1.Introduction
The need for ship modelling technology is continuously growing with the development of marine information technology. Ship modelling is necessary not only for conventional demands, such as early ship design stages and realtime ship simulation, but also for various forms of fast time simulation for education and training (Benedict et al., 2014). Given this rapid technological development, the importance of simple and efficient ship modelling method is still increasing.
Estimating hydrodynamic coefficients for a ship model is one important stage in determining ship manoeuvrability with high accuracy. Especially for the submerged part of the hull, the forces and moments at work can be presented via hydrodynamic coefficients. The International Towing Tank Conference (ITTC) summarized different methods (Fig. 1) to estimate hydrodynamic coefficients to determine ship manoeuvrability (ITTC, 2008). Each method has individual accuracy, effort and cost characteristics, but the captive model test and Computational Fluid Dynamics (CFD) method are commonnly used at the design stage (Oltmann, 2003; Seils, 1990). These methods are the most reliable source of hydrodynamic coefficients, excluding full scale trials, which require relatively high cost and calculation time compared to an empirical method with system identification.
This paper applies a system identification method using sea trial data. It estimates the hydrodynamic coefficients for a ship model via a mathematical optimization algorithm. This algorithm compares results of manoeuvre simulation with benchmark data, such as sea trial data, and it provides updated, optimized target variables.
Various ideas on system identification have been studied with the progress of computational calculations. The Extended Kalman Filter (EKF) has been widely used since the beginning of the development of this method (Abkowitz, 1980; Hwang, 1980), and System Based (SB) free running tests have also been carried out with the EKF algorithm (Rhee and Kim, 1999; Zahng and Zou, 2011). Other mathematical algorithms have also been introduced with the development of computers, such as Sequential Quadratic Programming (SQP) and BroydenFletcherGoldfarbShanno (BFGS) (Saha and Sarker, 2010; Tran et al., 2014).
Kim et al. optimized hydrodynamic coefficients with an interior point algorithm, based on simulation manoeuvre data as a preliminary study (Kim et al., 2016). This paper presents a second validation of the suggested optimization algorithm. The benchmark data set consists of sea trial results for the training ship Hanbada of the Korea Maritime and Ocean University. Comparison among the benchmark data, initial condition of the optimization process and finally optimized data are also presented in this paper.
2.Modelling ship and benchmark data
2.1.Mathematical model
The 3DegreesofFreedom (DOF) shipfixed and Earthfixed coordinate systems are applied in this study, and Fig. 2 presents the relevant concepts where the Earthfixedcoordinate O_{0}  x_{0}y_{0} plane and the shipfixedcoordinate O  xy plane lie on an undisturbed free surface, with the x_{0} axis pointing in the direction of the original heading of the ship, while the z_{0} axis and the z axis point vertically downwards. The angle between the x_{0} and x axes is defined as the heading angle, ψ.
where,

G : Center of gravity

Ψ : Heading

β : Drift angle

δ : Rudder angle

$\overrightarrow{V}$ : Ship speed

r : Yaw rate
The fast time simulation tool SIMOPT, of the ISSIMS Institute from Hochschule Wismar was used for simulation in the optimization process (Fig. 3). This tool uses almost the same ship dynamic features as the Ship Handling Simulator (SHS) systems (ANS5000) developed by Rheinmetall Defence Electronics (ISSIMS GmbH, 2013). In the mathematical model used for this tool, a ship is considered a massive and rigid body. Forces and moment acting on the hull are described as in equation (1), according to the Newtonian law of motion (Rheinmetall Defence Electronic, 2008).
Each value for force and moment in the model consists of multiple modules, as in Equation (2): hull, propeller, rudder and other external forces and moments. Environmental factors are considered in the following chapter.
Equation (3) shows the composition of the hydrodynamic forces and moment acting on the hull. In this model, empirical regression formulas by Norrbin and Clarke are applied to calculate initial hydrodynamic coefficients (Norrbin, 1971; Clarke et al., 1983). Each hydrodynamic coefficient can be expressed as a function of the ship’s main dimensions, as in Equation (4): length, beam, draught and displacement of the ship. Y_{non} and N_{non} are nonlinear components of sway force and yaw moment. These nonlinear components are dependent on the position of the ship’s turning point.
2.2.Benchmark data
The training ship Hanbada has been adopted as a benchmark vessel for the optimization process, and the particulars of this vessel are given in Table 1.
The environment is one of the biggest factors that influences manoeuvre characteristics between the towing tank model experiment and the fullscale sea trial. Controlling and calibrating environmental factors are important for obtaining accurate mathematical optimization results from the sea trial. Thus, a correction method provided by the International Maritime Organization (IMO) has been applied to calibrate track coordinates for the sea trial results. The detailed procedure is as follows (IMO, 2002).
To measure environmental influence, turning circle test results are required. The recorded data should include the ship’s track, heading and the time elapsed with at least a 720° change of heading. In terms of the data, two halfcircles can be obtained after a heading change of 180° from the beginning of the test. Local current velocity V_{i} can be defined using two corresponding positions $\left({{x}^{\prime}}_{1i},\hspace{0.17em}{{y}^{\prime}}_{1i},\hspace{0.17em}{{t}^{\prime}}_{1i}\right)\hspace{0.17em}\text{and}\hspace{0.17em}\left({{x}^{\prime}}_{2i},\hspace{0.17em}{{y}^{\prime}}_{2i},\hspace{0.17em}{{t}^{\prime}}_{2i}\right)$, from the halfcircles drawn as Equation (5):
From local velocity, estimated current velocity can be calculated, as in Equation (6):
The magnitude of current velocity can be calculated using Equation (7):
The final corrected trajectories from the measured data can be obtained from Equation (8):
where X(t) is the measured position vector and X′(t) is the corrected vector for the ship, with X′(t) = X(t) at t = 0.
Fig. 4 and 5 show a comparison of the measured sea trial trajectory and the calibrated trajectory. The magnitude and direction of current velocity are also applicable to other manoeuvres.
3.Optimization of hydrodynamic coefficients
3.1.Mathematical optimization
Mathematical optimization is a process that minimizes or maximizes an objective function value, subject to several variable constraints (Nocedal and Wright, 2006). This can be expressed as in Equation (9):
where,

