Abstract: A minimum time maneuvering method for berthing and parallel shifting were derived by Ohtsu et al.⁽¹⁾ using the sequential conjugate gradient- restoration method (SCGM)⁽³⁾, whose maneuvering model is a sophisticate nonlinear MMG maneuvering one. Using this technique, we can get optimal steering commands, engine operation, thruster's ones for obtaining minimum time maneuvering. This paper aims at applying this method to augment a maneuvering ability of ship by changing the rudder shape. The problems are formulated as two point boundary value problem (TPBVP) and solved numerically using the SCGM. The effectiveness of the minimum time maneuvering solution is confirmed by some actual ship experiments at sea at first. Then, the effect to give to course changing by changing the rudder aspect ratio is examined.

　

 There are so many phases that a ship must show her highest ability in ship handling in restricted sea room. In these phases, a mariner wants to know how maneuvering command is the best. One solution on this problem is to get the knowledge from his great stock of experiences in his career. Another one is to depend on disciplining by a maneuvering simulator. However, there are individual variations in these methods.

　

 On the other hands, mathematical optimal solutions getting from reliable mathematical maneuvering model yield reasonable and rigorous ship handling command. The authors have given some results on the minimum time maneuvering method in which we adopt a nonlinear sophisticate mathematical model as the ship's model and the time-to-go as a measure of goodness of maneuvering⁽¹⁾⁽²⁾.

　

 In this paper, the authors formulate the minimum time maneuvering problem and confirm that the mathematical solution calculated numerically yield a reasonable and rigorous answer for the actual ship handling phase, using the result of the actual minimum time berthing test. Lastly based on the result, we discuss whether we can apply the proposed method to check the ability of course changing maneuvering by changing a rudder surface shape.

　

2.1 Formulation

　

 A minimum time maneuvering problem is formulated to guide a ship from a certain point to another requested one in a minimum time. This kind of problem is considered as a two-point boundary value problem, which set an initial point as the approaching one of the ship and a terminal point as the requested one. Such maneuvering problem might be solved using the theory of calculus of variations. However, since ship motion has high non-linearity, it is difficult to find an analytical solution. Ohtsu et al.⁽¹⁾ Okazaki et.al⁽²⁾ solved this problem by using a numerical method called the sequential conjugate gradient-restoration (SCGR) method developed by Miele et al.⁽³⁾. In this section, we formulate the problem of the minimum time maneuvering problem as a TPBVP one.

　

 The minimum time maneuvering problem is formulated as follows:

　

 A performance index of this problem is defined by a functional

　

　

 where I is a scalar value, x is the state vector, u is the control vector, t is the actual final time value and τ is the normalized time value. And the solution of the problem is minimized the performance index with constrains as follows:

　

1) the differential constraints

　

　

where the function ψ denotes a nonlinear ship's motions model, and

　

2) the boundary conditions:

　

 [ω(x)]0 = 0　(3)

　

 [Ψ(x,τ)]1 = 0　(4)

　

where ω denotes initial constrain and Ψ denotes terminal constrain.

　

3) the non-differential constraints,

　

S(u,τ,t) = 0, 0≤t≤1　(5)

　

where the function S is useful to limit the control inputs.

　

2.2 Ship's Motion Model

　

 The model of ship's motion used in this paper was based on a MMG model. The coordinate system was denoted on Figure 1.

　

　

 The state values were position x, position y ship's heading Ψ, surge speed u, sway speed v, yaw rate r and rudder angle δ. Therefore, the equation of state is written by

　

　

　

where m and Izz are the mass and the turning moment inertia. mx, my, and Jzz are the added mass and added moment of inertia. Tδ and a are time constant. The subscripts H, P and R denote the hydrodynamic force induced by the hull, propeller and rudder.