*Article* **Simulating Ship Manoeuvrability with Artificial Neural Networks Trained by a Short Noisy Data Set**

**Lúcia Moreira \* and C. Guedes Soares**

Centre for Marine Technology and Ocean Engineering (CENTEC), Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

**\*** Correspondence: lucia.moreira@centec.tecnico.ulisboa.pt

**Abstract:** Artificial neural networks are applied to model the manoeuvrability characteristics of a ship based on empirical information acquired from experiments with a scaled model. This work aims to evaluate the performance of the proposed method of training the artificial neural network model even with a very small quantity of noisy data. The data used for the training consisted of zig-zag and circle manoeuvres carried out in agreement with the IMO standards. The wind effect is evident in some of the recorded experiments, creating additional disturbance to the fitting scheme. The method used for the training of the network is the Levenberg–Marquardt algorithm, and the results are compared with the scaled conjugate gradient method and the Bayesian regularization. The results obtained with the different methodologies show very suitable accuracy in the prediction of the referred manoeuvres.

**Keywords:** ship's manoeuvrability; model tests data; artificial neural networks

#### **1. Introduction**

The prediction of ship dynamics in the seaway is complicated as it depends on the joint effect of the various environmental factors, independently of whether the interest is to predict the ship dynamics in straight trajectories or when performing manoeuvres.

Methods are available to determine manoeuvring trajectories from a given manoeuvring mathematical model. Several popular mathematical models for ship manoeuvrability have been widely applied, such as the Abkowitz model [1–3], the Manoeuvring Mathematical Group (MMG) model [4], the well-known Nomoto model [5], or even more detailed models proposed recently [6]. Reviews covering several elected issues associated with vessel mathematical models employed in ship manoeuvring, principally for simulation purposes, are presented in [7,8]. The need to have an accurate and fast prediction of ship responses is associated with ship manoeuvring, in particular, in decision support systems for ship handling or manoeuvring simulators led to the development of several empirical models [9].

The manoeuvring model's parameters may be estimated using different methods, including different types of regressions when captive model tests are performed [10,11] or several system identification techniques, which typically are applied on free-running model tests or full-scale tests [12,13]. Various system identification methods for vessels are available, such as the ones presented in [14–18]. When a model has parameters already identified, then it can be used to simulate the ship trajectories [19].

Artificial intelligence methods have been used to model different types ofresponses [20,21]. Neural networks have been used to predict manoeuvring capabilities [22]. A ship's minimum time manoeuvring system based on artificial neural networks (ANNs) and a nonlinear model predictive compensator was presented in [23], allowing the user to execute the optimization for any desired set of equality and non-equality constraints. The work presented in [24] is focused on getting optimized ship trajectories in narrow waterways under wind

**Citation:** Moreira, L.; Guedes Soares, C. Simulating Ship Manoeuvrability with Artificial Neural Networks Trained by a Short Noisy Data Set. *J. Mar. Sci. Eng.* **2023**, *11*, 15. https:// doi.org/10.3390/jmse11010015

#### Academic Editor: Sergei Chernyi

Received: 14 November 2022 Revised: 18 December 2022 Accepted: 19 December 2022 Published: 22 December 2022

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

disturbances considering time as an objective function, i.e., the ship tends to sail by taking the optimized rudder output for minimum time manoeuvre. Later, in [25], the same authors studied the application of an ANN controller for ship course-changing manoeuvres. In [26], a method that uses genetic algorithms to simultaneously optimize the number and weights of backpropagation neural network neurons to predict the ship's trajectory is studied.

Another ANN class that has been used to model ship dynamics is the Recurrent Neural Network (RNN). In an RNN, the connections between nodes form a directed or undirected graph along a temporal sequence, allowing it to exhibit a temporal dynamic behaviour. RNNs have been used in different maritime applications, such as the study presented in [27] that presents the use of an RNN for the prediction of the propulsion power of a vessel. In [28], a real-time ship vertical acceleration prediction algorithm based on the long short-term memory (LSTM) and gated recurrent units (GRU) models of an RNN is proposed. In [22], an RNN is used to model the surface ships' manoeuvrability characteristics. In [22], an RNN with four inputs, one hidden layer, and two outputs was used to learn the manoeuvring model of a ship from data generated through simulations. Inputs to the model are the commands of rudder angle and ship's speed, in addition to the recursive outputs sway and yaw velocities. The outputs of the system were the rate of sway and yaw at the current time instant.

A posteriori, the model presented in [22] was used to analyse the potential of ANNs in ship simulation when the training data are corrupted with noise, as is usually the case in full-scale tests [29]. An RNN to simulate catamaran manoeuvres was presented as a different methodology from the conventional approach of developing manoeuvring mathematical models [30]. The work presented here aims to assess the performance of ship manoeuvrability models developed by applying RNNs trained with a low quantity of noisy data from zig-zag and circle experiments carried out in agreement with the IMO standards [31].

Later, deep structured learning architectures such as long short-term memory (LSTM) networks, which are a type of RNN able to process not only single data points but also entire sequences of data, have been applied to the dynamic model identification [32]. Other methods are also available and have been used in specific data-based motion predictor applications such as support vector machines (SVMs) [16,33–35], deep learning, or autoregressive (AR) methods. On the other side, there exist model-based predictors such as dynamic models [19]. Several applications have been developed in the scope of the improvement of manoeuvring performance.

The main objective of the development of the RNN model is to obtain an alternative to the usual manoeuvring simulators that use traditional mathematical models, which are a function of the hydrodynamic forces and moment derivatives. These values are normally achieved through captive experiments performed with models in tanks. This procedure is time-consuming and costly, requiring exclusive use of a large specialized, purpose-built facility. Another possibility is to use the trajectories of small or even large selfpropelled models to train neural networks or to identify the parameters of the traditional mathematical models. Furthermore, this is one of the valid methods that can be used in the design stage of a ship.

The alternative RNN model presented in this paper represents an implicit mathematical model for ships in which time histories of manoeuvring motions are previously known. The main advantage of the RNN consists in that the parameters used for the training are easily obtained from full-scale trials of existing ships or self-propulsion tests of models. RNNs can handle noisy data because they can generalize after training on noisy data instead of merely memorizing the noise.

The RNN model used in this paper for the manoeuvring simulation is based on the one used in [31], but it takes much less data for the network training using the methods presented in a previous study, which handled only simulated data [36], while here real measurements are used. The RNNs studied have the advantage of having very few parameters making them very fast to train. The performance of the network is analysed regarding the

limited data set used for training. RNNs are autonomous but highly susceptible to errors. If the data set is small enough not to be inclusive, biased predictions may come from a biased training set.

In recent years, ANNs have been effectively used in an extensive range of maritime applications. The vessel dynamics may also be considered as a black box and modelled using a proper tool. ANNs have been used for the problem of parameter estimation in [37], where the weights of the network correspond to the parameters of the Nomoto model. The network learns these parameters from data acquired experimentally. One more application, presented in [38], uses a feedforward ANN to learn the behaviour of the nonlinear terms of the manoeuvring model from data obtained through numerical simulations. This ANN is then used in simulations to replace the calculation of the nonlinear terms. In [39], RNNs were used as manoeuvring simulation tools. Inputs to the simulation, cast in the form of forces and moments, were redefined and extended in a manner that accurately captures the physics of ship motion.

In the present case study, the main innovation is to train the network using a different methodology, the Levenberg–Marquardt algorithm, instead of the backpropagation method. This methodology is used to solve nonlinear least squares problems, and it is a combination of two other methods: gradient descent and Gauss–Newton. As there are two possible options for the algorithm's direction at each iteration, the Levenberg–Marquardt is more robust than the Gauss–Newton. As an advantage, it shows to be faster to converge than either the Gauss–Newton or gradient descent. In addition, it can handle models with multiple free parameters that are not precisely known. If the initial guess is far from the mark, the algorithm can still find an optimal solution. In this paper, the results obtained with the Levenberg–Marquardt algorithm are compared with the ones obtained with the training performed with two different methods: the scaled conjugate gradient method and Bayesian regularization. The Levenverg–Marquardt algorithm allowed training the system with a relatively short training time series.

