximin problem is developed. Key to the new method is the observation that, at the maximin point, the time derivative of the performance index must vanish. With the zero derivative condition being treated as an inner boundary condition, the maximin problem can be converted into a Bolza problem in which the performance index, evaluated at the inner boundary, is being maximized wrt the controls. In turn, the Bolza problem with an added inner boundary condition can be solved via the multiple-subarc sequential gradient-restoration algorithm (SGRA). The new method is applied to two cases of the collision avoidance problem: collision avoidance between two ships moving along the same rectilinear course and collision avoidance between two ships moving along orthogonal courses. For both cases, we are basically in the presence of a two-subarc problem, the first subarc corresponding to the avoidance phase of the maneuver and the second subarc corresponding to the recovery phase. For stiff systems, the robustness of the multiple-subarc SGRA can be enhanced via increase in the number of subarcs. For the ship collision avoidance problem, a modest increase in the number of subarcs from two to three (one subarc in the avoidance phase, two subarcs in the recovery phase) helps containing error propagation and achieving better convergence results.

Suggested Citation

  • A. Miele & T. Wang, 2005. "Maximin Approach to the Ship Collision Avoidance Problem via Multiple-Subarc Sequential Gradient-Restoration Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 124(1), pages 29-53, January.
    • Handle: RePEc:spr:joptap:v:124:y:2005:i:1:d:10.1007_s10957-004-6464-y
    DOI: 10.1007/s10957-004-6464-y
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    References listed on IDEAS

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    1. A. Miele & T. Wang & C. S. Chao & J. B. Dabney, 1999. "Optimal Control of a Ship for Collision Avoidance Maneuvers," Journal of Optimization Theory and Applications, Springer, vol. 103(3), pages 495-519, December.
    2. A. Miele & T. Wang & C. S. Chao & J. B. Dabney, 1999. "Optimal Control of a Ship for Course Change and Sidestep Maneuvers," Journal of Optimization Theory and Applications, Springer, vol. 103(2), pages 259-282, November.
    3. A. Miele & T. Wang, 2003. "Multiple-Subarc Gradient-Restoration Algorithm, Part 2: Application to a Multistage Launch Vehicle Design," Journal of Optimization Theory and Applications, Springer, vol. 116(1), pages 19-39, January.
    4. A. Miele & T. Wang, 2004. "New Approach To The Collision Avoidance Problem For A Ship," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 6(01), pages 137-155.
    5. Y. Yavin & C. Frangos & T. Miloh & G. Zilman, 1997. "Collision Avoidance by a Ship with a Moving Obstacle: Computation of Feasible Command Strategies," Journal of Optimization Theory and Applications, Springer, vol. 93(1), pages 53-66, April.
    6. A. Miele & T. Wang, 2003. "Multiple-Subarc Gradient-Restoration Algorithm, Part 1: Algorithm Structure," Journal of Optimization Theory and Applications, Springer, vol. 116(1), pages 1-17, January.
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    Cited by:

    1. A. Miele & T. Wang & J. A. Mathwig & M. Ciarcià, 2010. "Collision Avoidance for an Aircraft in Abort Landing: Trajectory Optimization and Guidance," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 233-254, August.
    2. A. Miele & T. Wang, 2006. "Optimal Trajectories and Guidance Schemes for Ship Collision Avoidance," Journal of Optimization Theory and Applications, Springer, vol. 129(1), pages 1-21, April.

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