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 a ngle time rate are subject to upper and lower bounds. We formulate and solv e four Chebyshev problems of optimal control, the optimization criterion be ing the maximization with respect to the state and control history of the m inimum value with respect to time of the distance between two identical shi ps, one maneuvering and one moving in a predetermined way. Problems P1 and P2 deal with collision avoidance maneuvers without cooperat ion, while Problems P3 and P4 deal with collision avoidance maneuvers with cooperation. In Problems P1 and P3, the maneuvering ship must reach the fin al 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-stead y 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 algorit hm in conjunction with a new singularity avoiding transformation which acco unts automatically for the bounds on rudder angle and rudder angle time rat e. The optimal control histories involve multiple subarcs along which either t he 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-st eady state is imposed at the final point, there is a higher number of subar cs than in problems where quasi-steady state is not imposed; the higher num ber of subarcs is due to the additional requirement that the lateral veloci ty and rudder angle vanish at the final point.