4. APPLICATION OF THE MINIMUM TIME SOLUTION TO IMPROVE MANEUVERING ABILITY
4.1 Application for Deciding a Rudder Shape
It is important for ship designer to satisfy the maneuvering ability specified by owners. When we apply the proposed method to the problem, we can get one of solving tools for checking it, as we could confirm that the proposed method was reliable in the former section. In this section, we discuss how the timetogo is influenced by changing the rudder shape, in particular, aspect ratio.
Table 2 Principal dimensions of a container ship
Length 
202.0 [m] 
Breadth 
31.2 [m] 
Depth 
6.93 [m] 
Cb 
0.518 

The ship discussed here is a large container ship whose principal dimensions are shown in Table 2. And the angle of course change is assumed to be 45 degrees. Figure 5 shows the course changing problem treated here.
Fig.5 Course changing problem
4.2 The Effect of Rudder Aspect Ratio for Course Changing Ability
The aspect ratio of rudder is important element to decide the course changing ability which is one of important elements to maneuver a ship. In here, we discuss about the effects of the rudder aspect ratio on the course changing ability of ship.
The original aspect of the ship was 1.4. Therefore, we will change the ratio from 0.8 to 2.0 at every 0.2, keeping the area of the rudder to the original value.
Figure 6 shows on of the example of the rudder command in the minimum time 45 degrees' course changing maneuvering.
Fig.6 
The example of rudder command in minimum time 45 degrees' course changing (rudder limit 35degrees) 
We can detect that the optimal control commands in the minimum time course changing maneuvering are constituted of three bangbang controls from this figure.
Adopting the distance from the starting point to the deflection point as the measure of maneuverability in course changing maneuvering as shown in Figure 5, we calculated them in changing her rudder aspect ratio to various quantities. Figure 7 shows the results of the calculation. From the figure, it is clear that the maneuverability is improved as the aspect increases and the distance to the deflection point decreases. This result demonstrates the effectiveness in applying the proposed method not only for actual maneuvering but also for improvement of her maneuverability at restricted sea like channel, port.
Fig.7 
The distance to deflection point vs rudder aspect ratio in minimum time course changing 
5. CONCLUSION
In this paper, we discussed about a minimum time maneuvering problem of a ship. We formulated the problem as a two point boundary value problem, solved a minimum time berthing problem and confirmed that the solution is reasonable and rigorous method from the view points of ship handling theory, using the result of the actual sea test Lastly, we solved the minimum time course changing problem of a container ship, and demonstrated to be able to play an important role to improve the maneuverability by changing her control surface like rudder aspect ratio.
REFERENCES
[1] K.Ohtsu, K.Shouji and T.Okazaki "Minimum Time Maneuvering of Ship with Wind Disturbances" Journal of Control Engineering Practices, Vol.4.3, 1996.
[2] T.Okazaki, K.Ohtsu and K.Shouji, "A Study of Minimum Time Berthing Solutions", 5^{th} Maneuvering and Control of Marine Crafts (MCMC2000), 2000.
[3] A.K. Wu and A.Miel, "Sequential Conjugate GradientRestoration Algorithm for Optimal Control Problem with Non Differential Constraints and General Boundary Conditions,", Part 1,Journal of Optimal Control Applications and Method, Vol.1980.
AUTHOR'S BIOGRAPHY
Kohei Ohtsu : Dr. and Professor Ohtsu was born in Osaka in 1943. He graduated at Tokyo University of Mercantile Marine(TUMM). He was the captain of a training ship Shioji Maru in 1977. He gained the Award of SNAJ, JIN and so on. He is a member of JIN, SNAJ, IEEE.
