1。2Overview

In this paper, a simulation and visualization system is pre- sented to simulate ship motions for education and games。 Users can use our system as a tool for learning ship motions or they can utilize its numerical modules to develop computer games or more advanced simulation systems。 Since the cal- culation of ship motion is difficult, three basic assumptions are adopted in our system to simplify the computation:

1。A ship is a rigid body。 Its elastic deformation is neglected。

2。The sea is deep and sea waves are comprised with sinusoidal waves of different frequencies, speeds, and angles。

3。The magnitudes of motions are relatively small。 The 6 degrees  of  freedom  can  be  calculated  separately and superimposed to generate the overall movement of the ship。

Under these constraints, the physics models of ship motion are greatly simplified and a real-time simulation for ship motion is obtained。

This paper is organized as follows: in Sect。 2, funda- mental knowledge and definitions of ship motions are introduced。 Therefore, more advanced material can be presented in the following sections。 In Sect。 3, the mathe- matical models for heave, pitch, and roll are presented。 These motions are caused by sea waves。 The numerical methods of estimating other three motions are deduced in Sect。 4。 Implementation details and ship characteristics are discussed in Sect。 5。 Visualization and testing results are shown in Sect。 6 and followed by conclusion and future work。

2Fundamentals

2。1Coordinate system and ship  motions

To designated directions and positions, two coordinate systems are defined in our system。 The first one is the world

surge, heave, and sway are translate motions in Xs, Ys, and Zs directions while roll, yaw, and pitch are rotations about Xs, Ys, and Zs axes。 However, based on computational aspects, we pide the six motions into another two cate- gories。 The first category include the motions induced by waves。 They are heave, pitch, and roll。 The second group of motions is caused by propellers, rudders, currents, and winds。 Surge, sway, and yaw belong to this group。 The motions of the first group do not change the ship position in the X-Z plane。 They influence the ship’s posture only。 The course of a ship is decided by surge, sway, and yaw。 Totally, these six motions determine the trajectory and vibration of the ship on the sea   surface。

2。2Ship classification

The ship shape is an important factor for ship motions。 For example, a warship are less sensitive to sea waves than a small fishing boat。 We classify ships into three basic types based on their lengths L, widths B, drafts D and the wave length k。 The first type of ship is the thin ships (Newman 1961)。 Their shapes satisfy the following  constraints:

coordinate system whose origin is fixed at a position on the D D

sea surface。 The sea surface is spanned by the X and Z axes, and  the  Y  axis  points  vertically。  The  second coordinate

system is attached to the ship。 It is moving with the ship。 The origin, G, is at the geometrical center of the ship。 The Xs is aimed at the forward direction of the ship。 The Zs axis is pointed to the starboard (right hand side) of the ship, and the Ys axis directs upward。 The two coordinates systems are shown in Fig。 1。

When floating on the sea surface, a ship possesses six degrees of freedom。 The six ship motions are shown in Fig。 1。 They are heave, pitch, roll, surge, sway, and yaw。 In the view point of the local coordinate system of the ship,

Fig。 1 The coordinate systems and 6 degree of freedom of ship motion

For example, a fishing boat is a thin ship。 The second  type