Compared with other components, the physics engine module is immature in most VR systems。 A physics engine must calculate the behaviors of moving objects as realistic as possible under a severe constraint of time。 Therefore, pursuing accurate numerical solutions is impractical。 Instead, simplified but physically meaningful models should be adopted for simulating object motions。 In this article, we present effective mathematical models ship motions。 Our methods can be employed to develop physics engines for ship handling simulation systems, computer games, and other maritime applications。 Based on the models, we create a simulation program which can be directly used in the physics classes on floating body motions and the development of computer  games。

1。1Related work of ship motion  simulation

The hydro-dynamics for ship motions are difficult to solve。 Most numerical algorithms are based on strip theory developed in (Korvin-Kroukovsky and Jacobs 1957; Salvesen et al。 1970)。 However, it takes hours to produce a set of accurate solutions for just one or two motions by using modern computers (Aryanpour and Ghorashi 2001)。 These numerical procedures are impractical for real time simulation。 Others use simplified models for the estimation of ship motions。 In (Triantafyllou et al。 1983), Kalman filter techniques are utilized in ship motion estimation。 However, their method requires the knowledge of the ship, and if the ship parameters are unavailable their method is not applicable。 In (Lainiotis et al。 1992), an improved method is proposed。 It requires less information about the ship, though the accuracy of the method still depends on the availability of ship parameters。 In computer game design, ship models are fabricated。 No real data can be retrieved。 This method is not convenient either。 Zhang et al。 (2004) develop mathematical models for ship motions。 They estimate the total forces acting on the ship first, then based on Newton’s law, they deduce first order differential equations   to   model   the   relation   between   forces  and

accelerations。 The equations are solved by using Runge– Kutta method。 Since their applications focus on handling ships inside or near harbor areas, only the physics models for surge, sway, and yaw are  given。

Another work is presented in (Cieutat et al。 2001)。 The authors propose wave models based on the work of (Fournier and Reeves 1986)。 Then they compute heave, pitch, and roll of a ship by using sea surface height under the ship。 For estimating pitch and roll, the tangent plane of sea surface is computed first。 Then the ship is rotated such that its orientation is aligned with the tangent plane。 Their mathematical models are too simplified。 Since only the sea surface height is taken into consideration, ships of different shapes will produce the same behavior if the wave condi- tion is the same。 However, a war ship and a craft do not share the same maneuvering characteristics。 Thus their methods are not flexible enough to simulate the behaviors of different ship models。 Another research trend is to pre- dict ship motions based on the previous status of the ship。 In (Zhao et al。 2004), ship status and motions at previous steps are recorded and a tensor field is created by using these data。 Then the eigen values and eigen vectors of the tensor field are computed and a minor component analysis method is used for predicting ship motions。 This method is useful for short-term motion prediction but not suitable for simulating ship motions。 There are also some commercial software available for ship motion simulation, for example, Nord-Control and Transas。 These systems are usually expensive and complicated。 Users need a lot of maritime knowledge and training to operate them。 The other disad- vantage is that they are closed systems。 It is not easy for end-users to add their ship models or software modules into these systems。