Section 2 carries out the description of the manoeuvring tests, along with a summary of the acquired results and the pre-processing steps executed before using them for training and testing. Section 3 reports the configuration and training method of the presented RNN model. Section 4 presents the results acquired with the proposed model. Lastly, Section 5 outlines and analyses the results, comparing them with values attained with models being used for analogous assignments under identical situations.

#### **2. Description of the Manoeuvring Tests**

The manoeuvring experiments executed to collect the data used in this article paper are presented in [40,41]. The experiments were conducted on the "Piscina Oceaˆnica de Oeiras", Portugal, with the chemical tanker ship model in March 2016. This swimming pool has a length of 50 m and a breadth of 30 m. The model is a scaled model of a chemical tanker built at the "Estaleiros Navais de Viana do Castelo", Portugal.

The scaled (1/65.7) model of the chemical tanker is shown afloat in Figure 1, and its main dimensions are stated in Table 1, together with the ones of the real ship. The vehicle is built from single-skin glass-reinforced polyester with plywood framings, and its design speed is 0.98 m/s.

The hardware architecture comprises all the sensors and actuators that are used in the real-time navigation and control platform. The hardware structure is further split into a command and monitoring unit (CMU) and a communication and control unit (CCU).

The main goal of the shore-based CMU is to assist in the manual and autonomous control of the vehicle by providing a human–machine interface (HMI). The CMU mostly consists of various instrumentations: laptop, global positioning system (GPS) unit, industrial Wi-Fi unit, compact-RIO, main AC power supply unit, DC power supply unit, and an anemometer to measure the relative wind speed and direction.

**Figure 1.** Photo of the chemical tanker model. **Figure 1.** Photo of the chemical tanker model.



Propeller diameter (m) 5.4 0.082

Design speed (m/s) 8 0.984 Scaling coefficient - 65.7 The hardware architecture comprises all the sensors and actuators that are used in the real-time navigation and control platform. The hardware structure is further split into a command and monitoring unit (CMU) and a communication and control unit (CCU). The main goal of the onboard CCU is to execute real-time control algorithms that are related to the course and speed controls of the model. The CCU comprises the following instrumentations: laptop, CompacRIO units, industrial Ethernet switch (IES), GPS unit, inertial measurement system (IMS) (capable of measuring the 3-axis angles of heading, roll, and pitch, the 3-axis angular velocities of heading, roll, and 3-axis linear accelerations of surge, sway, and heave), industrial Wi-Fi unit, DC motors with encoders able to take the measurements of the 3-axis angles of heading, roll, and pitch, the 3-axis angular velocities of heading, roll, and 3-axis linear accelerations of surge, sway, and heave, position sensor, fibre-optic gyrocompass, laptop computer, batteries, and fuse units.

The main goal of the shore-based CMU is to assist in the manual and autonomous control of the vehicle by providing a human–machine interface (HMI). The CMU mostly consists of various instrumentations: laptop, global positioning system (GPS) unit, indus-Measurement and registration of the kinematical parameters listed in Table 2 were envisaged, and all parameters indicated in the table were measured during the tests. The uncertainty estimates are approximate and were obtained from the instruments' documentation.


the measurements of the 3-axis angles of heading, roll, and pitch, the 3-axis angular velocities of heading, roll, and 3-axis linear accelerations of surge, sway, and heave, position

uncertainty estimates are approximate and were obtained from the instruments' docu-

longitude *λ*. These are transformed to the standard Cartesian earth coordinates of the

= ( − <sup>0</sup>

)<sup>0</sup>

= ( − <sup>0</sup>

) (1)

(2)

Measurement and registration of the kinematical parameters listed in Table 2 were

The GPS unit generates instantaneous ship coordinates in terms of latitude *φ* and

sensor, fibre-optic gyrocompass, laptop computer, batteries, and fuse units.

ship's origin *ξ<sup>C</sup>* and *η<sup>C</sup>* for the manoeuvre's starting point (Figure 2):

trial Wi-Fi unit, compact-RIO, main AC power supply unit, DC power supply unit, and **Table 2.** Measured Parameters.

mentation.

The GPS unit generates instantaneous ship coordinates in terms of latitude *ϕ* and longitude *λ*. These are transformed to the standard Cartesian earth coordinates of the ship's origin *ξ<sup>C</sup>* and *η<sup>C</sup>* for the manoeuvre's starting point (Figure 2):

$$\mathfrak{F}\_{\mathbb{C}} = \mathfrak{x}(\mathfrak{\phi} - \mathfrak{\phi}\_{0}) \tag{1}$$

$$
\eta\_{\mathbb{C}} = \kappa (\lambda - \lambda\_0) \cos \phi\_0 \tag{2}
$$

**Figure 2.** Definition of kinematic parameters (all shown quantities are positive). **Figure 2.** Definition of kinematic parameters (all shown quantities are positive).

**3. Neural Network Training** The model used has the following six inputs: The subscript '0' denotes the initial values of the corresponding variables, and *κ* is the conversion coefficient from minutes to meters equal to 1852 m/min. In Figure 2 β is the drift angle, χ is the course angle provided by the GPS, ψ is the heading angle, δ<sup>R</sup> is the rudder angle, *r* is the yaw rate, *V* is the speed of the ship, *V*<sup>A</sup> is the relative wind speed, β<sup>A</sup> is the wind drift angle, and χ<sup>A</sup> is the wind course angle.

• Rudder angle *θ*(*k*); • RPM(*k*); • Sway velocity at previous time step *v*(*k* −1); After this initial transformation, the coordinate ξ is assumed to be measured along the true meridian while η is along the parallel. However, when analysing the trajectories, the coordinates are transformed further so that the origin of the earth axes matches the ship's position at the start of a manoeuvre, and the ξ-axis is directed along the approach path.

• Heading angle at previous time step (*k* −1); • *x* position at previous time step *x*(*k* −1); • *y* position at previous time step *y*(*k* −1); and three outputs: • Heading angle at current time step (*k*); • *x* position at current time step *x*(*k*); Altogether, six test runs with the model are used for the training of the network, namely, four zigzags and two circles. Table 3 presents a summary of the data collected. In total, 5229 data points are provided (2348 from turnings and 2881 from zig-zag tests). Since the forward speed was not recorded, because the GPS used to track the model position only feedbacks the position and time, it was replaced in the model with the revolutions per minute (RPM) values. In this case, the orders are rudder angle and RPMs for certain sailing conditions. Increasing the RPM value increased the model forward speed, and decreasing the RPM decreased the model speed. RPM means the rotations of the propeller, which is directly related to the speed imparted to the ship. Changing RPM changes the speed.

