Potential risk ship domain as a danger criterion for real-time ship collision risk evaluation

Ocean Engineering 194 (2019) 106610

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Potential risk ship domain as a danger criterion for real-time ship collision risk evaluation Namkyun Im a , Tu Nam Luong b ,∗ a b

Division of Navigation Science, Mokpo Maritime University 58628, Korea Department of Maritime Transportation System, Mokpo Maritime University 58628, Korea

ARTICLE Keywords: Ship domain Potential collision risk Risk evaluation Collision assessment Risk zone

INFO

ABSTRACT One of the most useful method to indicate the risk of ship collision is the ship domain. After decades of development, different ship domain models have been constructed from various aspects. Nevertheless, a shortcoming of the early research of ship domains is not able to describe the level of danger of any point in the area around the ship. Therefore, a Potential Risk Ship Domain (PRSD) model with clear meaning of risk degree is first proposed in this paper. The zone around a ship has been established which is based on concept of ship domain and consists of continuously risk levels, by using kernel-density algorithm. The influence of ship length and speed as well as navigation situation on the size and shape of ship domain by PRSD model are analyzed. Size of ship domain and potential collision risk indexes for each risk level will be identified by comparison with previous models. Simulations show that the PRSD model can effectively evaluate the danger of navigation and discovery potentially dangerous areas for navigation by indicating the dynamic and continuous index of potential risk degree in real time.

1. Introduction Efficient maritime navigation through dynamic obstacles is one of many challenges faced by mariners, especially in terms of determining the maneuvers necessary to avoid a probable collision. Ship collision refers to the physical event that two or more ships occupy the same point or area on the sea surface at the same time. Therefore, the evaluation of safe area around the ship is of major importance for navigation. When ships approach each other, they should ensure that a minimum spacing around them is clear of other vessels in order to pass each other safely and the shortest distance of the collision does not occur. It led to the appearance of ship domain concept. Ship domain is generally defined as a safety area around either the own ship or target ship which stays or should stay clear of other ships, not of other ship’s domains. If other ships or navigational obstacles do not exceed this area, navigational situation can be considered safe. In maritime traffic engineering, the concept of ship domain has been suggested and developed by a number of authors. The two-dimensional area, named ‘‘effective domain’’ was first proposed in 1971 by Fujii and Tanaka (1971) to deal with the safe navigation of ships in open waters off Japan which is an ellipse with a long radius of 8 L and short radius of 3.2 L (𝐿 is ship length) under ordinary navigation condition. Four years later, Goodwin (1975)

developed a sector-shaped domain model to investigate the maritime traffic based on data generated by a radar simulator from collision experiments. The ship domain was divided into segments of the circles: port, starboard and astern, which have different radii of 0.7 nautical miles (nm), 0.85 nm and 0.45 nm, respectively in southern North Sea waters and 2.4 nm, 2.4 nm and 0.5 nm in open ocean. Davis et al. (1980) considered the Goodwin (1975) model having much defects such as discontinuous boundaries with sudden jumps. By smoothing the domain edges, the domain as a circle around a ‘phantom’ ship which was at the center the circle, the real ship being fixed by a distance and an angle (relative to ship’s head) from the phantom. Coldwell (1983) specified different dimension values of ship domain for meeting (head-on and crossing) and overtaking encounters, for example 6.1 × 5 cables for head-on meetings with an offset (1.75 cables to port and 3.25 to starboard and 6 × 3.5 cables for overtaking situations. Fujii and Tanaka (1971), Goodwin (1975) and Coldwell (1983) all used radar observations in different areas with some differences in geometrical dimensions and distance measurement unit of ship domains. A source of data from the Automatic Identification System (AIS) makes it possible to overcome the shortcomings of statistical methods, primarily due to a large number of observations. Gucma and Marcjan (2012) built a probabilistic domain based on distance distribution into 8 sectors (every 45◦ ). Construction of domain shape was held together by linking

∗ Corresponding author. E-mail addresses: [email protected] (N. Im), [email protected] (T.N. Luong).

https://doi.org/10.1016/j.oceaneng.2019.106610 Received 22 June 2019; Received in revised form 4 September 2019; Accepted 18 October 2019 Available online 30 October 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.

Ocean Engineering 194 (2019) 106610

N. Im and T.N. Luong

𝑔𝑓 (𝑣), 𝑔𝑎 (𝑣), 𝑔𝑠 (𝑣), 𝑔𝑝 (𝑣) The speed function in fore, aft, starboard and port side, respectively 𝑘𝐴𝐷 Gain of the Advance of the ship 𝑘𝐷𝑇 Gain of the Tactical Diameter of the ship 𝐿 Ship length 𝐿𝑂𝐴 Length Overall 𝑃𝜑 Radius of ship domain of Pietrzykowski and Urias (2009) at bearing 𝜑 𝑃 𝐶𝑅 Potential Collision Risk 𝑃 𝑅𝑆𝐷 Potential Risk Ship Domain 𝑅 Risk diameter 𝑅𝑓 , 𝑅𝑎 , 𝑅𝑠 , 𝑅𝑝 Radii of ship domain of Wang and Chin (2016) in fore, aft, starboard and port side, respectively 𝑅𝑏𝑓 , 𝑅𝑏𝑎 , 𝑅𝑏𝑠 , 𝑅𝑏𝑝 Radii of blocking area of Kijima and Furukawa (2003) in fore, aft, starboard and port side, respectively 𝑅𝑤𝑓 , 𝑅𝑤𝑎 , 𝑅𝑤𝑠 , 𝑅𝑤𝑝 Radii of watching area of Kijima and Furukawa (2003) in fore, aft, starboard and port side, respectively 𝑇 𝐶𝑃 𝐴 Time to Closest Point of Approach 𝑇 𝐶𝑃 𝐴′ Non-dimensional Time to Closest Point of Approach 𝑣 Ship speed 𝑉 𝑇𝑆 Vessel Traffic Service 𝑥, 𝑦 Position of the ship 𝑥𝑝 , 𝑦𝑝 Position of point p

