Abstract In this  article,  efficient  computational models for ship motions are presented。 These models are used to simulate ship movements in real time。 Compared with traditional approaches, our method possesses the ability to cope with different ship shapes, engines, and sea conditions without the loss of efficiency。 Based on our models, we create a ship motion simulation system for both entertain- ment and educational applications。 Our system assists users to learn the motions of a ship encountering waves, currents, and winds。 Users can adjust engine powers, rudders, and other ship facilities via a graphical user interface to create their own ship models。 They can also change the envi- ronment by altering wave frequencies, wave amplitudes, wave directions, currents, and winds。 Therefore, numerous combinations of ships and the environment are generated and the learning becomes more amusing。 In our system, a ship is treated as a rigid body floating on the sea surface。 Its motions compose of 6 degrees of freedom: pitch, heave, roll, surge, sway, and yaw。 These motions are pided into two categories。 The first three movements are induced by sea waves, and the last three ones are caused by propellers, rudders, currents, and winds。 Based on Newton’s laws and other basic physics motion models, we deduce algorithms to compute the magnitudes of the motions。 Our methods can be carried  out in real time  and possess high    fidelity。86022

According to ship theory, the net effects of external forces on the ship hull depend on the ship shape。 Therefore, the behaviors of the ship are influenced by its shape。 To enhance our physics models, we classify ships into three basic types。 They are flat ships, thin ships, and slender ships。 Each type of ship is associated with some predefined parameters to specify their characteristics。 Users can tune ship behaviors by varying the parameters even though they have only a little knowledge of ship   theory。

Keywords    Ship motions · Physics engine   ·Computer games · Simulation

1Introduction

Ships have been used in traveling, trading, and war for thousands of years。 They are necessities in human socie- ties。 Simulating ship motions is an important issue in education systems and entertainment industries。 Ship motion simulation systems help students and game players to learn the maneuver of ships under different sea condi- tions。 The procedures for ship motion simulation can be attributed to two categories。 In academic institutes, com- plicated mathematical equations are solved to calculate ship motions。 This type of computation requires extensive computing resources and usually produces massive numerical results which need special expertise to compre- hend。 The process itself is always boring to ordinary people。 Learners may quickly lose their interests to know more about ship maneuvering。 Therefore, this type of method is inadequate for education and entertainment。 On the other hand, in computer game industries, simple physical models are used to simulate ship motions。 The computation  is  fast  and  the  results  can  resemble   some

simplest ship movements。 However, these models can not simulate complicate ship motions and they lack the flexibility to cope with different ship models and sea conditions。

Virtual reality (VR) techniques have long been used to create virtual scenarios for driving training and games。 As pointed out in (Cremer et al。 1996; Kuhl et al。 1995; Zhang et al。 2004), a good VR system must possess the following components:

•a physics engine for computing the vehicle   dynamics,

•a virtual environment  database,

•efficient graphics procedures for scene  rendering,

•integration procedures connecting the physics engine, the visual systems, and the user  interface。