• *y* position at current time step *y*(*k*). Connecting inputs and outputs is a single hidden layer with five neurons. A sigmoid-The analysis of the experimental data suggests that the trajectories have been modified by the effect of wind as the circles are not concentric and show a drift, which would be the effect of wind and current [42].

ture with the capability to provide smooth results. The function is given by:

( )

*f x*

*i*

based activation function is applied to every neuron in the hidden layer, creating a struc-

=

+1

(3)

*i*

*x*

*i*

*e*

*x*

*e*

It can be seen in Table 3 that the wind conditions during the zig-zag tests are very

Usually, multilayer perceptrons (MLPs) are trained with the backpropagation tech-

Although backpropagation is a gradient descent technique, the Levenberg–Mar-

nique, but in this work, the damped least-squares method, also known as the Levenberg– Marquardt algorithm, is employed, as well as the scaled conjugate gradient and Bayesian

quardt algorithmic rule is deduced from Newton's procedure that was defined to minimize functions that are additions of squares of nonlinear functions [43], as the configura-

similar but different from the wind conditions of the circles. Due to this difference in wind conditions, it is not appropriate to train a model on zig-zag data and validate it on circle data or vice versa. Two separate models are trained, one for each type of test. The training data points are all concatenated into two arrays, one for zig-zag tests and another one for circle tests. Each of these two arrays was then split according to the followingproportions:

80% of the data points used fortraining; 10% of the data points used forvalidation;

10% of the data points used fortesting.

regularization methods for comparison.

tion below:


**Table 3.** Data recorded.

## **3. Neural Network Training**

The model used has the following six inputs:


Connecting inputs and outputs is a single hidden layer with five neurons. A sigmoidbased activation function is applied to every neuron in the hidden layer, creating a structure with the capability to provide smooth results. The function is given by:

$$f(\mathbf{x}\_i) = \frac{e^{\mathbf{x}\_i}}{e^{\mathbf{x}\_i} + 1} \tag{3}$$

where *x* is the input of neuron *i*.

It can be seen in Table 3 that the wind conditions during the zig-zag tests are very similar but different from the wind conditions of the circles. Due to this difference in wind conditions, it is not appropriate to train a model on zig-zag data and validate it on circle data or vice versa. Two separate models are trained, one for each type of test. The training data points are all concatenated into two arrays, one for zig-zag tests and another one for circle tests. Each of these two arrays was then split according to the following proportions:

80% of the data points used for training;

10% of the data points used for validation;

10% of the data points used for testing.

Usually, multilayer perceptrons (MLPs) are trained with the backpropagation technique, but in this work, the damped least-squares method, also known as the Levenberg– Marquardt algorithm, is employed, as well as the scaled conjugate gradient and Bayesian regularization methods for comparison.

Although backpropagation is a gradient descent technique, the Levenberg–Marquardt algorithmic rule is deduced from Newton's procedure that was defined to minimize functions that are additions of squares of nonlinear functions [43], as the configuration below:

$$E = \frac{1}{2} \sum k(e\_k)^2 = \frac{1}{2} \|e\|^2 \tag{4}$$

where *e<sup>k</sup>* is the error in the *k th* exemplar and *e* is the vector of the elements *e<sup>k</sup>* . If the discrepancy between the preceding weight vector and the current one is small, the vector of the errors can be approximated to the first order using Taylor series expansion:

$$e(j+1) = e(j) + \frac{\partial e\_k}{\partial w\_l}(w(j+1) - w(j))\tag{5}$$

Therefore, the error function can be displayed as:

$$E = \frac{1}{2} \left\| \boldsymbol{\varepsilon}(\boldsymbol{j}) + \frac{\partial \boldsymbol{e}\_k}{\partial \boldsymbol{w}\_i} (\boldsymbol{w}(\boldsymbol{j} + 1) - \boldsymbol{w}(\boldsymbol{j})) \right\|^2 \tag{6}$$

Minimizing the error function in regard to the current weight vector:

$$w(j+1) = w(j) - \left(f^T f\right)^{-1} f^T e(j) \tag{7}$$

where (*J*)*ki* = *∂e<sup>k</sup> ∂w<sup>i</sup>* is the Jacobian matrix.

The Hessian matrix for the sum-of-square error function is expressed by:

$$\delta(H)\_{ij} = \frac{\partial^2 E}{\partial w\_i \partial w\_j} = \sum \left\{ \left( \frac{\partial e\_k}{\partial w\_i} \right) \left( \frac{\partial e\_k}{\partial w\_i} \right) + e\_k \frac{\partial^2 e\_k}{\partial w\_l \partial w\_j} \right\} \tag{8}$$

Neglecting the second term in (8), the matrix can be updated as:

$$\mathbf{H} = \mathbf{J}^T \mathbf{J} \tag{9}$$

The weights modification needs to take the inverse of the Hessian. The matrix is fairly uncomplicated to compute since it is grounded on first-order partial derivatives in regard to the network weights that are easily managed by the training algorithm. Although the updating equation is used repetitively to reduce the error function, this may generate a large step size, which could refute the linear approximation on which the equation is based. In the Levenberg–Marquardt algorithm, the error function is reduced to a minimum while the step size is remained low intending to guarantee the effectiveness of the linear approximation. This minimization is obtained through a modified error function of the following configuration:

$$E = \frac{1}{2} \|e(j) + \frac{\partial e\_k}{\partial w\_i}(w(j+1) - w(j))\|^2 + \lambda \|w(j+1) - w(j)\|^2 \tag{10}$$

where *λ* is a parameter governing the step size. Reducing the modified error to a minimum with regard to *w*(*j* + 1):

$$w(j+1) = w(j) - \left(f^T f + \lambda I\right)^{-1} f^T e(j) \tag{11}$$

When *λ* is null, (11) simply describes Newton's method, using the approximation to the Hessian matrix. Once *λ* is large, the formula converts to the steepest descent with a small step size. Newton's method is faster and more accurate when it is close to an error minimum; thus, the objective is to switch to Newton's method promptly. Consequently, *λ* is reduced after every successful step (reduction in performance function) and is increased just in case a tentative step would increase the performance function. Thus, the performance function is decreased every time at all procedure iterations. *λ* is reduced after every successful step (reduction in performance function) and is increased just in case a tentative step would increase the performance function. Thus, the performance function is decreased every time at all procedure iterations.

*J. Mar. Sci. Eng.* **2023**, *11*, x FOR PEER REVIEW 8 of 14

In this study, a single-hidden-layer MLP network is applied in MatLab and trained using the Levenberg–Marquardt algorithm. For the training mechanism, four input variables and two output variables were employed, as aforementioned. The quantity of hidden neurons of 5 was chosen after a methodical examination of the system convergence and generalization ability. In this study, a single-hidden-layer MLP network is applied in MatLab and trained using the Levenberg–Marquardt algorithm. For the training mechanism, four input variables and two output variables were employed, as aforementioned. The quantity of hidden neurons of 5 was chosen after a methodical examination of the system convergence and generalization ability.

A diagram of the designed framework is shown in Figure 3. Investigations concerning the training performance of different variants of the Backpropagation algorithms establish that the Levenberg–Marquardt algorithm is the fastest to converge. In addition, comparisons of predictions made by the different neural networks reveal that the neural network trained using the Levenberg–Marquardt algorithm gives the most accurate predictions. Results supporting these affirmations can be found in [44]. The fast convergence teamed with suitable predictive quality reported in the bibliography makes the Levenberg–Marquardt algorithm the primary suitable choice for training the neural network for the application developed in this work. A diagram of the designed framework is shown in Figure 3. Investigations concerning the training performance of different variants of the Backpropagation algorithms establish that the Levenberg–Marquardt algorithm is the fastest to converge. In addition, comparisons of predictions made by the different neural networks reveal that the neural network trained using the Levenberg–Marquardt algorithm gives the most accurate predictions. Results supporting these affirmations can be found in [44]. The fast convergence teamed with suitable predictive quality reported in the bibliography makes the Levenberg–Marquardt algorithm the primary suitable choice for training the neural network for the application developed in this work.

**Figure 3.** An illustration of the designed framework diagram. **Figure 3.** An illustration of the designed framework diagram.