List of Abbreviations and Symbols 𝛥𝑏

𝛥𝑏𝑓 , 𝛥𝑏𝑎 , 𝛥𝑏𝑠 , 𝛥𝑏𝑝

𝛥𝑏𝑖

𝛥𝐷

𝛥𝑑

𝛥𝑑𝑓 , 𝛥𝑑𝑎 , 𝛥𝑑𝑠 , 𝛥𝑑𝑝

𝛥𝐷𝑖

𝛥𝑑𝑖

𝛥𝑤

𝛥𝑤𝑓 , 𝛥𝑤𝑎 , 𝛥𝑤𝑠 , 𝛥𝑤𝑝

𝛥𝑤𝑖

𝜆 𝜎 𝜑 𝜑𝑝 𝐴𝐼𝑆 𝐶𝑂𝐿𝑅𝐸𝐺𝑆 𝐶𝑃 𝐴 𝐷𝜑 𝐷𝑓 , 𝐷𝑎 , 𝐷𝑠 , 𝐷𝑝

𝐷𝐶𝑃 𝐴 𝐷𝐶𝑃 𝐴′ 𝐸𝐶𝐷𝐼𝑆

Sum of squared distance differences with the blocking area of Kijima and Furukawa (2003) Squared distance differences with the blocking area of Kijima and Furukawa (2003) at fore, aft, starboard and port side, respectively Squared distance differences with the blocking area of Kijima and Furukawa (2003) number i Sum of squared distance differences with the ship domain of Pietrzykowski and Urias (2009) Sum of squared distance differences with the ship domain of Wang and Chin (2016) Squared distance differences with the ship domain of Wang and Chin (2016) at fore, aft, starboard and port side, respectively Squared distance differences with the ship domain of Pietrzykowski and Urias (2009) in the i bearing Squared distance differences with the ship domain of Wang and Chin (2016) number i Sum of squared distance differences with the watching area of Kijima and Furukawa (2003) Squared distance differences with the watching area of Kijima and Furukawa (2003) at fore, aft, starboard and port side, respectively Squared distance differences with the watching area of Kijima and Furukawa (2003) number i Lateral influence parameter Longitudinal influence parameter Ship heading of ship from north direction Heading difference between ship and point p Automatic Identification System International Regulations for Preventing Collisions at Sea Closest Point of Approach Radius of ship domain by PRSD model at bearing 𝜑 Radii of ship domain by PRSD model in fore, aft, starboard and port side, respectively Distance to Closest Point of Approach Non-dimensional Distance to Closest Point of Approach Electronic Chart Display and Information System

amount of AIS data in the Great Belt and in the Drogden Channel. The passing distance between ships was measured in Length Overall (𝐿𝑂𝐴) and was determined to have minor differences with the effective domain by Fujii and Tanaka (1971). In the investigation that revisits and proposes a method using AIS data in Swedish waters, Horteborn et al. (2019) concluded that the ship domain has the shape of an ellipse with half axis radii of 0.9 and 0.45 nm and recommended that LOA should be used as a determinant of the ship domain static ship domain in open waters and a dynamic ship domain in restricted waters. In the above models, ship domains are static, invariable of ship speed. Despite also being based on statistically processed empirical data, Wang and Chin (2016) domain brings more parameters and complexity than related older models. It is further formulated as a mathematical model by taking into account ship size and ship speed as well as a human factor component. However, the resulting free-form polygonal shape is actually quite close to an ellipse domain suggested by past works (Szlapczynski and Szlapczynska, 2017). Another shape of ship domain — hexagon was developed by Smierzchalski (2000) for a target ship on the basis of ship dimension and speed, and relative dynamic parameters. This domain was obtained which is dependent on actual value of safe distance, assumed by the navigator. The vertices of the irregular hexagonal-shaped domain were defined as a multiple of a unit governing safe distance. In the above models, ship domains, are assumed deterministic and the set of domain parameters has not been extended. It was (Zhao et al., 1993) who introduced the idea of a fuzzy boundary for domain based on Goodwin (1975) using fuzzy sets theory. It was assumed that ‘‘if the relative motion line of a target is outside of the fuzzy boundary, it is safe, no action need be taken; if the relative motion line is just inside the fuzzy boundary, it is not certainly safe, but not certainly dangerous either, action need not be taken; if the relative motion line is inside the fuzzy boundary, it is dangerous, action must be taken to keep it out of the fuzzy boundary’’. Unlike crisp domains, a fuzzy one can vary between different levels of safety. The domain model based on neural

distances plotted in the middle of each sector. They also concluded that the type of ship had no effect on the shape of the domain. Hansen et al. (2013) analyzed shape and size of the ship domain based on a large 2

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networks has been derived that can express the effect of visibility and a ship’s maneuverability and react quickly to a variety of situations (Zhu et al., 2001). Pietrzykowski and Uriasz (2004) followed the concept of Zhao et al. (1993) by developing a ship fuzzy domain defined as a fuzzy area around the ship which the navigator should keep clear of other vessels and objects. The shape and size of the domain depend on the adopted level of navigational safety which can be different for different navigators. The fuzzy domain has been applied in both narrow fairways (Pietrzykowski, 2008) and open waters (Pietrzykowski and Urias, 2009). The developed fuzzy domain is claimed as a universal criterion for the assessment of a current navigational situation. Kijima and Furukawa (2001, 2003) introduced a new ship domain which is a combination of two ellipses, named Blocking area and Watching area. Shapes of both areas are modeled by longitudinal radii in fore and aft domains and common transverse radius. According to the authors, existing ship in the watching area is regarded as target ship which should be watched out for by own ship; when the watching area of the target ship invades the blocking area, own ship should decide whether she will change or keep her course? Based on the formulas of Kijima and Furukawa (2003) and Wang et al. (2010) changed the crisp boundaries to fuzzy ones, resulting in a fuzzy quaternion ship domain (QSD) and continued as a Dynamic Quaternion Ship Domain (DQSD) (Wang, 2013). The shape is modeled by index parameters. The fuzzy possibility value is related to the size of the ship domain and the shape index is related to the state of the navigator. Qu et al. (2011) combined this model with a criterion of number of ship domains overlaps is applied in for assessment of collision risk in the Singapore Strait, based on the AIS data. Nevertheless, it is furthermore not clear which parameter choices would be valid since different choices result in significantly different collision risk evaluations (Goerlandt and Kujala, 2014). Dinh and Im (2016) also used the term ‘‘blocking area’’ for the most dangerous area that is around a target ship and should not be violated by the own ship. Its dimensions are based mainly on calculations of the advance distance of the own ship (for all encounters) and additionally the advance distance of the target (for head-on encounters only). Apart from the blocking area, a circular area, called ‘‘action area’’ is where the ship must perform a collision avoidance maneuver in order to resolve the encounter situation safely. Nevertheless, existing ship domains are lack of the ability to reflect continuously risk level. The problem of ship domain determination is still up to date due to many factors affecting its shape and size. In many experts’ opinion, the most important domain determinants include the size and speed of the vessel and the type of sea area open or restricted (Wielgosz, 2017). It should be mentioned that there are some successful applications of ship domains, namely,

