Optimal control of a ship for collision avoidance maneuvers

We consider a ship subject to kinematic, dynamic, and moment equations and steered via rudder under the assumptions that the rudder angle and rudder angle time rate are subject to upper and lower bounds. We formulate and solve four Chebyshev problems of optimal control, the optimization criterion being the maximization with respect to the state and control history of the minimum value with respect to time of the distance between two identical ships, one maneuvering and one moving in a predetermined way. Problems P1 and P2 deal with collision avoidance maneuvers without cooperation, while Problems P3 and P4 deal with collision avoidance maneuvers with cooperation. In Problems P1 and P3, the maneuvering ship must reach the final point with a given lateral distance, zero yaw angle, and zero yaw angle time rate. Sn Problems P2 and P4, the additional requirement of quasi-steady state is imposed at the final point. The above Chebyshev problems, transformed into Bolza problems via suitable transformations, are solved via the sequential gradient-restoration algorithm in conjunction with a new singularity avoiding transformation which accounts automatically for the bounds on rudder angle and rudder angle time rate. The optimal control histories involve multiple subarcs along which either the rudder angle is kept at one of the extreme positions or the rudder angle time rate is held at one of the extreme values. In problems where quasi-steady state is imposed at the final point, there is a higher number of subarcs than in problems where quasi-steady state is not imposed; the higher number of subarcs is due to the additional requirement that the lateral velocity and rudder angle vanish at the final point.