In summarizing, in this study, a single-hidden-layer MLP network is applied in MatLab and trained by making use of the three different algorithms for comparison: the Levenberg–Marquardt, the scaled conjugate gradient, and Bayesian regularization methods. For the training mechanism, six input variables and three output variables are employed, as aforementioned. In this case, the rule-of-thumb method to determine the number of neurons to use in the hidden layer was based on the number of hidden neurons that should be between the size of the input layer and the size of the output layer. Different trials were performed with a different number of neurons between 3 and 6 in the hidden layer taken for each trial to determine the sensitivity of the neural network to these number of hidden neurons on the training performance. Then, 5 was chosen for the number of neurons in the hidden layer. These results are omitted from the text because they do not present interesting information. In summarizing, in this study, a single-hidden-layer MLP network is applied in MatLab and trained by making use of the three different algorithms for comparison: the Levenberg–Marquardt, the scaled conjugate gradient, and Bayesian regularization methods. For the training mechanism, six input variables and three output variables are employed, as aforementioned. In this case, the rule-of-thumb method to determine the number of neurons to use in the hidden layer was based on the number of hidden neurons that should be between the size of the input layer and the size of the output layer. Different trials were performed with a different number of neurons between 3 and 6 in the hidden layer taken for each trial to determine the sensitivity of the neural network to these number of hidden neurons on the training performance. Then, 5 was chosen for the number of neurons in the hidden layer. These results are omitted from the text because they do not present interesting information.

#### **4. Results 4. Results**

Figures 4a, 5a, 6a, and 7a show the predicted and experimental heading angle for data sets ZigZag1 to 4, and Figures 4b, 5b, 6b, and 7b show the respective predicted and experimental trajectories. The parameter used to assess the model error in zig-zag tests is the average heading error, and the results are based on the entire data set (All) using the Figure 4a, Figure 5a, Figure 6a and Figure 7a show the predicted and experimental heading angle for data sets ZigZag1 to 4, and Figure 4b, Figure 5b, Figure 6b and Figure 7b show the respective predicted and experimental trajectories. The parameter used to assess the model error in zig-zag tests is the average heading error, and the results are based on the entire data set (All) using the scaled conjugate gradient method.

scaled conjugate gradient method. The correlation coefficient *r* is calculated to control how well the system output fits the desired output. The correlation coefficient between a network output *x* and the desired output *d* is stated by:

$$r = \frac{\frac{\sum\_{i}(\mathbf{x}\_{i} - \overline{\mathbf{x}})\left(d\_{i} - \overline{d}\right)}{N}}{\sqrt{\frac{\sum\_{i}(d\_{i} - \overline{d})^{2}}{N}} \sqrt{\frac{\sum\_{i}(\mathbf{x}\_{i} - \overline{\mathbf{x}})^{2}}{N}}} \tag{12}$$

where *N* is the number of observations.

The best *r* values acquired for the approximation of the heading angle for the zig-zag manoeuvres are registered in Table 4 for the training, validation, and test subsets, as well as for the entire data set (all) for the three different methods considered.

The predictions for the zig-zag manoeuvres are very suitable, as can be seen in Figures 4–7 and in the results listed in Table 4. It can be noticed that the predictions in almost all the runs are very similar, mainly because all the trials were performed under the same environmental conditions. From the obtained results presented in Table 4, it can be seen that it is possible to predict the heading angle with very suitable accuracy for all three studied methods. *J. Mar. Sci. Eng.* **2023**, *11*, x FOR PEER REVIEW 9 of 14 *J. Mar. Sci. Eng.* **2023**, *11*, x FOR PEER REVIEW 9 of 14 *J. Mar. Sci. Eng.* **2023**, *11*, x FOR PEER REVIEW 9 of 14

**Figure 4.** Trial #1—Zig-Zag 30–30. (**a**) Heading angle estimation; (**b**) trajectory prediction. **Figure 4.** Trial #1—Zig-Zag 30–30. (**a**) Heading angle estimation; (**b**) trajectory prediction. **Figure 4.** Trial #1—Zig-Zag 30–30. (**a**) Heading angle estimation; (**b**) trajectory prediction. **Figure 4.** Trial #1—Zig-Zag 30–30. (**a**) Heading angle estimation; (**b**) trajectory prediction.

**Figure 5.** Trial #2—Zig-Zag 30–30. (**a**) Heading angle estimation; (**b**) trajectory prediction. **Figure 5.** Trial #2—Zig-Zag 30–30. (**a**) Heading angle estimation; (**b**) trajectory prediction. **Figure 5.** Trial #2—Zig-Zag 30–30. (**a**) Heading angle estimation; (**b**) trajectory prediction. **Figure 5.** Trial #2—Zig-Zag 30–30. (**a**) Heading angle estimation; (**b**) trajectory prediction.

**Figure 6.** Trial #3—Zig-Zag 20–20. (**a**) Heading angle estimation; (**b**) trajectory prediction. **Figure 6.** Trial #3—Zig-Zag 20–20. (**a**) Heading angle estimation; (**b**) trajectory prediction. **Figure 6. Figure 6.** Trial #3 Trial #3—Zig-Zag 20–20. ( —Zig-Zag 20–20. (**aa**) ) Heading angle estimation; ( Heading angle estimation; **b** ( ) trajectory prediction. **b**) trajectory prediction.

**Figure 7.** Trial #4—Zig-Zag 20–20. (**a**) Heading angle estimation; (**b**) trajectory prediction. **Figure 7.** Trial #4—Zig-Zag 20–20. (**a**) Heading angle estimation; (**b**) trajectory prediction.

(

*d d*

*i*

−

The correlation coefficient *r* is calculated to control how well the system output fits **Table 4.** Zigzags error measures (*r*).


*N N i i* where *N* is the number of observations. The best *r* values acquired for the approximation of the heading angle for the zig-zag manoeuvres are registered in Table 4 for the training, validation, and test subsets, as well as for the entire data set (all) for the three different methods considered. In Figure 4b, it can be seen that for *x* values larger than 50 m, there is a significant deviation in the predicted value. This can be explained from the wind velocity plot for this trial, presented in Figure 8, where it can be seen that around 400 s of the trajectory, the wind speed decreases, which causes a slowdown in the trajectory. For this reason, the neural network that had learned the trajectory of the previous instants had the tendency to continue with the same progress on the *y-*axis. *J. Mar. Sci. Eng.* **2023**, *11*, x FOR PEER REVIEW 11 of 14

)

(

*x x*

*i*

−

)

2

studied methods. In Figure 4b, it can be seen that for *x* values larger than 50 m, there is a significant **Figure 8.** Trial #1—Wind velocity. **Figure 8.** Trial #1—Wind velocity.

**Figure 9.** Experimental and estimated trajectory for data set turning 1.

**Figure 10.** Experimental and estimated trajectory for data set turning 2.

deviation in the predicted value. This can be explained from the wind velocity plot for this trial, presented in Figure 8, where it can be seen that around 400 s of the trajectory, the wind speed decreases, which causes a slowdown in the trajectory. For this reason, the neural network that had learned the trajectory of the previous instants had the tendency For assessing the performance of the model in circle tests, the tactical diameter is of interest. The tactical diameteris defined as the distance between two points whose heading differs by 180°. Figures 9 and 10 show the predicted and experimental trajectories for data For assessing the performance of the model in circle tests, the tactical diameter is of interest. The tactical diameter is defined as the distance between two points whose heading differs by 180◦ . Figures 9 and 10 show the predicted and experimental trajectories for data sets Turning 1 and Turning 2 using the Lavenberg–Marquardt method.

to continue with the same progress on the *y-*axis.

sets Turning 1 and Turning 2 using the Lavenberg–Marquardt method.

For assessing the performance of the model in circle tests, the tactical diameter is of

interest. The tactical diameteris defined as the distance between two points whose heading

For assessing the performance of the model in circle tests, the tactical diameter is of

interest. The tactical diameteris defined as the distance between two points whose heading differs by 180°. Figures 9 and 10 show the predicted and experimental trajectories for data

differs by 180°. Figures 9 and 10 show the predicted and experimental trajectories for data

sets Turning 1 and Turning 2 using the Lavenberg–Marquardt method.

sets Turning 1 and Turning 2 using the Lavenberg–Marquardt method.