model is constructed based on kernel-density function, consists of three boundaries that divide the neighborhood around the ship into four zones with different potential risk levels in order to express the degree of risk level around the ship and show the ability to evaluate the potential collision risk in real time and in both restricted and open sea areas. The length and speed of the ship, as well as the type of water area have been fully considered and the parameters are determined by comparing with previous ship domains by fitting method. Obviously, this is a dynamic ship domain model which accounts for ship dimensions, maneuverability and navigational situations. The PRSD model is dramatically effective method in order to estimate the potential risk of collision and to identify the dangerous area around the ship. It can provide a reference for the evaluation of ship collisions and other traffic accidents and for the supervision of VTS centers. The main merits of this framework are: • propose a novel ship domain, name Potential Risk Ship Domain, which can calculate continuous and dynamic potential collision risk level in area around the ship • propose a Collision Assessment zone around a ship included four zones with different levels of potential collision risk using PRSD model for risk evaluation. Improvement of boundaries with values of PCR in PRSD model can not only reveals the different levels of ship collision hazard, but also overcome the deterministic problem of existing models. In PRSD model, the continuous values in range of (0,1) are used to indicate the collision risk. Furthermore, two ship domains can be overlapped when coming closer. It helps to predict the potential collision area. The paper is organized as follows. Section 2 presents a construction of PRSD model. Section 3 explains the applied methods for determining the values of parameters of PRSD. In Section 4, a Collision Assessment zone around a ship with different levels of potential collision risk index using PRSD model is introduced and in Section 5, the results of the reliability criteria for the proposed methods are shown in the case study. Finally, concluding remarks are provided in Section 6. 2. Potential risk ship domain modeling 2.1. Influence and density function Any data point in the neighborhood will be affected by other points surround it. The influence of each data point can be modeled formally using a mathematical function, called influence function (Hinneburg and Keim, 1998). The influence function can be defined as the function which describes the impact of a data point within its neighborhood. Any other data point in the vicinity will be affected. Examples for influence functions are parabolic functions, square wave function, or the Gaussian function. In order to construct such the new ship domain, firstly, an function that can continuously describe the risk of every point inside the ship domain by exact value is needed. This development can solve the limitation of the earlier models. The continuous values in range of (0,1) are used to indicate the collision risk. The core idea is that each spatial point has an influence on the space through the influence function. In this paper, the Gaussian influence function is used to construct the ship domain, because it is in line with the requirement that the closer to the ship, the larger the influence is. The influence function is applied to each data point. An estimate of the overall density of the data space is the sum of the influence function of all data points (Hinneburg and Keim, 2003). The kernel functions are a mathematical description of the influence a data object has within its neighborhood. 𝐹 𝑑 is denoted for the d-dimensional feature space. The density function at a point 𝑥 ∈ 𝐹 𝑑 is defined as the sum of the influence functions of all data objects at that point. The influence function of a

• collision avoidance: Applying a ship domain is also mentioned in Tsou et al. (2010) applied a ship domain in where a collision avoidance method based on genetic algorithm (GA) and took into account COLREGS. However, again a circular domain is applied and COLREGS are handled separately. This research is continued in Tsou (2016), where collision avoidance method is combined with the use of ECDIS. The domain remains a circle with a radius set arbitrarily to 1.5 nm. • near miss detection and trajectory processing: van Iperen (2015) applies the concept of a ship domain while at the same time identifying an empirical domain for the North Sea. Ship domain in Zhang et al. (1994) has been there supplemented by a number of other situational characteristics to obtain a greater precision of identifying near miss ship collisions. • waterway risk analysis: Various approaches collision risk analysis are compared and assessed, including previous domains in Goerlandt and Kujala (2014) and Goerlandt and Montewka (2015). This paper presents an alternative model of dynamic ship domain, Potential Risk Ship Domain (PRSD), for assessing the potential collision risk for surface ships in wide-range encounters. The proposed 3

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Fig. 1. Example of kernel-density function.

data object 𝑦 ∈ 𝐹 𝑑 is a function 𝑓𝐵𝑦 which is defined in terms of a basic influence function 𝑓𝐵 : 𝑓𝐵𝑦 (𝑥) = 𝑓𝐵 (𝑥, 𝑦)

