### 1. Introduction

_{0}coordinate, y

_{0}coordinate, heading angle, rudder angle, and propeller revolution, which are used as both training data and testing data for the ANN model. Once the ANN model is trained on the training data, validation is performed using the testing data. By inputting the initial ship motion state (x

_{0}coordinate, y

_{0}coordinate and heading angle) into the trained and validated ANN model, the next ship motion state (rudder angle and propeller revolution) is predicted. The x

_{0}coordinate, y

_{0}coordinate and heading angle in the next state are calculated using a mathematical model of ship maneuvering. The predicted ship states are continuously updated as input values of the ANN model. These predicted ship states are then compared to the final ship state at the dock. The prediction process ends when the error between the predicted state and the target state is within 5% in calm water and 10% in waves.

### 2. Docking problem and simulation model

### 2.1 Ship docking problem

^{1/6}(m/s), where ∇ in m

^{3}(ITTC, 2008). Small planing ship is often designed to operate at high speeds. However, the situation required operating at much lower speeds to ensure safe and precise maneuvering of ships docked in ports. This poses an interesting challenge for researchers who want to investigate the motion behavior of small planing ship. Therefore, adapting to the demands of low-speeds docking operations required careful consideration and adjustment of navigation strategies. Two requirements have been proposed for captains in maneuvering ships when docked (Kose et al., 1989). First, the ship's final docking location must be some distance from the dock, not entirely at the dock. Secondly, the captain must have enough time to plan the maneuvering of the ship in an emergency. Therefore, two phases of the ship docking process were proposed (Shuai et al., 2019), as depicted in Fig. 1. These phases include the ballistic phase and the side-push phase. In the ballistic phase, the ship is maneuvered to change course, speed and stops using the main propulsion and rudder. In the side-push phase, the ship is docked using tunnel thrusters to provide side thrust. In Fig. 1, two right-handed coordinate systems were used in ship maneuvering. The earth-fixed coordinate system (Oxy) was assumed to be at the midship at time t0. The body-fixed coordinate system (O

_{0}x

_{0}y

_{0}) was assumed to be at the midship. The heading angle is defined as the angle between the directions of the x

_{0}-axis and the x-axis.

_{n}, y

_{n}). The final position was considered the error range between the predicted motion state and the final target state, which is within 5% in calm water and 10% in waves. The range of the final position of the ballistic phase is depicted in Fig. 2. In addition, a leisure boat is a small planing ship that was chosen to study the ship maneuvering behavior in port. The principal dimensions of the ship which is considered the target ship are described in Table 1.

### 2.2 Ship mathematical model

_{G}and I

_{z}are the ship mass, the longitudinal center of gravity in the body-fixed coordinate system and the moment of inertia in yaw motion, respectively. u, v and r are the velocities on the x-axis, y-axis, and z-axis, respectively.

^{rd}-order function, these hydrodynamic forces acting on the ship's hull are described by Eq. (2). In Eq. (2), the 3

^{rd}-order polynomial function is replaced by a 2

^{nd}-order polynomial function and the absolute value of sway velocity and yaw rate.

##### (2)

_{0}, X'

_{vv}, X'

_{rr}, and X'

_{vr}are the surge derivatives. Y'

_{v}, Y'

_{v|v|}, Y'

_{r}, Y'

_{r|r|}, Y'

_{v|r|}, and Y'

_{v|r|}are the sway derivatives. N'

_{v}, N'

_{v|v|}, N'

_{r}, N'

_{r|r|}, N'

_{v|r|}, and N'

_{v|r|}are the yaw derivatives. p is the water density. The superscripts represent non-dimensional variables. In addition, the hydrodynamic coefficients for surge, sway and yaw motion were obtained based on the experiment. The hydrodynamic coefficients for surge, sway and yaw motion are described in Table 2.

_{t}is the longitudinal coordinates of the propeller to the ship's center of gravity and δ is the rudder angle. n is the propeller revolution. P is the pitch face propeller. m is the ship mass. s

_{A}is the apparent slip coefficient. C is the constant chosen for the type of ship being considered. W

_{f}is the Taylor wake factor.

### 3. Proposed approach

### 3.1 Artificial Neural Network

_{j}and b

_{j}are the output value and the bias corresponds to each node in the hidden layer, respectively. x

_{i}and w

_{i}are the input value and weight corresponding to each node in the input layer, respectively. f is the activation function. There are many activation functions used in regression problems such as Sigmoid, Tanh and ReLu described by Eq. (6).

_{k}and b

_{k}are the output value and the bias corresponds to each node in the output layer, respectively. X

_{j}and W

_{j}are the output value and weight corresponding to each node in the hidden layer, respectively. is the linear activation function.