Ship Collision Risk Assessment Algorithm Based on the Especial Cautious Navigation Angle Model

Abstract

To address the challenges posed by dense shipping traffic and the difficulty of identifying navigation risks in open waters, this paper introduces an Especial Cautious Navigation Angle (ECNA) model for ships, grounded in ship collision avoidance geometry. The ECNA model dynamically identifies the range of navigation angles where collision risks may arise between ships. Building upon this model, a comprehensive scoring algorithm is proposed to assess ship collision risks in open waters. This algorithm not only effectively tracks the evolving risk of collisions but also prioritizes ships with the most imminent danger of collision. Experimental results demonstrate that the ECNA model can accurately define the range of collision risk navigation angles. Furthermore, the scoring algorithm provides a quantitative analysis of the development trends in collision risks and offers continuous monitoring of these risks during navigation in open waters. The proposed model and algorithm exhibit strong practical applicability and operability in identifying ship collision risks in both open and dense navigable areas. These findings not only offer valuable guidance for real-world collision risk identification but also contribute to the theoretical advancement of ship collision risk analysis, presenting a novel solution to this pressing issue.

1. Introduction

Ship navigation safety is a critical factor in ensuring the efficient and sustainable development of maritime transportation. Ship collisions not only result in significant economic losses for shipowners but also pose a serious threat to the lives of those involved. As the water transportation industry continues to grow, the number of ships (particularly in congested port areas) has increased. This leads to a higher frequency of ship encounters and, consequently, an elevated risk of collision. Identifying and preventing such accidents has become increasingly challenging.

The development of ship collision risk identification has evolved through several key stages. Initially based on collision avoidance geometry, the field has progressed to incorporate advanced theories such as fuzzy set theory, evaluation methods, and intelligent systems. Research has also shifted from early, isolated collision avoidance techniques toward a more structured and systematic approach. Since the International Maritime Organization (IMO) introduced the International Regulations for Preventing Collisions at Sea in 1972, ship collision avoidance research has increasingly focused on the guidelines set forth in these regulations. The “Rules” clearly outline the fundamental principles and obligations for ship navigation and serve as the authoritative standard for modern maritime practitioners in preventing collisions. Their authority is widely recognized by both industry experts and practitioners.

As the number of ships continues to grow, the collision geometry theory becomes inadequate for identifying risks, particularly in distinguishing vessels that pose significant collision threats in highly trafficked areas. As a result, current approaches to ship collision risk identification and visualization fail to meet the practical demands of contemporary ship navigation supervision, underscoring the need for updated solutions.

To address the challenge of identifying collision risks in densely navigable waters, this paper proposes a novel collision risk identification model based on collision avoidance geometry. The model uses ship navigation data to assess collision risks. It integrates the fundamental principles of the traditional PAD model with a newly introduced approach, the Especial Cautious Navigation Angle (ECNA) model. The ECNA model utilizes the navigation data of two ships involved in an encounter to determine the range of navigation angles at which collision risks arise. To further quantify the collision risk between the two ships, a ship collision risk scoring algorithm is developed based on the ECNA model. This algorithm provides a standardized method for measuring collision risks between ships in navigable waters. Using statistical calculations, it quantifies and ranks the collision risk levels between the marked ship and other ships. The proposed algorithm effectively identifies the ship that poses the greatest collision risk to the marked ship in dense traffic areas, solving the practical problem of collision risk identification in such environments.

Compared to the Point of Possible Collision (PPC) [1] and the Predicted Area of Danger (PAD) [2], the ECNA model introduced in this paper offers a novel approach to collision risk identification and visualization. Traditional PPC and PAD models identify potential collision points or areas where two ships might collide. However, these models face significant challenges in effectively visualizing collision risks, particularly when ship data are dynamic, and multiple ships may encounter each other simultaneously. The ECNA model addresses these limitations by directly displaying the angular range of potential collision scenarios during navigation, offering a more intuitive visualization compared to PPC and PAD. This improved risk representation makes the ECNA model especially valuable in ship navigation applications and supervision, where clear and intuitive risk displays are essential.

The ECNA model effectively displays the range of navigation angles at which the marked ship is at risk of collision, providing valuable insights for real-world navigation and ship supervision. This visual representation plays a crucial role by offering intuitive collision risk warnings to operators and managers. Additionally, a collision risk scoring algorithm based on the ECNA model is proposed, which quantitatively calculates collision risks using the visual display. The algorithm not only tracks the changing trends of collision risk for individual ships but also identifies the ship with the highest collision risk among those posing a threat to the marked ship, providing actionable alerts for ship operators and supporting safer navigation.

The remainder of this paper is organized as follows: Section 2 provides a brief literature review on ship collision risk identification. Section 3 presents the basic principles of the ECNA model and the proposed solution method. In Section 4, the ship collision risk scoring algorithm based on the ECNA model is described. Section 5 outlines three groups of simulation experiments and an actual experiment, through which the model and algorithm are thoroughly validated. The conclusions are discussed in Section 6.

2. Related Work

The theory and technology underlying ship collision risk identification have evolved over many years, giving rise to several key research areas, including ship collision avoidance geometry, ship domain, collision risk assessment, and anomaly detection in navigation data. The main research branches and representative works are illustrated in Figure 1.