ship. This phantom ship receives the influence from the real ship but has no impact to the neighborhood. This effect is greater when near the real ship and smaller when far away. Each phantom ship will have different degree of influence and this influence can be called ‘‘potential collision risk’’. The area that consists of every phantom ship around the real ship is Potential Risk Ship Domain (PRSD). One of functions of PRSD model is to express the impact of the ship to neighborhood. The potential risk of the vicinity of the ship is taking the characteristics of the ship and the actual situation of navigation. Ship domain is a kind of image description of ship collision risk, it is reasonable to be changed with different ship speed and other navigation data (Wang et al., 2014). The important factor affecting the safety of a ship at fore side is the speed. To model a dynamic ship domain, it is key point to form the shape of ship domain changing with ship’s specific speed. When the ship is sailing, a circular zone is constructed at aft side, otherwise a half-elliptical zone at the fore side is used to extend the way to achieve the impact of the ship. The reason is that the area behind the ship is not affected by the forward motion of the ship, however, due to the forward movement of the ship, the area in front of the ship is also affected by the danger of collision of the ship. In order to construct PRSD, firstly, a function, namely risk diameter, is needed. The traditional method for judging collision risk, i.e., the combination of 𝐷𝐶𝑃 𝐴 (distance to the closest point of approach) and 𝑇 𝐶𝑃 𝐴 (time to the closest point of approach), is applied in this function. However, in this paper, 𝐶𝑃 𝐴 (closest point of approach) between the dynamic real ship and static phantom ship is adopted. Then, by integrating the risk diameter into the Gaussian influence function, the spatial distribution of potential collision risk into the area around the ship is obtained and the PRSD can be constructed. Suppose the position of the own ship is (𝑥, 𝑦) with ship length 𝐿, speed 𝑣 and the heading 𝜑 and point 𝑝(𝑥𝑝 , 𝑦𝑝 ). DCPA and TCPA are the two parameters usually to determine the degree of risk of ship collision at sea. For the calculation of 𝐷𝐶𝑃 𝐴 and TCPA between own ship and point 𝑝, a perpendicular from point 𝑝 is drawn to the heading line of the own ship having 𝐶𝑃 𝐴. When approaching point 𝑝, 𝐷𝐶𝑃 𝐴 is the shortest distance between the own ship and point 𝑝. While, 𝑇 𝐶𝑃 𝐴 is the time needed to reach the closest point of approach. Consequently, in the ship motion simulation system, the ship motion parameters can be expressed in Fig. 2. Mathematically, 𝐷𝐶𝑃 𝐴 and 𝑇 𝐶𝑃 𝐴 between the ship to point 𝑝 can be calculated by using the following equations: √ ⎧ = (𝑥𝑝 − 𝑥)2 + (𝑦𝑝 − 𝑦)2 × sin 𝜑𝑝 ⎪𝐷𝐶𝑃 𝐴 = 𝐷𝜑 × sin 𝜑𝑝 √ (6) ⎨ (𝑥𝑝 −𝑥)2 +(𝑦𝑝 −𝑦)2 ×cos 𝜑𝑝 ⎪𝑇 𝐶𝑃 𝐴 = 𝐷𝜑 ×cos 𝜑𝑝 = 𝑣 𝑣 ⎩

(1)

The density function is defined as the sum of the influence functions of all data points. Given N data objects described by a set of feature vectors 𝐷 = [𝑥1 , … , 𝑥𝑁 ] the density function is defined as: 𝑓𝐵𝑦 (𝑥) =

𝑁 ∑

𝑥

𝑓𝐵𝑖 (𝑥)

(2)

𝑖=1

For the definition of specific influence functions, a distance function 𝑑 determines the distance of two d-dimensional feature vectors. 𝑓𝐵𝑦 (𝑥1 ) ⩾ 𝑓𝐵𝑦 (𝑥2 ) if 𝑑(𝑥1 , 𝑦) ⩽ 𝑑(𝑥2 , 𝑦)

(3)

Gaussian Influence Function is as follows: 𝑓𝐺𝑎𝑢𝑠𝑠 (𝑥, 𝑦) = 𝑒

−

𝑑(𝑥,𝑦)2 2𝜎 2

(4)

The function which results from a Gaussian influence function is as follows: 𝑓𝐺𝑎𝑢𝑠𝑠 (𝑥, 𝑦) =

𝑁 ∑

𝑒

−

𝑑(𝑥,𝑦)2 2𝜎 2

(5)

𝑖=1

Fig. 1 shows an example of a set of data points in 2D space together with the corresponding overall density functions for a Gaussian influence function. 2.2. Potential risk ship domain modeling Ship domain has become a basic method and widely used in research on collision avoidance and traffic engineering. Ship safety domain is a generalization of a safe distance and its introduction to maritime navigation comes from the observation that the safe distance is not the same in all directions (Szlapczynski and Szlapczynska, 2017). Distance between ships has always been the most important factor contributing to the level of safety (Szlapczynski et al., 2018). Nevertheless, the problem of quantitative quantification directly affects the application and development of the ship domain. The cause of this limitation is that in the earlier models, the level of ship collision risk inside and outside the domain is deterministic. The ship collision risk is 1 within the domain and 0 when out of domain. Therefore, the next step in the development of the ship domain depends on the identification of different levels of ship collision risk and overcome the problem of quantitative quantification in the ship domain boundary. When a ship is navigating, she will have effect to the surrounding area. This paper assumes that any point in this area is a static phantom 4

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Fig. 2. The schematic diagram of motion parameters.

Fig. 3. Example of ship domain by PRSD model.

The non-dimensional of 𝐷𝐶𝑃 𝐴 and 𝑇 𝐶𝑃 𝐴 can be obtained as follows: ⎧ ⎪𝐷𝐶𝑃 𝐴′ = ⎨ ⎪𝑇 𝐶𝑃 𝐴′ = ⎩

𝐷𝜑 ×sin 𝜑𝑝 𝐿 𝐷𝜑 ×cos 𝜑𝑝 𝐿𝑣

√ = =

longitudinal and lateral influence ranges of ship domain, respectively. At fore side, the radius 𝐷𝜑 is governed by a function of ship length 𝐿 and speed 𝑣. While at aft side, it is only governed by a function of ship length 𝐿. The length of 𝐷𝜑 depends on the value of 𝑃 𝐶𝑅 as well. The higher the 𝑃 𝐶𝑅 value (maximum 1) is, the shorter 𝐷𝜑 is; the lower the 𝑃 𝐶𝑅 value is (minimum 0), the longer it is. By applying PRSD model, the area around the ship can be digitally generated based on value of 𝑃 𝐶𝑅, as is depicted in Fig. 3. Every point in this area has a value of 𝑃 𝐶𝑅 and points with same value of 𝑃 𝐶𝑅 will be shown as contour lines. The ship heading determines the direction of the longitudinal axis of the ship domain, the ship speed determines the degree of extension of the fore side. The coverage of the ship domain is decided by the influence parameters. Next step is to specify the size of ship domain by PRSD model.