 x is the variable to be optimized, which normally should be a vector

 f is an objective function which returns a scalar and contains information for minimization or maximization

 c_{i} are constraints, sets of equations and inequalities that variable x must satisfy throughout the optimization process.
The MATLAB Optimization Toolbox calculates various kinds of optimization problems, such as constrained, unconstrained, continuous and discrete problems, using popular optimization solvers and algorithms. Fig. 6 shows the whole process of mathematical optimization for hydrodynamic coefficients.
Solvers require an objective function to provide a minimum or maximum value to optimize of target values. In order to improve the reliability and accuracy of the result of the optimization process, additional constraints may be required. A lower and upper bound, linear and nonlinear equalities and linear and nonlinear inequalities are representational constraints that may be involved in the optimization process.
3.2.Sensitivity analysis
The estimated time required for mathematical optimization is highly dependent on the number of variables to be optimized. In this study, the variables are the hydrodynamic coefficients for the ship’s hull. Thus, it is important to check the sensitivity of each hydrodynamic coefficient in terms of how strongly they contribute to a ship manoeuvre, prior to conducting the optimization process. Summarized sensitivity analysis procedures are as follows:

1) Separate coefficients into three groups according to the manoeuvre tests: straight motion, manoeuvre with a small rudder angle and manoeuvre with large rudder angle.

2) Change the specific coefficient from a value close to 0 for each sign to 10 times the original value, and conduct simulation.

3) Find the derivative of non dimensional manoeuvre characteristics with respect to the change of coefficient and divide this by the greatest value of the manoeuvre characteristics values for normalization.

4) Repeat for all coefficients.
Fig. 7 to 9 show the results of the sensitivity analysis, and Table 2 shows the list of coefficients to be optimized. The optimization process in this study consists of three phases. Step 1 optimizes two coefficients that represent the force acting on the xaxis. Step 2 takes four linear sway and yaw coefficients using a result from zigzag manoeuvre with a rudder angle of 10 degrees, which represents a manoeuvre with a small rudder angle. Step 3 takes nonlinear coefficients using a result from turning manoeuvre with a rudder angle of 35 degrees, which represents a manoeuvre with a large rudder angle.Fig. 8
3.3.Optimization conditions
Table 3 shows the overall conditions for the optimization process. As mentioned in Subsection 3.2, a stepwise process is applied for optimization. Trajectory differences between the benchmark data and the simulation results based on optimized coefficients are selected as objective functions. The optimization results from the previous step are also applied as initial conditions for the next optimization step.
4.Verification of optimization results
Table 4 presents the optimization results, and Table 5 compares the manoeuvre characteristics from the benchmark data, the simulation results using coefficients found via Clarke estimation and the results of all the optimization steps. The coefficients from Step 1 are relatively similar to the Clarke estimations, compared to results from other steps. Fig. 10 shows that all of the simulation results indicate similar trajectories.
Fig. 11 and 12 present the trajectory and heading changes for a zigzag manoeuvre, based on the optimization results from each step and compare these outcomes with the benchmark data. This shows that the simulation results from Steps 2 and 3 are close to the benchmark data, and the nonlinear coefficients optimized in Step 3 had little effect on the results of the Step 2.
Fig. 13 shows the trajectory for a turning manoeuvre. The result of Step 2 indicated that linear coefficients have a big influence on turning manoeuvres, though this influence could be negative or positive. The results of Step 3 showed that selected nonlinear coefficients can help manage a ship’s manoeuvre characteristics, especially for manoeuvre with large rudder angle.
5.Conclusion
This paper focused on an optimization process for hydrodynamic coefficients when modelling ships. The approach to basic mathematical optimization used here was derived from the authors’ latest research, and benchmark data was gathererd from a sea trial. A short summary of the study is as follows:
First, the training ship Hanbada was used to gather measured benchmark data. A sea trial was performed using the procedures suggested by IMO. The measured data was calibrated to correct for environmental influences on the raw values measured.
Second, basic modelling and simulation for optimization were carried out using fast time simulation tool SIMOPT, provided by the ISSIMS Institute in Germany. This tool applied a mathematical model for an ANS 5000 simulator by Rheinmetall Defence and this estimates hydrodynamic coefficients acting on the hull by empirical regression following the Clarke and Norrbin method.
Finally, the optimization process itself was composed of three steps: a straight motion, a yaw checking manoeuvre with a small rudder angle and a turning manoeuvre with a large rudder angle. Prior to starting the optimization process, a sensitivity analysis was carried out and the coefficients to be optimized were chosen. Each step involved different coefficients to avoid interference, and the corresponding results were satisfactory in comparison with the benchmark data.
In future studies, more optimization results based on sea trial data should be collected to identify differences between existing empirical estimation formulas and to suggest new regression formulas.