**Figure 9.** Experimental and estimated trajectory for data set turning 1.

**Figure 8.** Trial #1—Wind velocity.

**Figure 8.** Trial #1—Wind velocity.

*J. Mar. Sci. Eng.* **2023**, *11*, x FOR PEER REVIEW 11 of 14

**Figure 10.** Experimental and estimated trajectory for data set turning 2.

**Figure 9.** Experimental and estimated trajectory for data set turning 1.

**Figure 10.** Experimental and estimated trajectory for data set turning 2.

**Figure 10.** Experimental and estimated trajectory for data set turning 2.

The best *r* results obtained for the positions *x* and *y* approximations for the circle manoeuvres are registered in Tables 5 and 6, respectively. The predictions for the circle manoeuvres are also very suitable, as can be seen from Figures 9 and 10 and through the results listed in Table 5, despite only having two data sets available, with the only difference between them being the rudder angle. The graphical results of Figures 9 and 10 indicate the evaluation of the predictive accuracy of the model. The plots show how well the model predicts and fit the values of the variables obtained in the real experiments. The plots are computed by using the "all" data results.

**Table 5.** Circles error measures (*r*)—*x* position.


**Table 6.** Circles error measures (*r*)—*y* position.


Again, from the obtained results presented in Tables 5 and 6, it can be seen that it is possible to predict the *x* and *y* positions with very suitable accuracy for all three studied methods.

Since neural networks are expected to be suitable interpolators, it can be assessed if the network generalizes well by giving input rudder angles between 20◦ and 26◦ because the circle tests were performed for these two values of rudder commands. Since no data are available, it is not possible to quantify the error of these simulations.

#### **5. Conclusions**

A method based on ANNs has been implemented to predict the heading angle and trajectories of a model ship from the output rudder angle command, the RPM of the propulsion shaft, the measurements of sway velocity, heading angle, and x and y positions at the previous time step. The training results were presented for the Levenberg–Marquardt algorithm and compared with the scaled conjugate gradient and Bayesian regularization methods. The information used to train and validate the system was acquired through manoeuvring tests with a chemical tanker model ship.

The obtained neural network system is suitable for producing precise approximations of the mentioned variables, showing that it is possible to obtain suitable results with an ANN trained using only five hidden neurons. The main feature of this study is to demonstrate that the ANN is able to learn even from a short and noisy data set. In addition, the method can be useful for predicting manoeuvring capabilities in the design stage of a ship. In future work, it is expected to be applied to different types of ships.

**Author Contributions:** Funding acquisition, C.G.S.; formal analysis, L.M.; writing—original draft, L.M.; writing—review and editing, C.G.S.; methodology, L.M.; software, L.M.; supervision, C.G.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research was performed within the NAVAD project "Simulation of manoeuvrability of ships in adverse weather conditions", which is co-funded by the European Regional Development Fund (Fundo Europeu de Desenvolvimento Regional—FEDER) and by the Portuguese Foundation for Science and Technology (Fundação para a Ciência e a Tecnologia—FCT) under contract 02/SAICT/032037/2017. This work contributes to the Strategic Research Plan of the Centre for Marine Technology and Ocean Engineering (CENTEC), which is financed by the Portuguese Foundation for Science and Technology (Fundação para a Ciência e Tecnologia—FCT) under contract UIDB/UIDP/00134/2020.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **Abbreviations**


RPM Revolutions per minute

#### **References**


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**Su-Hyung Kim <sup>1</sup> , Chun-Ki Lee 2,\* and Yang-Bum Chae <sup>2</sup>**


**Abstract:** The length between perpendiculars (LBP) of most fishing vessels is less than 100 m. Thus, they are not subject to the International Maritime Organization (IMO) maneuverability standards, affecting research on maneuverability. However, upon referencing the statistics of marine accidents related to vessel maneuvering, the number of marine accidents caused by fishing vessels is 3 to 5 times higher than that of merchant ships. Therefore, systematic and consistent research on the maneuverability characteristics of fishing vessels is surely required. In particular, a fishing vessel frequently enters and departs from the same port and often sails at high speed due to familiarity with the characteristics of the situation, which may cause maneuvering-related accidents. In this study, the maneuverability of a fishing vessel in shallow water was predicted using an empirical formula. The results of this study are expected to not only be of great help in conducting simulations when analyzing marine accidents involving fishing vessels, but will also provide unique parameters of fishing vessels that lead to developing autonomous vessels.

**Keywords:** fishing vessel; shallow water; maneuverability; empirical formula

#### **1. Introduction**

The International Maritime Organization (IMO) approved the ship's maneuverability standard in 2002 to prevent maritime accidents caused by the unique maneuverability problems of the ship itself. For this reason, a ship equipped with traditional propulsion and steering systems of length between the perpendiculars (LBP) of 100 m or more is subjected to a maneuverability test based on this standard after construction is complete [1]. On the other hand, most fishing vessels have an LBP of less than 100 m, so they are not subject to the IMO maneuverability standard. This has led to reduced demand for these predictions or trials, which means little data is available [2,3]. In other words, while the maneuverability studies for a vessel to which the IMO maneuverability criteria for 100 m length or longer are applied and are actively conducted from the design stage, this has not been the case with most types of fishing vessels of less than 100 m, and the results of studies conducted on merchant vessels have been accepted and applied as they are [4].

However, the shape of a ship depends on its specific purpose. In particular, fishing vessels and merchant ships have their own hull shape characteristics. For example, the block coefficient *C<sup>b</sup>* , which can have the maneuverability needed to quickly chase shoals of fish, and installing the fishing gear at the correct location, is similar to high-speed slender ships such as container ships or car carriers. The large length to beam ratio (L/B) to secure sufficient hull capacity and stability are similar to low-speed large ships such as Ultra Large Crude-oil Carriers (ULCC) or Very Large Crude-oil Carriers (VLCC) [2]. Therefore, when applying the results of research performed on merchant ships to fishing vessels, appropriate corrections are required [5]. Of course, the best would be to conduct a study similar or identical to the study performed on merchant ship types on fishing vessels. However, in reality, most fishing vessels with a length of less than 100 m are not required

**Citation:** Kim, S.-H.; Lee, C.-K.; Chae, Y.-B. Prediction of Maneuverability in Shallow Water of Fishing Trawler by Using Empirical Formula. *J. Mar. Sci. Eng.* **2021**, *9*, 1392. https://doi.org/10.3390/ jmse9121392

Academic Editor: Cristiano Fragassa

Received: 16 November 2021 Accepted: 4 December 2021 Published: 6 December 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

to conform to the IMO maneuverability standard. Also, when a marine accident occurs, the possibility of causing significant marine pollution is significantly lower than for merchant ships, and it is difficult to conduct certain types of research for various reasons, such as the high cost of constructing a model ship.

Nevertheless, it is necessary to conduct a study of the maneuverability of a fishing vessel. Referencing the statistics of marine accidents related to ship maneuvering over the past 5 years (2016–2020), out of a total of 2227 cases (excluding leisure craft and other vessels), 1673 cases (75.1%) of the collision accidents involved fishing vessels, more than three times higher than the 554 cases (24.9%) involving merchant ships. In the case of contact and grounding accidents, 665 cases (82.8%) were fishing vessels out of a total of 803 cases (excluding leisure crafts and other vessels), almost five times higher than that of 138 cases (17.2%) of merchant ships [6]. In other words, despite satisfying the criteria for IMO maneuverability, marine accidents such as those described above consistently occur. Of course, it is known that human factors play the biggest role in the cause of marine accidents [7–13], though the problem of the unique maneuverability of a ship cannot be underestimated.

In addition, systematic and consistent research on the maneuverability of fishing vessels can accumulate data that identify the unique hull shape characteristics of fishing vessels. Accumulation of data on these hull shape characteristics provides high accuracy of characteristic parameters for fishing vessel types in marine accident analysis and autonomous vessel development. Studies on the prediction of the maneuverability of fishing vessels have been carried out by Yoshimura [14–17], Dan [18], Lee [3,4,19], Kim [2,5,20], etc. [21–27], but a lot of research is still required.