(𝑥𝑝 −𝑥)2 +(𝑦𝑝 −𝑦)2 ×sin 𝜑𝑝

√

𝐿 (𝑥𝑝 −𝑥)2 +(𝑦𝑝 −𝑦)2 ×cos 𝜑𝑝

(7)

𝐿𝑣

The relative bearing of point 𝑝 is given by: 𝑥𝑝 − 𝑥 𝜑𝑝 = arctan −𝜑 𝑦𝑝 − 𝑦

(8)

This paper adopts the normalization of 𝐷𝐶𝑃 𝐴 and 𝑇 𝐶𝑃 𝐴 to determine degree of potential collision risk. The risk diameter 𝑅 based on the model of Kearon (1977) which is the combination of 𝐷𝐶𝑃 𝐴′ and 𝑇 𝐶𝑃 𝐴′ can be found as follows: √ 𝑅 = 𝜆𝐷𝐶𝑃 𝐴′2 + 𝑇 𝐶𝑃 𝐴′2 (9) The dimension size of 𝐷𝐶𝑃 𝐴 and 𝑇 𝐶𝑃 𝐴 are different. Therefore, 𝜆 is used as a weighting to keep these variables matching. The potential collision risk (𝑃 𝐶𝑅) index at any point 𝑝(𝑥𝑝 , 𝑦𝑝 ) can be calculated as follows:

3. Ship domain by PRSD model In this step, the size of ship domain by PRSD model that was formulated earlier will be specified. As mentioned above, the size of ship domain by PRSD model can be measured by Eq. (11). It depends on an adequate selection of the parameters. It is seen that varying choices of parameters lead to important difference sizes of ship domain. In order to determine the size of ship domain by PRSD model, three important factors, namely 𝜎, 𝜆 and 𝑃 𝐶𝑅 should be identified. With each set of 𝜎, 𝜆 and 𝑃 𝐶𝑅, the ship domain by PRSD model will have a different size (as can be seen in Fig. 4). With same values of 𝜎 and 𝜆, smaller value of PCR will lead to larger size than greater one (PRSD 1 and PRSD 2). However, with different values of 𝜎 and 𝜆, PRSD will have different range with same value of PCR (PRSD 1 and PRSD 3). In general, the waters at sea can be divided into open sea areas and restricted areas. The traffic densities in these areas are significantly different. The dimension of ship domain is affected by different operating environment and ship density (Goodwin, 1975). It has been observed that, generally, the ship domain size changes distinctly depending on available maneuverable area dimensions (Wielgosz, 2017). Quite clearly, the ship domain in open sea and restricted areas should be treated differently. In the following, we describe how the size of ship domain by PRSD model is obtained in both areas by determination of parameters.

2

𝑃 𝐶𝑅 = 𝑓𝑝𝑜𝑖𝑛𝑡 (𝑝) = 𝑒

𝑅 − 𝑖2 2𝜎

(10)

Each point has a different range of effects from the ship. 𝑃 𝐶𝑅 is used as the value to reflect the potential collision risk distribution of the ship of one point in its surrounding environment. It describes the degree of potential risk of collision, the bigger this value is, the more dangerous. A small value of 𝑃 𝐶𝑅 leads to a large influence range of the ship, but a larger 𝑃 𝐶𝑅 makes the influence range smaller. The larger the distance between the own ship and the point 𝑝 is, the smaller the effect of the ship is. 𝑃 𝐶𝑅 value will reach the maximum value 1. This value approaches 0 at places far away from the ship. Each value of 𝑃 𝐶𝑅 corresponds to a contour line that joints points of equal value of 𝑃 𝐶𝑅. Although these points share same value of 𝑃 𝐶𝑅, the distances from the ship to these points are different. The distance depends on an angle 𝜑𝑝 clockwise from the ship heading, named 𝐷𝜑 . From Eq. (10), if 𝑃 𝐶𝑅 is identified (in range 0 to 1), 𝐷𝜑 at each angle 𝜑𝑝 can be calculated. The size of ship domain by PRSD model is measured by the radius 𝐷𝜑 from the ship center to the different vertices of the ship domain as follows: √ −2×ln(𝑃 𝐶𝑅) ⎧ ×𝑣×𝜎×𝐿 ⎪ 𝜆𝑣2 𝑠𝑖𝑛𝜑𝑝 2 +𝑐𝑜𝑠𝜑𝑝 2 ⎪ if 0◦ ⩽ 𝜑 ⩽ 90◦ or 270◦ ⩽ 𝜑 ⩽ 360◦ 𝑝 𝑝 (11) 𝐷 𝜑 = ⎨√ ⎪ −2×ln(𝑃 𝐶𝑅) × 𝑣 × 𝜎 × 𝐿 𝜆 ⎪ ⎩ if 90◦ < 𝜑𝑝 < 270◦

3.1. In restricted areas Restricted areas are high-density traffic areas where a ship is likely to collide with others if ship navigation is not carried out in the right

where 𝜎: longitudinal influence parameter; 𝜆: lateral influence parameter. Longitudinal and lateral influence parameters (𝜎, 𝜆) determine the 5

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N. Im and T.N. Luong

Fig. 4. PRSDs of same ship with different values of parameters (from the left: PRSD 1, PRSD 2, PRSD 3).

Fig. 5. Process of approximation method.

manner. They have dense traffic in the form of fishing boats, small crafts in addition to larger merchant ships which, by default, means that a higher degree of precaution, situational awareness, and decision making is to be exercised under such circumstances. Since there are several ships present in the vicinity, chances of a collision are very high. Attention is increasingly paid to the problem of using a ship domain in the process of safe ship conduct, particularly in restricted areas. A study on ship domain in restricted areas is promising and the problem and models of ship domain in the restricted areas have been described in many publications lately. In this section, the size of ship domain by PRSD model in restricted areas will be suggested by the identification of parameters. In formulating the PRSD model, parameters have been expected to be different under different conditions. In order to decide the size of ship domain by PRSD model, we adapt the approximation method. The aim of the approximation is to generate three parameters of ship domain in restricted areas: 𝜎, 𝜆 and 𝑃 𝐶𝑅. Five models of ship domain in restricted areas are reviewed and compared, i.e. Fujii and Tanaka (1971), Coldwell (1983), Hansen et al. (2013), Wang and Chin (2016) and Kijima and Furukawa (2003), then one of them will be employed as subject model for approximation process. For the technique of the approximation, Least Squares method is used. This is a tool for determination of a solution that is generally very close, often identical with the optimum. The stop condition is set on achieving the minimum fitness function (Wielgosz, 2017). The bestfit model can be obtained by defining an optimization problem with the objective to minimize the difference between the proposed PRSD model and a previous model. The size of ship domain by PRSD model will be differently generated by each set values of 𝜎, 𝜆 and 𝑃 𝐶𝑅. The best fitted domain is assumed to take that of minimum value of fitness function. The process of the approximation is presented in Fig. 5. The procedure in Fig. 5 is to prescribe a framework for the parameters approximation in order to determine the size of ship domain by PRSD model in restricted areas. By adopting this approach, the approximation will be done sequentially searching for the best-fitted parameters of the ship domain size. Radii at four directions of ship domain with each set value of 𝜎, 𝜆 and 𝑃 𝐶𝑅 are calculated, then they will be compared with four radii of subjected domain. Fitness function is the sum of squared differences between four radii of ship domain and the subjected domain. The approximation process will stop until the minimum value of fitness function is found out and then the parameters for the ship domain by PRSD model in restricted areas can be output. In order to conduct approximation process, the range of parameters

