### 1. Introduction

### 2. Characteristics of Waterjet-Powered Ships

### 2.1 Principle of Waterjet Thruster

### 2.2 Characteristics of Waterjet Thruster

### 3. Crabbing Mathematical Model of Ships

### 3.1 Estimating the Location of the Center of Gravity of a Ship using the Vector Decomposition Method

### 3.2 Calculation of Ship Crabbing Control Forces using Force and Moment Equilibrium

*f*. The thrust force of the central thruster is denoted as

_{L}*f*, and the thrust force of the starboard thruster is denoted as

_{C}*f*. The angle of the port thruster is represented by

_{R}*θ*, and the transverse distance from the thruster to the centerline is denoted as

*l*. The distance from the stern to the longitudinal center of gravity is denoted as

_{T}*l*. The equilibrium equations for the force and moment required for a ship to move only in the horizontal direction without rotating or moving vertically were derived using only the thrust force of the ship and no other external forces, such as currents or wind.

_{G}*f*) and the starboard thruster ((

_{C}*f*) of the hull were calculated. The equations of motion for hull crabbing can be solved by expressing the given values and the values to be found as follows:

_{R}### 4. Sea Trials with Real Ships

### 4.1 Specifications of a Navy Ship

### 4.2 Crabbing Method using Three Stern Thrusters

*l*) from the center of the hull to the thruster was 2.35 m. The center of gravity (

_{T}*G*) of the vessel was assumed to be located on the centerline of the hull, as there was no lateral inclination. Subsequently, the method of determining the longitudinal center of gravity of the hull described in section 3.1 was applied to calculate the center of gravity of the non-rotating PKG. As a result, the distance (

*l*) from the longitudinal center of gravity to the stern thruster was approximately 27.5 m.

_{G}### 4.3 Test Method and Conditions

*x*

_{0}and

*y*

_{0}axes in the Earth-fixed coordinate system, and

*x*and

*y*axes in the ship-fixed coordinate system. In the geocentric coordinate system, the velocity of a ship was defined as

*V*, while the velocity in the ship-fixed coordinate system was defined as

*u*and

*v*for the

*x*and

*y*directions, respectively. The rate of turn (ROT), which is defined as the speed at which a ship rotates around the origin

*O*in the fixed coordinate system, is denoted by r. The bow angle (

*β*) is defined as the angle formed between the

*x*

_{0}axis of the Earth-fixed coordinate system and the x-axis of the ship-fixed coordinate system. In the Earth-fixed coordinate system, the speed of the ship is defined as

*V*. The angle between the direction of the ship speed

*V*and the

*x*

_{0}axis of the Earth-fixed coordinate system is defined as

*β*

_{0}. The angle between the direction of the ship speed and the x-axis of the ship-fixed coordinate system is defined as the drift angle

*ψ*. The wind and current are defined as 0°, 90°, 180°, and 270° when coming from the north, east, south, and west, respectively.

*ψ*) must be determined to achieve this. The drift angle (

*ψ*) can be obtained by subtracting the gyro course (

*β*) of the ship from the direction of the ship velocity (

*β*

_{0}), as shown in Eq. (5).

### 5. Test Results and Analysis

### 5.1 Rotation of the Hull

*ψ*), ROT, wind direction, and wind speed. The x-axis on this graph represents time. Fig. 8(a), which depicts the ship trajectory, shows that the hull gradually rotates and moves from side to side due to the influence of environmental force (wind). The drift angle (

*ψ*) observed in Fig. 8(b) exhibited a difference of 10° to the left and right from 90°. In addition, while the hull moved laterally by 10.8 m, it also moved slightly longitudinally by 0.26 m. Furthermore, in Fig. 8(c), the hull rotated by approximately 0.4°, albeit very slowly.

*ψ*) of the hull exhibits the most dramatic variation in the 20- to 40-s interval.

### 5.2. Determining the Success of Crabbing

*ψ*) should be small at 90°, and the forward and backward velocities should be small compared to the transverse velocity to determine normal crabbing. The test results showed that the ROT was insignificant, while the drift angle (

*ψ*) fluctuated within a range of 10° in both directions. In this trial, however, it is challenging to fully control the rotation of the hull because of the absence of a bow thruster on the vessel, and the reliance solely on the stern thruster for crabbing. Furthermore, the vessel used in this trial does not have crabbing as its primary objective. The crabbing test was conducted to verify the ability to perform crabbing during port entry or exit or in emergencies, even in the absence of a bow thruster. Thus, in this trial, the focus was placed on determining the success of the crabbing test by assessing the longitudinal displacement in relation to the actual transverse displacement of the vessel. The test results indicate that a cumulative distance of 10.8 m was covered in the lateral direction over 180 s, while a maximum displacement of 0.26 m was achieved in the longitudinal direction (Fig. 9(a)). As it moved 0.4%, considering the length of the ship is 63 m, the crabbing test using only the stern thruster of the waterjet-powered ship was considered to be successful. Furthermore, the longitudinal velocity was virtually nonexistent, with a maximum of 0.02 m/s (Fig. 9(b)).