According to the ITTC recommendation, high-speeds ships are defined as ships with a Froude number above 0.45, and/or a speed above 3.7∇^1/6 (m/s), where ∇ in m³ (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 (O₀x₀y₀) was assumed to be at the midship. The heading angle is defined as the angle between the directions of the x₀-axis and the x-axis.

In our study, the first phase of the docking operation was conducted. The ship departed from the stationary state and maneuvered toward the port at low speed at various initial heading angles. The ship docked at the final position of the ballistic phase (x_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.

During maneuvering, ship motion is described by 6DOF including surge, sway, heave, roll, pitch and yaw. For surface ships, ship motion is simplified into 3DOF, including surge, sway and yaw. The ship motion equation for small planing ship was described as Eq. (1) (Katayama et al., 2009).

where, m, x_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. u˙, v˙ and r˙ are the accelerations on the x-axis, y-axis, and z-axis, respectively. H, P and W are symbols representing the hull, propeller and wave, respectively. X, Y and N are denoted for the hydrodynamic forces acting on the ship's hull as the x-axis, y-axis, and z-axis, respectively. Applying the Taylor-series expansion extended to the 3^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.

where X'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.

Hydrodynamic forces due to propeller are described by Eq. (3). For our small planing ship, the leisure boat uses two outboard motors as the main propulsion. Outboard motors include the engine, gearbox and propeller. The propellers used are coupled twin propellers with bravo three drives. The thrust of outboard motors is calculated using Eq. (4) (Gerr, 1989).

where T(N) is the thrust, x_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.

Wave force was obtained by performing the seakeeping test in the experiment. The wave force data was stored as a database and is described in Fig. 2.

ANN are computational models inspired by the structure and function of biological neural networks in the human brain. They consist of interconnected nodes, or neurons, organized in layers, including input, hidden, and output layers depicted in Fig. 3. During the training process, the ANN iteratively adjusts the weights and biases of its connections to minimize the error between the predicted and target values. The error is calculated using the mean square error (MSE) method. This optimization process, often performed using algorithms like backpropagation, allows ANN to learn complex nonlinear relationships and make accurate predictions based on new input data.

The predicted values of each node in the hidden layer can be written as shown in Eq. (5).

where y_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).

Similarly, the predicted value of each node in the output layer can be rewritten as Eq. (7).

where y_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.

The objective function of the training model is written as:

where n is the number of nodes in the output layer. y is the target output value.

The process of training and validating the ANN model involves splitting the data set into training and testing sets. This is typically done using a ratio such as 75% of the data set for the training set and 25% of the data set for the testing set. The training set was used to optimize the model's parameters through backpropagation, during which the model learns to minimize the difference between predicted and target values. Meanwhile, the testing set was used to assess the model's performance on data that was not observed during training, which helped avoid overfitting and underfitting phenomena. This iterative process continues until satisfactory performance is achieved on the testing set, ensuring the model's ability to generalize to unseen data.

In terms of ship motion prediction, the ANN model was trained and validated using historical data from control simulations. After training and validation, the ANN model was used to predict ship motion states, as depicted in Fig. 4. The input data can include variables such as x0 coordinate, y0 coordinate and heading angle, while the output data represents ship control parameters such as rudder angle and propeller revolution. Using mathematical models in ship maneuvering and the 4th-order Runge-Kutta method, ship motion states (surge velocity, sway velocity, yaw rate, x0 coordinate, y0 coordinate and heading angle) were calculated and continuously updated as input data for the ANN model.

The entire process of control simulation and ship motion state prediction was performed using the visualization interface shown in Fig. 5. This approach provides users with direct observation of the ship motion behavior and makes it easy to adjust to parameters or algorithms when required. The use of the visualization interface for simulation enables rapid responses to ship motions in real-time.

By adjusting the adjustment bar of the control simulation interface, the rudder angle and propeller revolution values were continuously changed and updated in the ship motion model. Furthermore, the interface provides immediate feedback by displaying the ship motion states during control. The ship docking control simulation aims to generate a variety of data as training data for the ANN model. The richer the training data, the easier it is for the ANN model to capture the complex relationship between input values and output values. To achieve this, the control simulation was performed with the same origin coordinates but with various initial heading angles both in calm water and in waves. The test conditions are described in Table 3.