Ship collision avoidance geometry is a foundational theoretical method in the early research on ship collision avoidance and was one of the first methods applied in maritime navigation practice. The principle of the collision avoidance geometry model is to use analytical geometry to assess the risk of ship collisions, offering the advantages of simplicity and broad applicability. The Distance of Close Point of Approaching (DCPA) and Time to Close Point of Approaching (TCPA) are widely used methods for identifying collision risks in maritime practice. This theory relies on accurate ship position and navigation data. With the widespread use of early shipborne radar, collision avoidance geometry algorithms were first applied in navigation practice. However, the rapid increase in the number of ships navigating open waters has significantly heightened the frequency of collision risk alerts in ship collision avoidance systems, thereby diminishing the effectiveness of collision risk identification.

The ship domain model refers to the surrounding water area around a ship that is essential for ensuring safe navigation and preventing collision accidents. The concept was first introduced by the Japanese scholar Fujii in 1971, who proposed the symmetrical elliptical model [28]. Subsequently, various scholars have developed ship domain models with different boundary types tailored to specific situations [29,30,31,32]. As research has progressed, ship domain models have evolved into several branches, including statistical-based models [3,4,33,34], intelligent algorithm-based models [5,6,7], and analytical models [8,9,10]. Ship domain models proposed by scholars for various application scenarios perform well in addressing specific collision risk identification challenges. However, these models lack advantages in predicting and visualizing potential future collision accidents, limiting their effectiveness in proactive collision risk management.

The development of collision avoidance geometry has been constrained by various factors, and the direct calculation of ship collision probability using navigation data is increasingly insufficient for both theoretical and practical applications. To better address the complexities of modern navigation environments, and with the ongoing advancement of ship domain theory, ship collision risk assessment methods based on both ship domain models and collision avoidance geometry models have gradually emerged. These methods include collision risk calculations based on collision avoidance geometry [11], ship domain models [12,13,14,15], fuzzy theory [16,17], and evaluation methods [18,19]. Collision avoidance geometry has been a foundational tool in ship navigation for many years, with its stability and reliability widely recognized by professionals in the field. Many of its principles continue to be applied in navigation practices and regulatory frameworks. However, as the number of ships increases, the method’s visual representation faces significant challenges, necessitating the development of a new and improved visualization approach.

With the successful application of big data and artificial intelligence in ship research, some scholars have attempted to directly apply ship trajectory big data to ship navigation anomaly detection, achieving promising results. These methods include anomaly detection based on machine learning algorithms [20,21,22,23], data mining algorithms [24,25], and data features [26,27,35]. Scholars have employed technologies such as artificial intelligence and big data to identify ship collision risks. With the ongoing advancement of these technologies, their potential applications continue to expand. However, when applied in regulatory contexts, the visual effectiveness of these technologies remains suboptimal.

Existing theoretical research has provided some solutions for visualizing ship collision risks and identifying them in both open and congested waters. However, in actual navigation supervision, current theoretical approaches do not offer effective visualization tools. To address this gap, a novel ship collision risk identification model, named the Especial Cautious Navigation Angle (ECNA) Model, has been proposed. To further quantify the collision risk between two ships, a ship collision risk scoring algorithm based on the ECNA model is introduced, enabling a quantitative analysis and comparison of collision risks between ships.

3. The Especial Cautious Navigation Angle Model

3.1. The Principle of the Especial Cautious Navigation Angle Model

The PAD model is a geometric model used in collision avoidance. It is an advancement of the PPC model. PAD addresses the limitation of PPC, which fails to account for ship size when assessing collision risk. By expanding the predicted collision point to a predicted collision area, PAD significantly enhances the accuracy and reliability of collision risk identification for ships.

The mechanism of PAD involves calculating the potential collision area between two ships, taking into account their direction and speed. In ship navigation collision risk identification, it is essential not only to prevent ships from entering the PAD area but also to monitor ships heading toward it. This is particularly challenging in open waters with multiple ships, where navigation conditions are complex and dynamic. Both the PPC and PAD models struggle to effectively assess the risk of multi-ship collisions.

In ship navigation, it is important to calculate an angle range that indicates a potential collision risk for the ship. When the ship’s heading falls within this range, it is more likely to move toward the PAD. At this point, the navigator must exercise particular caution in selecting the heading or adjusting the ship’s speed. This range of sailing angles is referred to as the Especial Cautious Navigation Angle (ECNA). The principle of ECNA, in line with the PAD principle, is illustrated in Figure 2.

For simplicity in the calculation, the central axis $M_1{M}_2$ of the PAD is used as the reference for the cautious navigation angle. The left and right boundaries of the ECNA are defined by $P_A{M}_1$ and $P_A{M}_2$, respectively. If a second ECNA is defined, its boundaries are $P_A{M}_3$ and $P_A{M}_4$.

3.2. The Triggering Conditions for the ECNA

The ECNA is designed to dynamically represent the angle range of collision risk between two passing ships. Consequently, the calculation of the cautious navigation angle depends on the navigation information of both ships. Suppose Ship A and Ship B are present simultaneously in a given water area. The coordinates of Ship A are $(x_A,y_A)$, while those of Ship B are $(x_B,y_B)$. The speed of Ship A is $V_A$ and the speed of Ship B is $V_B$. The course of Ship A is ${\psi }_A({\psi }_A\in [-180°,180°)$) and the course of Ship B is ${\psi }_B({\psi }_B\in [-180°,180°)$).

Two ships are sailing in the same water area, and the ECNA is triggered when certain conditions are met. The positional relationship between Ship A and Ship B is illustrated in Figure 3.

$D_{minCPA}$ is the safety limit value of the CPA in ship navigation, a key parameter used in marine radar systems to assess collision risk. $D_{minCPA}$ represents the minimum encounter distance required for a safe encounter between two ships. ${\phi }_B$ is the angle formed by the centerline of Ship B and the azimuth line of Ship A.