Based on a review of the state of the art, the authors conducted a study to devise an empirical maneuverability prediction formula more suitable to fishing vessels [2,7]. Through the verification process, it has been confirmed that improved results were obtained in predicting the maneuverability of a fishing vessel [5]. In this study, by applying the empirical formula of Kijima et al., which includes correcting factors of the hydrodynamic coefficients of fishing vessel hull shape obtained from the corrected empirical formula, a study was performed to predict maneuverability in shallow water.

#### **2. Mathematical Model**

#### *2.1. Coordinate System and Motion Equations*

The equation of motion used in this study was derived from the right-handed orthogonal coordinate system shown in Figure 1. o<sup>0</sup> − x0y<sup>0</sup> z<sup>0</sup> is the earth-centered fixed coordinate system, o − xyz and is the hull body-fixed coordinate system with the mid-ship fixed at the origin (o). Here, z<sup>0</sup> is oriented vertically downwards in the x<sup>0</sup> − y<sup>0</sup> plane, and similarly, z is oriented vertically downwards in the x−y plane.

The maneuvering equation of motion can be expressed in various ways. In the Kijima et al. empirical formula used in this study, the drift angle β and the nondimensionalized angular velocity r0 were used, as shown in Equation (1).

r

0 = rL/U

(m0 + m0 x ) L U . U U cos β − . β sin β + m0 + m0 y r 0 sin β = X 0 − m0 + m0 y <sup>L</sup> U . U U sin β + . β cos β + (m0 + m0 x )r 0 cos β = Y 0 I 0 zz + i 0 zz L U 2 . U L r <sup>0</sup> + <sup>U</sup> L . r 0 = N0 (1) m0 , m0 x , m0 <sup>y</sup> = m, mx, my/ 1 2 ρL 2d I 0 zz, i<sup>0</sup> zz = Izz, izz/ 1 2 ρL 4d X 0 , Y<sup>0</sup> = X, Y/<sup>1</sup> 2 ρLdU<sup>2</sup> N<sup>0</sup> = N/<sup>1</sup> 2 ρL <sup>2</sup>dU<sup>2</sup>

**Figure 1.** Coordinate system.

The Kijima et al. empirical formula is based on the maneuvering modeling group (MMG) model [17,28]. It can be expressed as Equation (2) by dividing the external force terms X 0 , Y 0 , N0 on the right side of Equation (1) into hull, rudder, and propeller components, respectively. Here, the subscripts H, R, and P denote hull, rudder, and propeller [29].

X 0 = X 0 <sup>H</sup> + X 0 <sup>R</sup> + X 0 P Y 0 = Y 0 <sup>H</sup> + Y 0 R N0 = N0 <sup>H</sup> + N<sup>0</sup> R (2)

#### *2.2. Forces and Moment Affecting the Hull*

The forces X 0 <sup>H</sup>,Y 0 <sup>H</sup> and moment N<sup>0</sup> <sup>H</sup> affecting on the hull are calculated using the drift angle β and the nondimensionalized angular velocity r 0 and it can be expressed as Equation (3) [30].

$$\mathbf{X}'\_{\rm H} = \mathbf{X}'\_{\beta\rm r} \mathbf{r}' \sin \beta + \mathbf{X}'\_{\rm uu} \cos^2 \beta$$

$$\mathbf{Y}'\_{\rm H} = \mathbf{Y}'\_{\beta} \beta + \mathbf{Y}'\_{\rm r} \mathbf{r}' + \mathbf{Y}'\_{\beta\,\beta} \beta \left| \beta \right| + \mathbf{Y}'\_{\rm rr} \mathbf{r}' \left| \mathbf{r}' \right| + \left( \mathbf{Y}'\_{\beta\,\beta\,\mathbf{r}} \beta + \mathbf{Y}'\_{\beta\,\mathbf{r}\,\mathbf{r}} \mathbf{r}' \right) \mathbf{\beta} \mathbf{r}' \quad \text{(3)}$$

$$\mathbf{N}'\_{\rm H} = \mathbf{N}'\_{\beta} \beta + \mathbf{N}'\_{\rm r} \mathbf{r}' + \mathbf{N}'\_{\beta\,\beta} \beta \left| \beta \right| + \mathbf{N}'\_{\rm rr} \mathbf{r}' \left| \mathbf{r}' \right| + \left( \mathbf{N}'\_{\beta\,\beta\,\mathbf{r}} \beta + \mathbf{N}'\_{\beta\,\mathbf{r}\,\mathbf{r}} \mathbf{r}' \right) \mathbf{\beta} \mathbf{r}' \quad \text{(4)}$$

#### *2.3. Forces and Moment from the Propeller*

In general, the force generated by the propeller is the forward and backward force X 0 P , if left-right force Y 0 P and moment N0 P are omitted under the assumption that they are minute, they can be expressed as Equation (4) [29].

$$\mathbf{X}'\_{\rm P} = \mathbf{C}\_{\rm tP} (1 - \mathbf{t}\_{\rm P0}) \mathbf{K}\_{\rm T} (\mathbf{J}\_{\rm P}) \mathbf{n}^2 \mathbf{D}\_{\rm P}^4 / \frac{1}{2} \mathbf{L} \mathbf{d} \mathbf{U}^2$$

$$\mathbf{K}\_{\rm T} (\mathbf{J}\_{\rm P}) = \mathbf{C}\_1 + \mathbf{C}\_2 \mathbf{J}\_{\rm P} + \mathbf{C}\_3 \mathbf{J}\_{\rm P}^2$$

$$\mathbf{J}\_{\rm P} = \mathbf{U} \cos \boldsymbol{\beta} (1 - \mathbf{w}\_{\rm P}) / (\mathbf{n} \mathbf{D}\_{\rm P})$$

## *2.4. Forces and Moment from the Rudder*

The forces X 0 R , Y 0 R and moment N0 R affecting the rudder is expressed as Equation (5), and tR, aH, x 0 H are the main interaction coefficients affecting the rudder, propeller, and

hull, and the normal force affecting the rudder F 0 <sup>N</sup> also has a strong correlation with the interaction coefficients, as shown in Equation (6) [29].

X 0 <sup>R</sup> = −(1 − tR)F 0 <sup>N</sup> sin δ Y 0 <sup>R</sup> = −(1 + aH)F 0 <sup>N</sup> cos δ N0 <sup>R</sup> = −(x 0 <sup>R</sup> + aHx 0 <sup>H</sup>)F 0 <sup>N</sup> cos δ (5) F 0 <sup>N</sup> = (AR/Ld)CNU 02 R sin a<sup>R</sup> C<sup>N</sup> = 6.13KR/(K<sup>R</sup> + 2.25) U 02 <sup>R</sup> = (1 − wR) 2 {1 + Cg(s)} g(s) = ηK{2 − (2 − K)s}s/(1 − s) 2 η = DP/h<sup>R</sup> K = 0.6(1 − wP)/(1 − wR) s = 1.0 − (1 − wP)Ucosβ/nP w<sup>R</sup> = wR0·wP/wP0 a<sup>R</sup> = δ − γ·β 0 R β 0 <sup>R</sup> = β − 2x<sup>0</sup> R ·r 0 , x0 R ∼= −0.5 (6)

#### **3. Empirical Formula**

The method for predicting maneuverability at the design stage can be classified as a method using a database of similar or identical ships, constructing and testing a model, and a numerical simulation which is a mathematical method. Among them, the empirical formula, one of the numerical simulation methods, is mainly utilized in the case of fishing vessels. Although the accuracy is lower than that of the model test, it has the advantage of the execution process being relatively simple.

There are various empirical formulae developed for hull-shape merchant ships designed by each ship design laboratory, and they are corrected through consistent researches and revisions. However, since the process of deriving an empirical formula similar to this is technology belonging to each research institute, only a small portion of it is disclosed. On the other hand, the Kijima et al. empirical formula has been published using the specifications of the target ships used in the research for the process of deriving the empirical formula [29,31,32]. Also, since the proposed empirical formula concerning the shape of the stern for shallow water has been developed [33], the authors performed a study selecting the subjects based on the Kijima et al. empirical formula.