The initial and final states of the ship motion can be entered into the ship motion prediction interface. The interface then predicts and displays the next states of ship motion. This integrated approach streamlines workflow and enhances efficiency in directly analyzing and responding to ship maneuvers. The test conditions for predicting ship motion states are designed as shown in Table 4.

In the case of a small planing ship docking, ship speed was 2 knots. The ship moved from a stationary state until it reached the target speed. The rudder angle was adjusted to maneuver the ship toward the dock. When the ship reached the dock, the propeller revolution was reduced to 0 rpm, but ship speed did not suddenly decrease to 0 knots. Based on the ship's existing momentum according to the inertial principle and the geometric characteristics of the small planing ship, the ship continued to move slowly over time. The simulation results of ship docking control in calm water and waves are shown in Fig. 6 and Fig. 7, respectively.

In the wave case, the wave direction of 180° and the wavelength over ship length ratio (λ/L) of 0.5 were chosen for analysis. Notably, the presence of waves significantly influences the docking process. Due to the wave forces, the ship's existing momentum and ship speed decrease faster compared to calm water conditions.

Before training an ANN model, the structure off the ANN model must be determined. The neural network structure includes activation functions, the number of hidden layers, the number of nodes in each hidden layer, and the number of epochs. A well-defined structure ensures that the ANN has the necessary complexity to adequately solve a given problem while optimizing computational efficiency and resource utilization. Moreover, determining the model structure is crucial to achieving optimal generalization performance. The Multi-Layer Perceptron Regressor (MLPRegressor) method was employed to determine the model structure. The structure is selected based on the minimum MSE value between the predicted and target values in the training data. To choose an effective and suitable model structure for any given problem, it is necessary to test different model structures on the same training data. A variety of ANN model structures are assumed. Activation functions include ReLu, Tanh, Sigmoid and Linear. The number of nodes in each hidden layer is assumed to be the same. The number of hidden layers is set to [1,2,3,4,5] and the number of nodes in each hidden layer is set to [3,6,9,12,15]. Each component of the ANN model structure is permuted and continuously updated into the MLPRegressor model. The results of the MSE value for the ANN model structure are shown in Fig. 8. The components of the ANN model structure with the minimum MSE value were selected. The optimized ANN model structure includes 5 hidden layers, 15 nodes in each hidden layer, 5000 epochs and the ReLu activation function.

Once the ANN model structure is determined, model validation should be performed using test data to ensure the model's ability to generalize to unseen data. The validation results of the ANN model are presented in Fig. 9, depicting the predicted rudder angle and propeller revolution. The comparison of the predicted values and target values in the testing data shows a good fit for the target values. In addition, the model loss, which measures the disparity between training and validation values, is minimal. Therefore, these results indicate the ANN model's effectiveness in the prediction process.

The ANN model was trained and then used to predict ship motion states using the results for ship docking control simulation in calm water in Fig. 6. As shown in Table 4, the next motion states were predicted at the initial heading angles of 23.3° and -38.8°. The prediction process is complete when the error between the predicted motion state and the target final state is within 5%. Similarly, in the case of waves, the ANN model was trained based on simulated results for ship docking control simulation in Fig. 7. Predictions off next motion states are made at initial heading angles of 55.2° and -10.4°, with the prediction process ending when the error is within 10%. The results for motion state prediction in calm water and waves are shown in Fig. 10 and Fig. 11, respectively. It's worth noting that under the influence of the training data, ship speed in the predicted final state has not yet reached zero. However, automatic navigation while docking the ship in the ballistic phase has been completed.

The performance of the ANN model and the quality of the predicted results depend on the quality of the training data. To achieve this, data obtained from free-running model tests in the experiment or sea trial data should be applied. These data sets can provide valuable insight into the dynamics of small planing ship's docking operations, providing a more comprehensive representation of real-world scenarios. In addition, maneuvering small ships at low speeds also causes yaw rotating motion. This rotation increases the complexity of the docking process, which must be accounted for in the training data and model structure.