To facilitate the calculation of the ECNA during the navigation of two ships, the method of coordinate rotation is employed. In the Cartesian coordinate system, the positions of the two ships are represented by $P_A\left(x_A,y_A\right)$ and $P_B\left(x_B,y_B\right)$. The coordinates of Ship B are then rotated around $P_A$ so that they align with the same vertical axis as Ship A, with $P_A$ as the center of rotation. Ship B is placed to the right of Ship A. The rotation angle is $\mathsf{\omega }$ ($\mathsf{\omega }\in [-\mathsf{\pi },\mathsf{\pi })$).

The angle $\mathsf{\gamma }$ between $P_B{P}_1$ and $P_A{P}_B$ can be calculated as follows:

$$\mathsf{\gamma }=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\left({\displaystyle \frac{D_{minCPA}}{{Dis}^{\left(4\right)}\left(\left(x_A,y_A\right),\left(x_B,y_B\right)\right)}}\right)$$

(1)

where ${Dis}^{\left(4\right)}\left(\right)$ is the function used to calculate the distance between two points.

As shown in Figure 3, when the positions of Ship A and Ship B, the heading of Ship B, and the speed of Ship B are known, the minimum encounter range around Ship A is drawn with a radius of $D_{minCPA}$. Tangent lines $P_B{P}_1$ and $P_B{P}_2$ are drawn from point $P_B$ to the minimum encounter range of Ship A. When the relative heading $V_R$ of the two ships lies between $P_B{P}_1$ and $P_B{P}_2$, Ship B will be within the minimum encounter range of Ship A. This means that the two ships will encounter each other at a distance of less than $D_{minCPA}$, which endangers their navigational safety. Therefore, the criteria for triggering the display of the caution angle are whether the straight line corresponding to the relative heading intersects the minimum encounter range of Ship A.

To analyze the relationship between the relative heading and the minimum encounter range of Ship A, a speed circle for Ship A is drawn. This circle can be used to calculate the number of PAD models generated by two ships and serves as a criterion for determining the number of ECNA models triggered. This circle is centered at the forward point of the speed vector of Ship B, with the speed of Ship A as its radius. The intersection of the speed circle with the tangent lines $P_B{P}_1$ and $P_B{P}_2$ of the minimum encounter range determines whether the ECNA has been triggered. The relative speed ratio $K_V$ between the two ships is an important factor in classifying whether the relative speed crosses the minimum encounter range of Ship A. This can be expressed as follows:

$$K_v={\displaystyle \frac{V_A}{V_B}}$$

(2)

Based on the value of $K_V$, three distinct situations can be identified.

$$\left\{\begin{array}{c}{K}_v<1 \mathrm{S}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}1\\ K_v=1 \mathrm{S}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}2\\ K_v>1 \mathrm{S}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}3\end{array}\right.$$

(3)

Situation 1: When the speed of Ship A is lower than that of Ship B, three scenarios can be identified. The schematic for relative speed judgment is shown in Figure 4.

Situation 2: When Ship A and Ship B have the same speed, three scenarios can be identified. The schematic for relative speed judgment is shown in Figure 5.

Situation 3: When the speed of Ship A is faster than that of Ship B, a single scenario can be identified. The schematic for relative speed judgment is shown in Figure 6.

The intersection of the tangent lines on both sides of the minimum encounter range with the speed circle of Ship A can be used to determine the number of ECNAs for the three situations. The judgment criteria are outlined in

Table 1. Determination of triggering conditions for ECNA.

Situation	Scenes	$\mathbf{Number} \mathbf{of} \mathbf{Intersections} \mathbf{with} {\mathit{P}}_{\mathit{B}}{\mathit{P}}_1$	$\mathbf{Number} \mathbf{of} \mathbf{Intersections} \mathbf{with} {\mathit{P}}_{\mathit{B}}{\mathit{P}}_2$	Danger Range	Number of Triggered ECNAs
1	I	0	0	None	0
	II	2	0	$\mathrm{Between} V_A^{\left(1\right)}$$\mathrm{and} V_A^{\left(2\right)}$	1
	III	2	2	$\mathrm{Between} V_A^{\left(1\right)}$$\mathrm{and} V_A^{\left(2\right)}$$\mathrm{Between} V_A^{\left(3\right)}$$\mathrm{and} V_A^{\left(4\right)}$	2
2	I	0	0	None	0
	II	1	1	$\mathrm{Between} V_A^{\left(1\right)}$$\mathrm{and} V_A^{\left(2\right)}$	1
	III	2	0	$\mathrm{Between} V_A^{\left(1\right)}$$\mathrm{and} V_A^{\left(2\right)}$	1
3	I	1	1	$\mathrm{Between} V_A^{\left(1\right)}$$\mathrm{and} V_A^{\left(2\right)}$	1

.

In Table 1, several situations require special explanation. In Scenario III of Situation 1, since Ship A and Ship B have the same speed, the other intersection point between the speed circle of Ship A and $P_B{P}_1$ coincides with the position of Ship B. The corresponding speed is $V_A^{\left(2\right)}$, and both its magnitude and heading are identical to those of Ship B. In Situation 3, since the speed of Ship A is greater than that of Ship B, only one scenario exists in this case.