## *3.1. Kijima et al. Empirical Formula*

The empirical formula used in this study is an empirical formula that does not consider the shape of the stern among all of the proposed Kijima et al. empirical formulas. This formula is suitable for predicting the ship's maneuverability in deep water and is suitable for ships with conventional hull shapes, in particular those that have conventional stern shapes [29]. A typical equation is shown in Equation (7) below to derive linear coefficients among the hydrodynamic forces affecting the hull in the even keel state.

$$\begin{aligned} \mathbf{Y}'\_{\beta} &= \frac{1}{2}\pi \mathbf{k} + 1.4 \mathbf{C}\_{\mathbf{b}} \mathbf{B}/\mathbf{L} \\ \mathbf{Y}'\_{\mathbf{r}} - (\mathbf{m}' + \mathbf{m}'\_{\mathbf{x}}) &= -1.5 \mathbf{C}\_{\mathbf{b}} \mathbf{B}/\mathbf{L} \\ \mathbf{N}'\_{\beta} &= \kappa \\ \mathbf{N}'\_{\mathbf{r}} &= -0.54\kappa + \kappa^2 \end{aligned} \tag{7}$$

## *3.2. Corrected Empirical Formula*

As stated, research on the maneuverability of fishing vessels is lacking compared to merchant ships due to various reasons. The model test is rarely carried out due to cost and time-consuming considerations during the design stage, and there have also been many difficulties in securing performance data (resistance, self-propulsion, and propeller open water test) required to predict the maneuverability.

For these reasons, the authors conducted a study to derive a corrected empirical formula. It is expected to further improve the prediction of the maneuverability of fishing vessels by including the unique hull shape characteristic parameters of fishing vessels in the Kijima et al. empirical formula, which is widely used in shipbuilding practice [2,5,20]. The schematic processes are as follows:



**Table 1.** Hull shape parameters of 18 ships.

**Figure 2.** Hull shape parameters of 18 ships.


**Table 2.** Linear hydrodynamic coefficients of 18 vessels.

**Figure 3.** Correlation diagram between linear hydrodynamic coefficients and characteristics of hull shape parameters.

(3) The correlation between the selected characteristics of hull shape parameters and the maneuvering hydrodynamic coefficients are shown and averaged using a trend line to derive a corrected empirical formula expected to be more suitable for fishing vessels (Figure 4). The equations for deriving the coefficients in the even keel state are shown in Equation (8) below.

Y 0 <sup>β</sup> = −1.5747{1 − Cb/(L/B)} + 0.4488 Y 0 ββ = 0.0417 × (L/B) + 0.541 Y 0 r − m0 + m 0 *x* = 0.0432 × (L/B) − 0.4276 Y 0 rr = −0.7946 × {1 − Cb/(L/B)} + 0.0563 Y 0 <sup>β</sup>rr = 0.0993 × (L/B) + 0.0975 Y 0 ββ<sup>r</sup> = 2.7467 × k − 0.6316 N 0 <sup>β</sup> = 0.238 × Cb/(B/d) + 0.0663 N 0 ββ = −0.016 × (L/B) + 0.0503 N 0 <sup>r</sup> = 0.0515 × {1 − Cb/(L/B)} − 0.0537 N 0 rr = −0.0144 × (L/B) + 0.0525 N 0 <sup>β</sup>rr = −0.9156 × k + 0.0439 N 0 ββ<sup>r</sup> = −3.399 × {1 − Cb/(L/B)} − 0.0737 1 − t<sup>R</sup> = −0.0127 × (L/B) + 0.8122 a<sup>H</sup> = −0.1107 × (L/B) + 1.1421 x 0 <sup>H</sup> = −0.258 × (L/B) + 0.4603 1 − wP0 = 0.0227 × (L/B) + 0.5818 ε = −1.4308 × {1 − Cb/(L/B)} + 0.9453 γ = 0.1608 × (L/B) − 0.5764 

(8)

**Figure 4.** Trend line of correlation between linear hydrodynamic coefficients and characteristics of hull shape parameters.

(4) The results for maneuvering hydrodynamic coefficients of the 5 model fishing vessels were derived from the corrected empirical formula (Table 3), and the validity was verified by performing a turning movement simulation in 4 vessels, except for F(E), which was under construction at the time (Figure 5).

**Table 3.** Linear coefficients of the model fishing vessels derived from the corrected empirical formula.


**Figure 5.** Comparison of turning movement test results in model fishing vessels.

It can be confirmed that the corrected empirical formula in Figure 5 shows an improved result compared to the empirical formula developed for merchant ships in predicting the maneuverability of fishing vessels. However, since the corrected empirical formula proposed by the authors is limited to the hull shape parameters of the limited stern fishing trawlers, prediction errors may occur in predicting the maneuverability of fishing vessels having hull shape parameters out of the range presented below.

0.574 ≤ C<sup>b</sup> 0.616, 4.93 ≤ L/B ≤ 5.76, 2.64 ≤ B/d ≤ 2.9

#### *3.3. Kijima et al. Empirical Formula in Shallow Water including Correcting Factors*

In modern times, all ships, including fishing vessels, have grown in size and often do not satisfy the operating conditions in ports. Accordingly, not only maneuvering in deep water but also maneuvering in shallow water has become an important research subject. However, since it is difficult to verify research results using real ships in shallow water, research is mainly conducted through numerical simulations or model tests. Studies using model tests such as free-running sailing obtain the most reliable results, though in reality, the method of estimating maneuverability by calculating hydrodynamic force coefficients is widely used. However, all hydrodynamic force coefficients are not linearized, and these values may cause a large error in estimating maneuverability. Fortunately, studies are being conducted to minimize these errors [34–36], and it is expected that a more accurate estimation of the nonlinear coefficients will be possible in the future. In this study, the maneuverability of a target fishing vessel was estimated using numerical simulation, which is somewhat uncertain, but results can be derived more easily. After that, the scope of the study will be gradually expanded through comparative analysis with the model tests.

To predict the maneuverability of ships in shallow water, correcting factors concerning the effect of ship-draft to water-depth ratio and the value of the maneuvering hydrodynamic coefficients used for predicting the maneuverability of the ships in deep water is required. Accordingly, Kijima and Nakiri [33] proposed an empirical formula for predicting ship maneuverability in shallow water, including correcting factors regarding the effect of ship-draft to water-depth ratio as shown in Equation (9). The equations for deriving the typical linear coefficients are shown in Equation (10).

$$\begin{aligned} \mathbf{D}\_{\text{shallow}} &= \mathbf{f}(\mathbf{h}) \times \mathbf{D}\_{\text{deep}} \\ \mathbf{f}(\mathbf{h}) &= 1/(1-\mathbf{h})^{\text{n}} - \mathbf{h} \\ \mathbf{f}(\mathbf{h}) &= 1 + \mathbf{a}\_{1}\mathbf{h} + \mathbf{a}\_{2}\mathbf{h}^{2} + \mathbf{a}\_{3}\mathbf{h}^{3} \end{aligned} \qquad \tag{9}$$

$$\begin{aligned} \mathbf{Y}'\_{\beta} &: \mathbf{n} = 0.40 \mathbf{C\_b} \mathbf{B/d} \\ \mathbf{Y}'\_{\mathbf{r}} - (\mathbf{m}' + \mathbf{m}'\_{\mathbf{x}}) \begin{bmatrix} \mathbf{a}\_1 = -5.5(\mathbf{C\_b} \mathbf{B/d})^2 + 26 \mathbf{C\_b} \mathbf{B/d} - 31.5 \\ \mathbf{a}\_2 = 37(\mathbf{C\_b} \mathbf{B/d})^2 - 185 \mathbf{C\_b} \mathbf{B/d} + 230 \\ \mathbf{a}\_3 = -38(\mathbf{C\_b} \mathbf{B/d})^2 + 197 \mathbf{C\_b} \mathbf{B/d} - 250 \\ \mathbf{N}'\_{\beta} : \mathbf{n} = 0.425 \mathbf{C\_b} \mathbf{B/d} \\ \mathbf{N}'\_{\mathbf{r}} : \mathbf{n} = -7.14 \mathbf{\kappa} + 1.5 \end{bmatrix} \end{aligned} \tag{10}$$