should be considered. Each parameter has special features in the PRSD model. It is implied that these parameters have a different range of effects to the size of ship domain. Longitudinal and lateral radii of PRSD model are required in this step. Applying 𝜑𝑝 = 0◦ , 90◦ , 180◦ and 270◦ to Eq. (12), four radii of PRSD model can be given by: √ { 𝐷𝑓 = −2 ln(𝑃 𝐶𝑅) × 𝑣 × 𝜎 × 𝐿 √ (12) 𝐶𝑅) 𝐷𝑎 = 𝐷𝑠 = 𝐷𝑝 = −2 ln(𝑃 ×𝜎×𝐿 𝜆 where 𝐷𝑓 , 𝐷𝑎 , 𝐷𝑠 , 𝐷𝑝 : radius of ship domain by PRSD model in fore, aft, starboard and port side, respectively. As explained in Section 2.2, 𝑃 𝐶𝑅 value varies from 0 to 1. The range of influence parameters should be decided to reduce time of approximation process. The appropriate value of influence parameter is which make the size of ship domain reasonable with existing ship domains in restricted areas, neither too small nor too large. First, the forward radius 𝐷𝑓 of the ship domain is independent of lateral influence parameter 𝜆, so to determine the range of longitudinal influence parameter 𝜎, 𝐷𝑓 was calculated with each value of 𝜎 from 0.1 (interval 0.1) by Eq. (12) until it was realized that if the value of 𝜎 is greater than 1, the radius of ship domain by PRSD model in fore side 𝐷𝑓 is significantly larger when comparison with fore radii of other ship domains in restricted area. In cases (𝜎1 = 0.1, 𝜎2 = 1) and (𝑃 𝐶𝑅1 = 0.1 and 𝑃 𝐶𝑅2 = 0.9), 𝐷𝑓 was compared with the forward radii of previous ship domains in restricted areas, which is presented in Figs. 6–7. The only requirement on 𝜎 is that it generates a reasonable fore size of ship domain. As we can see, the longest range of forward radii in restricted areas is around 6 times of 𝐿𝑂𝐴. Forward radii of dynamic ship domains (Wang and Chin, 2016; Kijima and Furukawa, 2003) are small at low speeds but greater when speed increase. In case 𝜎1 = 0.1, even the length of 𝐷𝑓 of ship domain by PRSD model at 𝑃 𝐶𝑅1 = 0.1 is significantly smaller than those of previous ship domains. It will lead to the fore zone of ship domain is too small. While with 𝜎2 = 1, the length of radius in fore of previous ship domains correspond to longitudinal radius 𝐷𝑓 of ship domain by PRSD model at 𝑃 𝐶𝑅2 = 0.9. At speed of 10kn and higher, they are even smaller than 𝐷𝑓(0.9) . In this case, the fore zone of ship domain by PRSD model is overwhelmingly larger than other ship domains. Therefore, the suitable value of 𝜎 for restricted area should be in range 0.1 to 1. For lateral influence parameter, the similar comparison is conducted and the range of 𝜆 for ship domain by PRSD model in restricted areas is achieved from 0 to 1. After obtaining the ranges of parameters, one ship domain should be chosen for comparison. Ship domains of Fujii and Tanaka (1971), 6

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Table 1 Dimension of ship domains in restricted areas in case 𝑣 = 12𝑘𝑛. Ship domain

Size

Fujii Hansen Coldwell (head-on) Coldwell (head-on) Wang Kijima (blocking area) PRSD

Fore

Aft

Starboard

Port

3L 4L 6.1L 6L 4.06L 5.6L 2.8L

3L 4L 0 6L 2.82L 3.3L 1.34L

0.8L 1.58L 3.25L 1.75L 1.48L 3.4L 1.34L

0.8L 1.58L 1.75L 1.75L 1.44L 2.6L 1.34L

used empirical data in other waters (Japan, England and Denmark, respectively). Therefore, ship domain in Wang and Chin (2016) is selected for the subject domain of approximation. Now the problem is to define a fitness function to estimate the model parameters. As a measure of fitness, the selected fitness function is defined by summing up of squared distance differences between four domain radii of proposed model and model of Wang and Chin (2016) at same speed. Radii of ship domain in Wang and Chin (2016) are linear functions of ship’s length and quadratic functions of its speed. Radial distances from the ship center to the vertex of the polygon ship domain are calculated as follow:

Fig. 6. Comparisons between 𝐷𝑓 at 𝜎1 = 0.1 of PRSD and previous ship domains in restricted areas.

⎧ ⎪𝑅𝑓 = 𝛼𝑓 × 𝐿 × 𝑔𝑓 (𝑣) ⎪𝑅𝑎 = 𝛼𝑎 × 𝐿 × 𝑔𝑎 (𝑣) ⎨𝑅 = 𝛼 × 𝐿 × 𝑔 (𝑣) 𝑠 𝑠 ⎪ 𝑠 ⎪𝑅𝑝 = 𝛼𝑝 × 𝐿 × 𝑔𝑝 (𝑣) ⎩