3.3. Solution to the Boundary of the Range of ECNAs

The methods for calculating the boundaries of ECNAs for various scenarios are illustrated in Figure 7. For convenience in determining the boundaries of ECNAs and subsequent collision risk scoring, an attached polar coordinate system is established at the center of the ship. The ship’s heading serves as the polar axis, with the clockwise direction defined as the positive direction. The polar angles of the left and right boundaries of the ECNA in this attached coordinate system are denoted as ${\theta }_L$ and ${\theta }_R$, respectively.

When the method of coordinate rotation is employed, the calculation of polar coordinates is simplified. Based on the ship’s heading ${\psi }_A$ and the rotation angle $\mathsf{\omega }$, the boundary of the ECNA is converted into the polar angle in the ship’s attached polar coordinate system. The boundaries of the ECNA are then calculated separately based on the grouping situation.

After the ECNA is triggered, the polar coordinates of its boundaries can be determined according to

Table 2. Calculation of polar coordinates of the boundary of the ECNA.

Situation	Scenario	Number of Triggered ECNAs
1	II	1
	III	2
2	II	1
	III	1
3	I	1

. ${\delta }_A$ is the value converted from angle to radian by ${\psi }_A$.

The boundary calculation process of the ECNA model is illustrated in Figure 8.

In summary, after obtaining the navigation information of both ships, the ECNA can assess whether there is a collision risk during their encounter based on the triggering rules. The ECNA can dynamically update the display as the ship’s navigation information changes. Compared to the PPC and PAD, the ECNA offers a more intuitive method for displaying collision risks. Additionally, a comprehensive solution method for calculating the ECNA is proposed.

4. Collision Risk Assessment Algorithm Based on the ECNA Between Ships

The ECNA can display the range of heading angles that pose a collision risk with other ships as the monitored ship navigates. To accurately assess the evolution of collision risk during ship navigation, a collision risk scoring algorithm based on the ECNA is proposed. The ECNA is drawn in the attached polar coordinate system, with the heading set to 0 radians and angles increasing clockwise, as shown in Figure 9.

The range angle of the ECNA can be expressed as

$$\left\{\begin{array}{ll}\left[{\theta }_L,{\theta }_R\right]\hfill & \mathrm{E}\mathrm{C}\mathrm{N}\mathrm{A} \mathrm{e}\mathrm{x}\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g} 0 \mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{s}\hfill \\ \left[0,{\theta }_R\right]\cup \left[{\theta }_L,2\pi )\right.\hfill & \mathrm{E}\mathrm{C}\mathrm{N}\mathrm{A} \mathrm{i}\mathrm{n}\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g} 0 \mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{s}\hfill \end{array}\right.$$

(4)

Based on the fundamental principle of the ECNA, when two ships encounter each other and generate PADs of the same scale, the closer a ship is to the PAD ($d_P^{\left(1\right)}>d_P^{\left(2\right)}>d_P^{\left(3\right)}$), the larger the ECNA angle (${\beta }_1<{\beta }_2<{\beta }_3$), as shown in Figure 10. Additionally, the shorter the remaining time for the ships to take evasive maneuvers, the higher the probability of a collision, which can be represented by the collision risk score (${\chi }_1<{\chi }_2<{\chi }_3$). Therefore, when evaluating the risk of ship collisions, the ECNA angle should be positively correlated with the collision risk score.

When ECNAs with the same angle are generated at different positions around the ship, the degree of collision risk varies for the following reasons:

(1) The time required for the focused ship to adjust its heading to different angles varies. When five ECNAs with the same angle range are generated around the focused ship, they can be expressed as ${ECNA}^{\left(1\right)}$, ${ECNA}^{\left(2\right)}$, ${ECNA}^{\left(3\right)}$, ${ECNA}^{\left(4\right)}$, and ${ECNA}^{\left(5\right)}$. The distribution of these ECNAs is shown in Figure 11.

The focused ship is heading toward 0 degrees in the attached polar coordinate system, and its heading does not require adjustment to remain within the angle range of ${ECNA}^{\left(1\right)}$. To enter the ranges of ${ECNA}^{\left(2\right)}$, ${ECNA}^{\left(3\right)}$, ${ECNA}^{\left(4\right)}$, and ${ECNA}^{\left(5\right)}$, the ship must adjust its heading to at least ${\theta }_L^{\left(2\right)}$, ${\theta }_L^{\left(3\right)},{\theta }_L^{\left(4\right)}$, and ${\theta }_L^{\left(5\right)}({\theta }_L^{\left(2\right)}$<${\theta }_L^{\left(3\right)}$<${\theta }_L^{\left(4\right)}$<${\theta }_L^{\left(5\right)}$). The time $t_c$ required for the ship to achieve the heading adjustment satisfies $t_c^{\left(2\right)}<t_c^{\left(3\right)}<t_c^{\left(4\right)}<t_c^{\left(5\right)}$. Therefore, there are significant differences in the collision risks represented by the five ECNAs. The direction indicated by the angle bisector of each ECNA is considered the direction of that ECNA. The closer the direction of the ECNA is to π, the lower the collision risk for the ship.

(2) The ECNA with the same direction and angle range can represent collision risks with different remaining times. When three ships have the same speed but different distances to the PAD, the ECNAs will have the same angle range. With the same speed, the closer a ship is to the PAD, the earlier it will enter the PAD, as shown in Figure 12.

In summary, when the ECNA is used to express the risk of ship collision, the direction, angle range, and expected remaining time of the collision are key factors that influence the quantitative analysis of collision risk between two ships. These three factors satisfy the following relationship:

(1)

When the expected remaining time for collision and the range of the ECNA are the same, the collision risk gradually decreases as the ECNA moves from 0 to $\pi$. Conversely, the collision risk gradually increases as the ECNA moves from $\pi$ to $2\pi$.

