Optimal control of a ship for course change and sidestep 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 Mayer problems of optimal control, the optimization criterion being the minimum time. Problems P1 and P2 deal with course change maneuvers. In Problem P1, a ship initially in quasi-steady state must reach the final point with a given ya w angle and zero yaw angle time rate. Problem P2 differs from Problem P1 in that the additional requirement of quasi-steady state is imposed at the fi nal point. Problems P3 and P4 deal with sidestep maneuvers. In Problem P3, a ship init ially in quasi-steady state must reach the final point with a given lateral distance, zero yaw angle, and zero yaw angle time rate. Problem P4 differs from Problem P3 in that the additional requirement of quasi-steady state i s imposed at the final point. The above Mayer problems are solved via the sequential gradient- restoratio n algorithm in conjunction with a new singularity avoiding transformation w hich accounts automatically for the bounds on rudder angle and rudder angle time rate. 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.