(13)

where 𝑅𝑓 , 𝑅𝑎 , 𝑅𝑠 , 𝑅𝑝 : radius of Wang and Chin (2016) model in fore, aft, starboard and port side, respectively; 𝛼𝑓 , 𝛼𝑎 , 𝛼𝑠 , 𝛼𝑝 : the normalized radial distance of the ship domain when the ship is stationary; 𝑔𝑓 (𝑣), 𝑔𝑎 (𝑣), 𝑔𝑠 (𝑣), 𝑔𝑝 (𝑣): function of ship speed at fore, aft, starboard and port side, respectively. The fitness function thus becomes: ⎧ 2 ⎪𝛥𝑑𝑓 = (𝐷𝑓 − 𝑅𝑓 ) 2 ⎪𝛥𝑑𝑎 = (𝐷𝑎 − 𝑅𝑎 ) ⎨𝛥𝑑 = (𝐷 − 𝑅 )2 𝑠 𝑠 ⎪ 𝑠 ⎪𝛥𝑑𝑝 = (𝐷𝑝 − 𝑅𝑝 )2 ⎩ 𝛥𝑑 =

4 ∑∑

𝛥𝑑𝑖

(14)

(15)

𝑣 𝑖=1

where 𝛥𝑑𝑓 , 𝛥𝑑𝑎 , 𝛥𝑑𝑠 , 𝛥𝑑𝑝 : squared distance difference at fore, aft, starboard and port side, respectively; 𝛥𝑑: sum of squared distance differences. The fitness function of the approximation for the size of ship domain by PRSD model in restricted areas involving a set of three parameters: 𝜎, 𝜆 and 𝑃 𝐶𝑅 and calculated by Eq. (15). The parameters values can be estimated such that the sum of squared distance differences between radii the proposed model and the subjected model (Wang and Chin, 2016) is minimal. It means that after finding the minimum value of fitness function (𝑚𝑖𝑛𝛥𝑑), and the corresponding parameters can be selected. The ship domain with a set values of (𝑃 𝐶𝑅 = 0.8, 𝜎 = 0.35, 𝜆 = 0.03) is best fitted with ship domain of Wang and Chin (2016) in restricted areas. Applying 𝑃 𝐶𝑅 = 0.8, 𝜎 = 0.35, 𝜆 = 0.03 to Eq. (13), four radii of ship domain by PRSD model in restricted areas can be calculated. The result in case 𝑣 = 12𝑘𝑛 and radii of ship domains in restricted areas is represented in Table 1 and Fig. 8. The resulting past ship domain in restricted areas are reproduced in Fig. 8 along with the ship domain by PRSD model. The comparison shows that ship domain of Fujii and Tanaka (1971) is reasonably compatible with the proposed ship domain on the fore side but overestimating the space requirement on the aft side while underestimating the lateral sides. Hansen et al. (2013) and Wang and Chin (2016) ship domains match well with the proposed ship domain on the lateral sides

Fig. 7. Comparisons between 𝐷𝑓 at 𝜎1 = 1 of PRSD and previous ship domains in restricted areas.

Coldwell (1983) and Hansen et al. (2013) remain static ones, only focus on the size of ship regardless of the dynamic features. A threat zone which accounts for dynamic changes in speeds. The size of the threat zone is dynamically adjusted throughout the navigational process (Tran et al., 2002). Among past ship domains for restricted areas, only ship domains in Kijima and Furukawa (2003) and Wang and Chin (2016) are dynamic ones. However, size of ship domain in Wang and Chin (2016) is smaller than the one in Kijima and Furukawa (2003) when compared with each other. Furthermore, most of the previous models either completely lack empirical support in deriving the ship domains. An empirical investigation on the appropriate shape of the domain would be useful. The traffic movement data in Wang and Chin (2016) model was obtained from the database in the water of Singapore port and straits. This is busy and high-density waterway. Using empirical data shows that the size of this domain is considered as minimum safety distance for restricted area. Furthermore, the resulting polygon is actually quite close to an ellipse – a shape suggested by past works (Fujii and Tanaka, 1971; Coldwell, 1983; Hansen et al., 2013), which 7

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Fig. 8. The ship domain by PRSD and previous ship domains in restricted areas in case 𝑣 = 12𝑘𝑛.

Fig. 9. The ship domain by PRSD and previous ship domains in open areas in case 𝑣 = 15𝑘𝑛, 𝐿𝑂𝐴 = 200 m.

but is greater on the longitudinal sides. It should be noted that the Fujii and Tanaka (1971) and Hansen et al. (2013) models still cannot take into account the effect of change of speed while Wang and Chin (2016) model does not enlarge significantly with increasing speed. Coldwell (1983) modeled for a head-on and an overtaking encounter, resulting in a half-elliptical ship domain and a symmetrically full-elliptical ship domain, respectively. Their port sides are closer matches to the one of ship domain by PRSD model, however, three other sides appear overestimated. Kijima and Furukawa (2003) overestimated the size of ship domain along the longitudinal and lateral sides. It would be appropriate for risk assessment described in Section 4.

and size dynamically change with various navigational situations. As mentioned before, the objective is to minimize the difference between the effective clear areas described by the proposed model and the subjected model. The difference between them can be treated as an error function, given by Eq. (16). The sum of the squared distance differences between the domain boundaries 𝐷𝜑 after applying 𝑃 𝐶𝑅 = 0.8 and the minimum ship domain for open sea area of Pietrzykowski and Urias (2009) for each bearing is chosen as fitness function, given by Eq. (17). 𝛥𝐷𝑖 = (𝐷𝜑 − 𝑃𝜑 )2

3.2. In open sea areas 𝛥𝐷 =

Unlike restricted areas, open sea areas have lesser traffic and ample sea room. As expected, the ship domain for the open ocean is greater than that in restricted waters due to more degrees of freedom. Similarly, the ship domain in high-density water is smaller than that in a water with less traffic. It means that larger size of ship domain by PRSD model in open sea areas is required, then another set value of parameters needs to be identified. To determine the size of ship domain by PRSD model in open sea areas, the similar procedure described in Section 3.1 is adopted for the parameters estimation. However, one issue should be mentioned that our objective is to propose a danger criterion for collision assessment for the area around the ship, which divided into four zones by three boundaries (in Section 4) based on the value of 𝑃 𝐶𝑅, that can be used in both restricted and open sea areas. Therefore, after 𝑃 𝐶𝑅 = 0.8 was determined for restricted area, this value was also directly applied for open sea area as the value for minimum area in both areas. Only 𝜎, 𝜆 are estimated and 𝑃 𝐶𝑅 = 0.8 is set in advance for the process of approximation. To find out an appropriate for size of ship domain by PRSD model in open sea areas, a dynamic model that proposes a general boundary of ship domain in these areas is required. Furthermore, ship length also should be considered in this model. For this purpose, the ship domain in Pietrzykowski and Urias (2009) is subject to an approximation. Pietrzykowski and Urias (2009) extended and generalized the fuzzy boundary of ship domain: an area surrounding the ship that the navigator should keep clear of other vessels and objects, whose shape