(2)

When the expected remaining time of the collision and the direction of the ECNA are the same, a larger angle range of the ECNA corresponds to a higher collision risk.

(3)

When the direction and range of the ECNA are the same, a smaller expected remaining time for the collision corresponds to a higher collision risk.

By combining the above three relationships, the mapping method can effectively address the interactions between the key factors when the ECNA is used to analyze the trend of collision risk between ships. The map must provide efficient query and calculation methods for the three factors: direction, angle range, and expected remaining time of collision. Based on the constraints of these three key factors, the basic form of the collision risk scoring map based on the ECNA can be determined, as shown in Figure 13.

The horizontal axis of the collision risk map for the ECNA represents the angle $\theta$, corresponding to the boundary polar angle range of the ECNA in the ship’s attached body polar coordinate system, spanning the range [0, 2π). The vertical axis represents the cardinal number $\mathsf{\zeta }$ for ship collision risk assessment, which quantifies the impact of the expected remaining time for a collision on the collision risk score. The smaller the expected remaining time for the collision, the larger the value of the ship collision risk cardinal number $\mathsf{\zeta }$. When the distance between the passing ships is less than $D_{minCPA}$, a collision risk between the two ships is considered. The cardinal number $\mathsf{\zeta }$ for ship collision risk satisfies a proportional relationship, which can be expressed as

$$\mathsf{\zeta }\propto {\displaystyle \frac{\overline{V_R}}{D-D_{minCPA}}}$$

(5)

where $\overline{V_R}$ is the average relative speed and $D$ is the distance between the two ships. Therefore, the collision risk scoring map can be expressed as

$$\left\{\begin{array}{l}f\left(\theta \right)=-{\displaystyle \frac{\zeta }{\pi }}\theta +\zeta \\ f\left(\theta \right)={\displaystyle \frac{\zeta }{\pi }}\theta -\zeta \end{array}\right.$$

(6)

Based on the basic form of the collision risk scoring map, there are several methods to quantify the impact of the ECNA on collision risk scoring. According to the relationship between the range of the ECNA and collision risk, it can be inferred that the ECNA range extends symmetrically on both sides based on its direction. The degree of collision risk expressed by the ECNA with the same direction but varying angle ranges differs. The goal of establishing ship collision risk scoring rules for the ECNA is to quantitatively analyze the development trend of ship collision risk under a defined scoring framework. Therefore, based on Figure 13, there are two methods for calculating collision risk scores, including but not limited to the following:

(1)

Coefficient Function Method

According to Equation (5), the cardinal number of ship collision risk can be solved. The direction of the ECNA, ${\theta }_0$, is used to locate the corresponding ship collision risk cardinal number ${\zeta }_0$ on the map. The range of the ECNA is ${\beta }_0$. The collision risk score can then be expressed as

$${\chi }_0=h\left({\theta }_0,{\beta }_0\right){\zeta }_0$$

(7)

where $h\left({\theta }_0,{\beta }_0\right)$ is the coefficient function. This function expresses the coefficients for different ECNA ranges (${\beta }_0$)) in various directions (${\theta }_0$). The coefficient function, along with the ship collision risk cardinal number ${\zeta }_0$, is used to calculate the collision risk score.

(2)

Integration Method

The cardinal number for ship collision risk is integrated based on Equation (5). The polar angles ${\theta }_L$ and ${\theta }_R$ correspond to the boundaries on both sides of the ECNA. By integrating the collision risk scoring map within this angle range, the result represents the current ship navigation collision risk score for the ECNA. This can be expressed as

$$\left\{\begin{array}{ll}{\chi }_S={\int }_{{\theta }_L}^{{\theta }_R}f\left(\theta \right)d\theta \hfill & \mathrm{E}\mathrm{C}\mathrm{N}\mathrm{A} \mathrm{e}\mathrm{x}\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g} 0 \mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{s} \hfill \\ {\chi }_S={\int }_0^{{\theta }_R}f\left(\theta \right)d\theta +{\int }_{{\theta }_L}^{2\pi }f\left(\theta \right)d\theta \hfill & \mathrm{E}\mathrm{C}\mathrm{N}\mathrm{A} \mathrm{i}\mathrm{n}\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g} 0 \mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{s}\hfill \end{array}\right.$$

(8)

Compared to the previous two methods, the Integration Method more effectively quantifies the relationship between the ECNA range and direction when the ECNA is in a specific direction. It also aligns the scoring results with the actual collision risk expressed by the ECNA. The cardinal number $\mathsf{\zeta }$ for ship collision risk is inversely proportional to the expected remaining time before collision, meaning that the larger the expected remaining time, the lower the collision risk score, reflecting a reduced collision risk level. In contrast to the coefficient function method, the integration method does not require additional design of the coefficient function. Instead, it calculates the area under a segmented function within a specific range in the map. Integrating the piecewise function shown in Figure 13 involves solving the area between the function and the coordinate axis. When the integration interval $∆\theta$ remains constant and the integration starting point ${\theta }_L$ gradually moves to the right from 0, the integration result decreases, which is consistent with the relationship between the factors. This behavior supports the rationality of the integration method.

In summary, the proposed ECNA collision risk scoring map effectively quantifies the impact of three factors (direction, angle range, and expected remaining time of the collision) on the development trend of collision risk.