#### *3.4. Discriminant for Course Stability*

The course stability depends on the ship-draft to water-depth ratio, and the conditions for quantitatively determining the course stability of a ship using the maneuvering hydrodynamic coefficients are as shown in Equation (11) below. If the value of stability index C is (+), the course is stable, and if it is (−), it is determined as unstable. Since −Y 0 β Y 0 <sup>r</sup> − (m<sup>0</sup> + m<sup>0</sup> x ) always represents a (+) value, so if the value of l 0 <sup>r</sup> − l 0 β is (+), the course is stable, and if it is (−), the course is unstable [33].

$$\begin{aligned} \mathcal{C} &= -\mathbf{Y}'\_{\mathbb{B}} \left\{ \mathbf{Y}'\_{\mathbf{r}} - (\mathbf{m}' + \mathbf{m}'\_{\mathbf{x}}) \right\} \times \left\{ \frac{\mathbf{N}'\_{\mathbf{r}}}{\mathbf{Y}'\_{r} - (\mathbf{m}' + \mathbf{m}'\_{\mathbf{x}})} - \frac{\mathbf{N}'\_{\beta}}{\mathbf{Y}'\_{\beta}} \right\} \\ &= -\mathbf{Y}'\_{\mathbb{B}} \left\{ \mathbf{Y}'\_{\mathbf{r}} - \left( \mathbf{m}' + \mathbf{m}'\_{\mathbf{x}} \right) \right\} \times \left\{ \mathbf{l}'\_{\mathbf{r}} - \mathbf{l}'\_{\beta} \right\} \end{aligned} \tag{11}$$

#### **4. Maneuverability Prediction of a Fishing Trawler**

*4.1. Target Fishing Vessel*

The target fishing vessel is the F(E) in Table 1. The reason for selecting F(E) as a representation for the target fishing vessel was because it had been included in the corrected empirical formula derivation process, though the verification process could not be executed as it was under construction at the time. However, through the previous study, the verification of the turning motion test, 10/10 zig-zag test, etc., was verified, so based on the results, successive studies have been conducted to determine whether it is effective in shallow water. Outlined results related to the previous studies are briefly introduced in Section 4.2.

The target fishing vessel is a fisheries training ship having the hull shape of typical stern fishing trawlers, and the main specifications and body plan are shown in Table 4 and Figure 6.


**Table 4.** Main specifications of Target fishing vessel.

#### *4.2. Prediction of Maneuverability in Deep Water*

4.2.1. Derivation of Maneuvering Hydrodynamic Coefficients

The basic purpose of this study is to verify whether the accuracy of the prediction for the maneuverability of a fishing vessel could be improved only by adding the characteristic parameters of the fishing vessels to the empirical formula developed for the merchant vessel type. To verify the validity of the corrected empirical formula, the maneuvering hydrodynamic coefficients of the target fishing vessel have been derived from the empirical

formula of Kijima et al., and the corrected formula is shown in Equations (7) and (8). Therefore, all the hydrodynamic coefficients, including the interaction coefficients, were derived only through Equations (7) and (8). The typical linear coefficient values among the hydrodynamic forces affecting the hull are shown in Table 5 below.



#### 4.2.2. Conditions for Maneuverability Evaluation

In order to evaluate the performance of a ship, a maneuvering trial must be executed on both port and starboard under the conditions specified below [1].


However, in an actual ship test, because the external force is affecting as an irresistible force, it is impossible to perfectly harmonize the conditions of the simulation, and such a factor was taken into consideration. Table 6 shows the actual ship test and simulation conditions of the target fishing vessel [5,37].

**Table 6.** Maneuverability evaluation conditions in deep water.


4.2.3. Results of Maneuverability Evaluation

The simulation results of the maneuverability prediction of the target fishing vessel in deep water (Turning motion and 10/10 Zig-zag) are shown in Figure 7 below. It can be confirmed that the corrected empirical formula is closer quantitatively and qualitatively via actual ship test results than the Kijima et al. empirical formula. On this test, both evaluations were performed [5,37].

**Figure 7.** Comparison of turning motion (**a**) and 10/10 zig-zag (**b**) simulation results.

First, in the case of the turning motion test, both empirical formula results met the IMO maneuverability criterion. However, it can be confirmed that the corrected empirical formula result is way further quantitatively and qualitatively than the actual ship test result. At the same time, the shape of the port and starboard trajectories in the actual ship test are somewhat different, and based on the port turn it can be analyzed as follows:

a. The effect of the wind blowing from the port stern direction (relative bearing 205◦–206◦ ) is shown in Figure 7. During those time, the actual ship test temporarily blocked the turning of the bow just before the turn started, and the results tended to be slightly longer than when turning to starboard. Also, from the point when the bow started to turn, it helped the process, and it became rapidly faster than starboard turning (Figure 7a).

Next, in the case of the 10/10 Zig-Zag test, it can be confirmed that the result of the corrected empirical formula is closer to the actual ship test result compared to the result of the Kijima et al. empirical formula in connection to the tendency of the occurring location and trajectory of the overshoot angle. In the case of 1st overshoot angle, in the actual ship test and the corrected empirical formula, it occurred around 15 s, and for about 10 s, the slope became sharp. On the other hand, in the Kijima et al. empirical formula, 1st overshoot angle started to occur around 27 s, and a slight turning occurred. In the case of the 2nd overshoot angle, it also shows a tendency similar to the 1st overshoot angle (Figure 7b).

However, compared to the results of the two empirical formulas, the overshoot angle of the actual ship test was larger to a degree, which can be seen as a result of reflecting the unique characteristics of the target fishing vessel.

Many factors contribute to maintaining the stability of the ship course, and rudder area ratio plays a large role. When the rudder is large the rudder effect is better, though there are also disadvantages such as a decrease in speed due to resistance generation. Thus, the surface area ratio of the rudder is different depending on the purpose of construction. In the case of fishing vessels, the rudder area ratio is generally larger than that of merchant ships because it requires quick maneuverability to track the fish group. The rudder area ratio can be expressed as the ratio of the longitudinal cross-sectional surface area of the hull subsidence to the rudder area. At maximum draft, merchant ships are around 1/60 to 1/70, fishing vessels are 1/35 to 1/40, and warships are 1/30 to 1/50 [38].

Based on the above-mentioned theory, the results of comparing the rudder area ratio in 5 fishing trawler models (Tables 1 and 2), which were used to derive a corrected empirical formula to check the effect of the rudder area ratio on the overshoot angle, is as shown in Table 7 and Figure 8 below.

**Table 7.** Rudder area ratio of model fishing trawlers.


**Figure 8.** Rudder area ratio of model fishing trawlers.

F(A)~F(C) satisfy the other rudder area categories of fishing vessels, but it can be confirmed that F(D) and target fishing vessel F(E) belong to the rudder area ratio category of general merchant ships. This is because in the case of F(D) a flap rudder was installed, and the target fishing vessel was equipped with a thruster at the bow and stern, a function supporting the role of the basic rudder.

Meanwhile, such a result is an important basis for confirming that the maneuvering characteristics may vary depending on the propeller and rudder characteristics. Even though the vessel types have a similar hull shape, it is found that a systematic updating process is required by consistently adding new characteristic parameters as well as developing empirical formulas specific to each vessel type.
