A Study on Ship Path Following System Utilizing a Small Model Boat

The maneuvering of a ship is represented by the yaw angular velocity, which is the output, corresponding to the rudder angle, the input. By measuring both values, the mathematical model of the system can be obtained. Commonly used model structures for linear systems include Auto Regressive (AR) models, AutoRegression with eXtra input (ARX) models, Output Error (OE) models, Box-Jenkins (BJ) models, etc.

In this study, the ARX model was employed to obtain the system model through input and output data, allowing for relatively easy determination of values of system parameters compared to other models. The ARX model expresses the output at the current time as the sum of a function of the previous time's output, current and previous time's inputs, and the white noise.

In the time domain, when a single input data u(t) at time t results in a single output data y(t), an ARX model, including possible errors e(t) other than the input and output, can be represented using the delay operator q as shown in Eq. (1) (Ljung, 1987).

• Here, $$ A\left(q\right)=1+a_1{q}^{-1}+a_2{q}^{-2}+\cdots +a_{na}{q}^{-na}$$
• 　　  $$ B\left(q\right)=b_1+b_2{q}^{-1}+b_3{q}^{-2}+\cdots +b_{nb}{q}^{-nb+1}$$

Where, y(t) is the output at time t, u(t) is the input at time t, e(t) is the white-noise disturbance value, na is the number of poles, nb is the number of zeros, nk is the dead time in the system.

To achieve the optimal ARX model, it is essential to carefully choose the model's order. Excessive order may cause overfitting, yielding a poorly generalized model. Conversely, overly simplified models may lead to underfitting, hindering accurate representation.

To determine the optimal order, various criteria like Akaike's Information Criterion (AIC), Bayesian Information Criterion (BIC), Minimum Description Length (MDL), etc., can be used. These methods calculate scores for different models, facilitating the selection of the optimal one and preventing overfitting by imposing penalties on complexity.

AIC tends to favor complex models and may be less accurate with larger sample sizes, while BIC generally selects simpler models, but its outcomes can be influenced by sample size. In contrast, MDL aims to prevent overfitting by choosing simple and explanatory models, enhancing generalization to the data, albeit with the trade-off of complex computations.

The study aims to derive a concise and comprehensive maneuvering model for ships, utilizing Rissanen's MDL criterion, Eq. (2), to select the model that succinctly explains the data (Rissanen, 1978; The MathWorks Inc., 2023).

Where, V is the loss function, d is the total number of parameters in the structure, N is the number of data points used for the estimation. The orders of parameters na and nb in the ARX model are varied to calculated the MDL values. The model with the smallest calculated MDL value is selected as the optimal model. The parameters a_na and b_nb are estimated using the least squares method.

As for the predictive performance evaluation of the ARX model, commonly used methods include Root Mean Square Error (RMSE) and Normalized Root Mean Square Error (NRMSE). RMSE reflects the absolute error magnitude, influenced by measurement units. Conversely, NRMSE normalizes observed values, providing a measure of relative prediction accuracy independent of units. Utilizing both metrics allows a comprehensive evaluation of the model's performance from different perspectives. RMSE and NRMSE are expressed as Eq. (3) and Eq. (4), respectively.

Where, y_i represents the actual measured values, $$ \widehat{y_i}$$ corresponds to the values estimated through the model, Max(y_i) is the maximum value of y_i, MIN(y_i) is the minimum value of y_i, and n signifies the number of measured samples. RMSE and NRMSE approaching 0 indicate that the values calculated through the model are similar to the actual values.

The ship's course controller should adapt to factors such as the shallow water effect and various environmental conditions that may influence the ship's maneuvering performance, such as turning ability and course keeping ability. It should ensure precise, quick course changes, avoiding overshoot, particularly during significant yaw angle changes (Fossen, 1999).

In this study, Proportional-Derivative (PD) controller was used to eliminate integral action, preventing system divergence and error accumulation. PD control, known for reducing overshoot, facilitates simpler coefficient settings.

Additionally, due to the small size of the experimental boat, significant drift angles result from external forces like wind and currents, causing a notable difference between the heading angle and the Course Over Ground (COG). To ensure accurate path following, COG was used as the control variable for the controller instead of the heading angle. [Fig. 2] illustrates the block diagram of the PD controller utilized in this study.

[Fig. 2]

Block diagram of PD controller.