5. Experiments and Analysis

5.1. Experimental Scheme

To verify the effectiveness of the ECNA and the collision risk scoring method for passing ships in different encounter scenarios, three groups of simulation experiments were conducted, as summarized in

Table 3. Experiment list.

No.	Purpose of the Experiment	Experimental Variables
1	Verify the effectiveness of the ECNA for two navigation ships in an overtaking situation	Speed
2	Verify the effectiveness of the ECNA for two navigation ships in a head-on situation	Speed
3	Verify the effectiveness of the ECNA for two navigation ships in a crossing situation	Speed

.

Experiment 1: Two ships were used to verify the effectiveness of the ECNA in an overtaking situation. The experiment involved two ships with different speeds, fully simulating a real sailing environment.

Experiment 2: Two ships were used to verify the effectiveness of the ECNA in a head-on situation. The experiment involved two ships with different speeds, fully simulating a real sailing environment.

Experiment 3: Two ships were used to verify the effectiveness of the ECNA in a crossing situation. The experiment involved two ships with different speeds, fully simulating a real sailing environment.

5.2. Experimental Platform

The simulation experiments were conducted using software named NavigationSafetyPlatform.exe (V1.0) developed by the first author of this paper. The ship motion in this platform was implemented using the MMG simulation model and the motion equation was solved using the Runge–Kutta method. This platform has the ability to add multiple ships for simultaneous simulation. However, the limitation of this platform is that it currently does not have the ability to run wind, wave, and current data, but these factors are also not relevant to the ECNA model proposed in this article. But, in future in-depth research, the platform can be further developed to ensure that experimental needs can be met. The simulation platform was built in C++ using the MFC 6.0 framework. The interface of the simulation platform is shown in Figure 14a.

To verify the effectiveness of the proposed model, an actual experiment was conducted. The experimental platform was developed in C++ using the MFC framework. The software platform receives AIS and GPS messages from AIS-receiving equipment via serial communication and is capable of decoding, displaying, and saving the message data. Additionally, the platform includes a function for AIS data scenario reproduction.

The platform is capable of reading AIS messages stored in a database, utilizing the AIS data decoding module to effectively reconstruct a ship’s operational scenario. This functionality enables the repeated simulation of ship navigation processes and the verification of algorithm effectiveness. However, the platform lacks the capability to integrate radar data, preventing the real-time superposition of AIS and radar data. Despite this limitation, experiments involving AIS-based models or algorithms can be effectively conducted on this platform. The interface of the experimental platform is shown in Figure 14b. The configuration of the experimental computer was as follows: Windows 7, Intel i7 5700HQ, 32GB RAM.

5.3. Experimental Data

The parameters of the ships used in the simulation experiment are shown in

Table 4. Parameters of the experimental ships.

No.	Length (m)	Width (m)	Block Coefficient	Draft (m)	Displacement (t)
1	117.00	18.00	0.79	7.00	11,000.00
2	160.00	24.40	0.80	9.80	23,000.00

.

The Pearl River Estuary features a complex water depth profile, characterized by the intersection and coexistence of shallow- and deep-water areas. Therefore, the chart data and ship AIS data from the Pearl River Estuary were used for the actual experiments.

5.4. Simulation Experimental Results and Analysis

To thoroughly verify the effectiveness of the ECNA and the collision risk scoring method in different encounter situations, three groups of experiments were conducted. The experimental details are shown in

Table 5. Experimental grouping details.

Group	Encounter Situation	No.	Speed Situation
I	Overtaking Situation	1	$V_1=V_2$
II	Head-on Situation	1	$V_1>V_2$
		2	$V_1=V_2$
		3	$V_1<V_2$
III	Crossing Situation	1	$V_1>V_2$
		2	$V_1=V_2$
		3	$V_1<V_2$

.

5.4.1. Experiment in Overtaking Situation

The simulation results for the overtaking situation are shown in Figure 15. The simulation ran for a total of 300 steps, with the ship’s position and the ECNA results being recorded on the interface every 20 steps.

This experiment simulated the visual warning scenario of the ECNA for two ships in an overtaking situation. Ship 1 was the overtaking ship and Ship 2 was the overtaken ship, maintaining a constant speed and heading. During the overtaking process, the ECNA was continuously triggered, displaying the range of headings within which Ship 1 could collide with Ship 2. Before Ship 1 initiated the overtaking maneuver, Ship 2 was directly ahead of Ship 1, and the ECNA warning range encompassed Ship 1’s current heading. As Ship 1 began the overtaking maneuver, its heading gradually approached the edge of the ECNA range. Once the ship’s turning maneuver became significant, it moved outside the ECNA range, indicating that the overtaking maneuver effectively eliminated the risk of collision with Ship 2. Throughout the entire process, the ECNA remained triggered. Even though Ship 1’s heading moved outside the ECNA range, the model continued to indicate the range of heading angles where a collision with Ship 2 could occur. This suggests that even if Ship 1 promptly adjusted its course to fall within the ECNA range, there still remained a potential risk of collision with Ship 2.

Based on practical ship maneuvering, when a ship engages in a turning and avoidance maneuver, it cannot instantly execute a sharp turn. During the latter half of the pursuit (or resumption process), the angle range of the ECNA is typically distributed on the port side of the ship and, in some cases, may even extend toward the stern on the port side. To accurately reflect the risk during these maneuvers, the collision risk score results were extracted for retaining the ECNA, as shown in Figure 16.

According to Figure 16, when Ship 1 was far behind Ship 2 before initiating the overtaking maneuver, with Ship 1’s speed being faster than that of Ship 2, a collision was likely if both ships maintained their current direction and speed. In this phase, the collision risk score was higher compared to the latter part of the overtaking process. As Ship 1 progressed into the first half of the overtaking maneuver, it gradually approached the starboard rear side of Ship 2, and the distance between the two ships decreased, leading to an increase in collision risk. Consequently, the collision risk score rose during this phase. When the bow of Ship 1 surpassed that of Ship 2 and the ECNA shifted away from the port side of Ship 1, the collision risk score dropped significantly. This gradual change in the collision risk score throughout the overtaking process further validates the scientific rationale behind the ECNA and its associated collision risk scoring method.

For triggering the alarm was set to 2000 in this study. However, this value can be adjusted based on specific circumstances. When the collision risk score reached 2000, the alarm was activated, providing a warning during experimental navigation or supervision. As shown in Figure 16b, the ECNA model was continuously triggered in this scenario.

5.4.2. Experiment in Head-On Situation

The simulation results for the head-on situation are shown in Figure 17. The simulation ran for a total of 300 steps, with the ship positions and ECNA results updated and displayed on the interface every 20 steps.

This experiment simulated the visual warning situation of the ECNA for two ships in a head-on situation. During the collision avoidance process between Ship 1 and Ship 2, as shown in Figure 17, the ECNA was not fully triggered. The non-triggered areas indicated that Ship 1 and Ship 2 would not experience a collision, assuming they maintained their speeds. When the ECNA was triggered, it displayed the heading range where a collision could occur between Ship 1 and Ship 2 during their encounter. By comparing Figure 17a–c, it can be observed that the speed differences between Ship 1 and Ship 2 were positive, zero, and negative, respectively. The corresponding ECNA triggering time decreased from longest to shortest as the speed difference increased. This demonstrates that the ECNA model could effectively address the impact of speed differences between the two ships on collision risk identification. The collision risk score results for retaining the ECNA are shown in Figure 18.

In the early stages of a potential collision between Ship 1 and Ship 2, the distance between the two ships was relatively large. Although the risk of collision was low at this point, there was still a possibility of collision if the ships maintained their current speed and direction. As the distance between the ships gradually decreased, the collision risk score increased. When the ships began to adopt avoidance maneuvers, the rate of increase in the collision risk score slowed down, and it quickly dropped after reaching its peak. In a head-on collision scenario, as the distance between the ships decreased, the collision risk gradually shifted from a frontal risk to a lateral collision risk as the ships came closer to each other.

In ship maneuvering, due to the hysteresis of ship turning, ships cannot adopt real-time steering control at large angles. During the close encounter between two ships (stern-to-stern), when the speed of Ship 1 exceeded that of Ship 2, the resulting Estimated Collision Navigation Area (ECNA) was distributed from the port side toward the stern (as shown in Figure 17a). In this situation, Ship 1 needed to immediately alter its course to have any chance of catching up to Ship 2, although the risk of collision remained relatively low. The collision risk score, shown in Figure 18a, decreased sharply, indicating that the risk of collision had been largely mitigated. When the speed of Ship 1 was less than or equal to that of Ship 2, even if Ship 1 changed course to pursue Ship 2, a collision was unlikely as long as both ships maintained their respective speeds. In a stern-to-stern configuration, the ECNA was not triggered and the collision risk score dropped to zero, as depicted in Figure 18c,e. Thus, the scientific validity of the ECNA and its collision risk scoring method was confirmed by the behavior of the ships in this head-on scenario.

The collision risk threshold to trigger an alarm was set at 2000 in this study. In other scenarios, this value can be adjusted based on specific requirements. When the collision risk score reached 2000, the alarm activated, as illustrated in Figure 18a,c,e. This served as a warning during experimental navigation or monitoring processes. In this scenario, the ECNA model was not fully triggered. Once the two ships ceased colliding, the ECNA model immediately stopped triggering, as shown in Figure 18b,d,f.

5.4.3. Experiment in Crossing Situation

The simulation results for the crossing situation are shown in Figure 19. The simulation consisted of 300 steps, with the ship’s position and ECNA results recorded every 20 steps on the interface.

This experiment simulated the visual warning scenario of the ECNA for two ships in a crossing situation. During the crossing maneuver between Ship 1 and Ship 2, the ECNA was not fully activated. When the ECNA was not triggered, no collision could occur between Ship 1 and Ship 2, provided that Ship 1 maintained its speed and Ship 2 maintained both its direction and speed.

When the speed of Ship 1 exceeded that of Ship 2, the ECNA displayed the range of angles at which Ship 1 could potentially collide with Ship 2. If Ship 1 followed Ship 2 within this ECNA range, there remained a risk of collision between the two ships. However, when the speed of Ship 1 was less than or equal to that of Ship 2, Ship 1 could pass behind Ship 2 during the later stages of the crossing maneuver. Even if Ship 1 changed course to pursue Ship 2, the two ships could not collide. The collision risk score results, based on the retained ECNA, are shown in Figure 20.

When the speed of Ship 1 exceeded that of Ship 2, the ECNA was triggered throughout the entire process. In the first half of the crossing maneuver, as the distance between the two ships gradually decreased, the collision risk score increased. Due to the hysteresis in ship turning, it is difficult for a ship to complete a turn immediately. As the ECNA shifted from the bow to the port side of the ship, the rate of increase in the collision risk score slowed down.