Where, K_P is the proportional gain, K_D is the derivative gain, ψ_d is the target course, ψ is the current course, e is the course error, r is the yaw angular velocity, δ_c is the target rudder angle and δ is the current rudder angle.

The synthesis of hull motion due to steering and external forces results in current course(ψ), obtained from GPS. The current course(ψ) is fed back to the controller, which calculates the target rudder(δ_c) angle to reduce the course error(e). The target rudder angle(δ_c) is input to the servo motor, which moves the rudder to the angle commanded by the controller. Eq. (5) represents the PD controller formula for calculating the target rudder angle.

The parameters of a PD controller must be appropriately tuned based on the system's characteristics and operating environment. This tuning process establishes suitable parameter values, enabling the design of a responsive controller tailored to the controlled system's characteristics.

One of the representative methods for tuning the parameters of a PD controller is the two methods proposed by Ziegler and Nichols (1942). Setting parameters through these tuning methods usually yields satisfactory controller performance, though optimal performance is not guaranteed. It is crucial to validate the controller's performance post-design using these methods, and if needed, fine-tuning of parameters may be necessary (Chung, 2018).

The first method in this tuning technique uses a step input to measure the output response of the target plant, extracting characteristic coefficients for setting the PD controller parameters. Since its applicability is limited to stable systems or those not including an integral component, it is not suitable for this study. Therefore, the second method, the ultimate sensitivity method, was chosen for controller tuning.

In the ultimate sensitivity method, a closed-loop circuit is created by introducing proportional action to the system. Applying a step input and progressively increasing the proportional gain from zero results in sustained oscillations in the output. The proportional gain value at which sustained oscillations occur is termed the ultimate gain (K_u = K_P), and the output's oscillation period is the ultimate period (T_u). These values are then used to determine the controller parameters, as shown in Eq. (6) and Eq. (7) (Ziegler and Nichols, 1942; McCormack and Godfrey, 1998).

The obtained parameters are used as the initial values for controller tuning, and fine-tuning is performed to set the parameters that represent the optimal response for the system.

To validate the ship maneuvering model, turning simulations were performed with rudder angles set to 25 degrees starboard and 30 degrees port. The results were compared and analyzed alongside the actual turning experiment results from section 1 of this chapter, as shown in [Fig. 5].

[Fig. 5]

Comparison between simulation and actual turning trajectories.

The blue solid line represents the simulated turning trajectory based on the identified maneuvering model, while the red line depicts the actual turning trajectory from the experimental boat turning. When turning starboard with a 25-degree rudder angle, the simulation's turning radius was about 1.2 m (1.3 L) larger than the experimental turning radius. In contrast, when turning port with a 30-degree rudder angle, the simulation's turning radius was approximately 1.7 m (1.9 L) smaller than the experimental turning radius.

The turning circle disparity between the experiment and simulation is rooted in the linear regression model employed for ship maneuvering in the simulation. This model suggests equal turning circles on both port and starboard sides when turning at the same rudder angle. However, real single-screw vessels exhibit slight variations in the turning circle due to factors like suction current, discharging current, and sidewise pressure, depending on whether the turn is to port or starboard (Yoon, 2019).

Typically, in single-screw vessels, the turning circle is larger on the side of the propeller rotation(Jung, 2008; Kim, 2005). Notably, our experimental boat, with the rudder positioned to the right of the propeller, exhibits a smaller than usual starboard turning circle, deviating from this general pattern.

Although there is a slight difference in turning radii between the simulation and the experiment, about 1 m (1.1 L), it is considered negligible when analyzing the overall yaw motion of the experimental boat.

Using the identified maneuvering model and designed controller, a path following simulation was performed for the experimental boat. Considering the size of the area where the sea trial took place, the simulation was set to navigate in a clockwise direction, starting from the departure point and following seven waypoints, as shown in <Table 2>.

<Table 2>

List of waypoints for simulation

[Fig. 6] depicts the simulation trajectory of the experimental boat. Star-shaped symbols represent waypoints, dashed lines connect waypoints in a straight line, and the blue solid line represents the trajectory of the experimental boat.

[Fig. 6]

Path following simulation trajectory of the experimental boat.

In the path following simulation, the experimental boat deviated approximately 1 m (1.1 L) at waypoint 2 and 1.2 m (1.3 L) at waypoint 4. However, overall, the boat navigated well along the designated path.