When Ship 1 passed behind the stern of Ship 2, the collision risk score dropped sharply, indicating a significant decrease in the likelihood of a collision between the two ships. When the speed of Ship 1 was less than or equal to that of Ship 2, a collision could no longer occur once Ship 1 had passed behind Ship 2, provided that Ship 1 maintained its speed and Ship 2 maintained both its speed and direction. As a result, the ECNA was no longer triggered and its corresponding collision risk score was reduced to zero. This confirmed the scientific validity of the ECNA and its collision risk scoring method, as demonstrated by the behavior of the two ships in the crossing situation.

The collision risk threshold to trigger an alarm was set at 2000 in this study. In other scenarios, this value can be adjusted based on specific requirements. When the collision risk score reached 2000, the alarm activated, as illustrated in Figure 18a,c,e. This served as a warning during experimental navigation or monitoring processes. In one scenario, when the speed of Ship 1 exceeded that of Ship 2, the ECNA model was triggered throughout the entire process because Ship 1 could potentially catch up to Ship 2, increasing the risk of a collision, as shown in Figure 16b. In other scenarios, the ECNA model did not trigger throughout the entire process. Once the two ships ceased colliding, the ECNA model immediately stopped triggering, as shown in Figure 20b,d,f.

5.5. Actual Experimental Results and Analysis

To thoroughly assess the effectiveness of the ECNA, an actual experiment was conducted in the Pearl River Estuary, where ship encounters are particularly complex. These actual ship experiments in this area effectively validated the ECNA’s performance.

The ship with MMSI number 41322XXXX was designated as Ship 3 for testing the ECNA. The experimental platform collected data, including the position, speed, and heading of Ship 3 and surrounding ships, triggering the ECNA for Ship 3 automatically. The experimental results are presented in Figure 21.

Based on the triggering and calculation principles of the ECNA, multiple ECNAs were activated for Ship 3, displaying the angular ranges where collision risks with surrounding ships could occur. Through several rounds of experimental verification, the ECNA demonstrated its capability to visualize ship collision risks, confirming its effectiveness as proposed in this paper.

The simulation experiments and actual experiments indicate that the ECNA model proposed in this paper significantly enhances the visualization of collision risks compared to traditional PPC and PAD models. However, the model has certain limitations. It relies on accurate ship navigation data, a dependency common to other geometric collision-avoidance models. As the number of ships increases, even in open waters, traffic density becomes a critical factor, intensifying the need for efficient updates to navigation data. Moreover, denser traffic or more frequent updates impose substantial demands on the hardware systems supporting the ECNA model, potentially causing display lags or delays. Addressing these challenges through optimization and hardware improvements is a crucial direction for future research.

The ECNA model presented in this paper demonstrates strong performance in identifying collision risks during experimental trials. It effectively visualizes the angular range of potential collision risks between the target ship and surrounding ships. However, in densely populated maritime areas, the model may activate frequently, necessitating a higher processing speed for systems utilizing it. Additionally, the model’s accuracy depends heavily on reliable ship navigation data. External environmental interference with navigation data can lead to errors in the model’s outputs. Future research will focus on examining the model’s robustness under extreme environmental conditions.

In summary, the ship collision risk identification method proposed in this paper effectively identifies collision risks in various encounter situations. Through both simulation and actual experiments, the ECNA demonstrated its ability to visualize the collision risks between two ships. Compared to classical models such as PPC and PAD, the proposed ECNA not only identifies collision risks but also clarifies the range of heading angles associated with these risks. This range provides visual risk warnings to sailors or ship navigation safety officers during actual navigation. Additionally, the model allows for the overlay and display of ECNAs generated by the marked ship and other ships. For the marked ship, the overlay display of risky heading angles offers a comprehensive view of the surrounding risk zones, demonstrating significant practical value.

6. Conclusions

A novel model for identifying collision risks during ship navigation called the Especial Cautious Navigation Angle (ECNA) was proposed in this paper. This model enables the visual display of ship navigation collision risk angles, building on the PAD and PPC frameworks. The advantages of the proposed model are as follows:

(1)

This paper proposed a novel collision risk identification model that incorporates the navigation information of two encountering ships, combined with traditional ship collision avoidance principles, to assess collision risks under various encounter situations.

(2)

After identifying the risk of ship collision, this paper proposed a novel method for visualizing the range of heading angles associated with collision risks. Compared to traditional methods, this approach offers broader applicability. Whether displayed in the ship’s navigation visualization system on the bridge or on the ship navigation safety supervision platform, the heading angle range of the ship or the marked ship at risk of colliding with surrounding ships can be shown, providing a more direct and effective display for crew or safety personnel.

(3)

After visualizing ship collision risks, a novel collision risk scoring algorithm was proposed. The algorithm calculates the collision risk score between two ships based on navigation parameters and the generated ECNA, determining the level of collision risk between ships and aiding in collision risk identification.

The proposed ECNA model effectively visualizes the range of angles associated with collision risks between two ships. The model can overlay the ECNA generated by one ship with those of multiple ships during encounters. This approach offers a clearer display of collision risks compared to traditional methods. Additionally, the proposed method includes collision risk algorithms that allow for quantitative comparison, making it particularly advantageous in densely trafficked waters, where its benefits are more pronounced compared to the PPC and PAD models.

As the number of ships continues to increase, ship density in coastal waters has been rising annually, making the identification of collision risks in congested areas increasingly challenging. Both sailors and navigation supervisors need effective methods to identify and visualize ship collision risks. Therefore, the novel approach proposed in this paper holds significant practical value. Additionally, the collision risk scoring algorithm introduced offers a new approach to the quantitative assessment of ship collision risks.