∑

𝛥𝐷𝑖

(16)

(17)

where 𝐷𝜑 : length of radius of ship domain by PRSD model at bearing 𝜑; 𝑃𝜑 : length of radius of minimum ship domain of Pietrzykowski and Urias (2009) at bearing 𝜑; 𝛥𝐷𝑖 : squared distance difference in the i bearing; 𝛥𝐷: sum of squared distance differences for all analyzed bearings. The radius 𝐷𝜑 of ship domain by PRSD model is then compared with the known radius of minimum ship domain of Pietrzykowski and Urias (2009) at same bearing 𝜑. This provides a measure of the error between them. The fitness function of the optimization problem for ship domain by PRSD model in open sea areas involving a set of two parameters: 𝜎, 𝜆. The parameter values can be estimated such that sum of the squared distance differences is minimal (𝑚𝑖𝑛𝛥𝐷). Detailed results obtained in open sea area after approximation process are 𝜎 = 1.25 and 𝜆 = 0.02. In order to valid the generated size of ship domain by PRSD model, the case of ship with 𝐿𝑂𝐴 = 200 m at the speed of 15 knots is examined. After applying 𝑃 𝐶𝑅 = 0.8, 𝜎 = 1.25, 𝜆 = 0.02 to Eq. (12), the size comparison with existing ship domains in open sea areas is represented in Table 2 and Fig. 9. Goodwin (1975) has three unequal sectors with different radii, which are invariant with ship speed and length. There are clearly distinctive differences between the proposed ship domain and model of Goodwin (1975). Goodwin (1975) oversimplifies the space domain with large discontinuities at the sector boundaries. In addition, it overestimates the space requirements, particularly on the port and starboard sides. Compared to Smierzchalski (2000), the ship domain by PRSD 8

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N. Im and T.N. Luong Table 2 Dimension of ship domains in open sea areas in case 𝑣 = 15𝑘𝑛, 𝐿𝑂𝐴 = 200 m. Ship domain

Size

Smierzchalski Pietrzykowski (min) Pietrzykowski (mean) Pietrzykowski (max)

Fore

Aft

1.5 1.5 1.5 1.5

0.5 0.5 0.5 0.5

Goodwin

nm nm nm nm

nm nm nm nm

Starboard

Port

0.75 0.75 0.75 0.75

0.5 0.5 0.5 0.5

nm nm nm nm

nm nm nm nm

Starboard sector: 2.4 nm; Port sector: 2.4 nm; Aft sector: 0.9 nm

PRSD

1.35 nm

0.64 nm

Table 3 Recommendation for value of influence parameter for different sea areas.

0.64 nm

0.64 nm

4. Collision assessment zone using potential collision risk

Area

𝑃 𝐶𝑅

𝜎

𝜆

Restricted areas or high-density traffic area Open areas or low-density traffic area

0.8 0.8

0.35 1.25

0.03 0.02

The area around a ship is simply divided into two zones in most of existing ship domains: inside the ship domain: ‘‘dangerous area’’ and outside the ship domain: ‘‘safe area’’. However, this kind of ship domain with a clear boundary is limited in applications. When two ships in sight of each other are approaching with no change of compass bearing, so that when there is risk of collision one of them, there may be four stages relating to the permitted or required action for each ship (Cockcroft and Lameijer, 2003).

model also compatible. There is very good match for minimum ship domain of Pietrzykowski and Urias (2009), particularly in the fore side and aft side. This means that our ship domain by PRSD model for open waters is compatible with Pietrzykowski and Urias (2009) model when the safety level is high. It is well suggested that the ship domain by PRSD model is suitable in open sea areas.

• at long range, before risk of collision exists, both ships are free to take any action • when risk of collision first begins to apply the give-way ship is required to take early and substantial action to achieve a safe passing distance and the other ship must keep her course and speed • when it becomes apparent that the give-way ship is not taking appropriate action in compliance with the Rules the stand-on ship is permitted to take action to avoid collision by her maneuver alone • when collision cannot be avoided by the give-way ship alone the stand-on vessel is required to take such action as will best aid to avoid collision.

3.3. Summary In this step, the size of ship domain by PRSD model that is identified. First, the parameters for ship domain in restricted areas were determined. To do that, it necessary to conduct a comprehensive comparison related to the shape of previous ship domains. Due to the number of parameters in the ship domain model, the Least Squares method is chosen as the approximation technique. The best-fit model can be obtained by defining a fitness function with the objective to minimize the difference between the boundaries of proposed domain and the selected domain Wang and Chin (2016). The fitness function for determining the size of ship domain by PRSD model is formulated as the sum of the squared distance differences between radii of two domains. There is a minimum value for this fitness function and the most suitable parameters are obtained. The same procedure is also carried out in order to obtain the size of ship domain by PRSD model in open sea areas. The minimum ship domain in open sea areas of Pietrzykowski and Urias (2009) is selected as a subject domain for approximation. The results of the size of PRSD model seem reasonable in accordance with existing ship domain models.

The distances at which the various stages begin to apply will vary considerably. They will be much greater for high speed ships. In past, the determination of the two boundaries or limits for action, close quarters range and critical distance were considered together (Hilgert, 1983). Ship domains with two areas such as blocking area and watching area (action area) were developed by Kijima and Furukawa (2003) and Dinh and Im (2016). If the neighborhood of a ship can be split into many zones, it would be more convenient for navigators to distinguish the danger from wide range. The navigator needs criteria like those in navigation. Such criteria are necessary for the navigator to make decisions in collision avoidance. By using the value of 𝑃 𝐶𝑅, the concept of PRSD model with three boundaries that divide the area around a ship, named Collision Assessment zone, into four zones with different potential risk levels are represented in Fig. 10. These four zones are: Safe Zone (SZ), Cautious Zone (CZ), Dangerous Zone (DZ) and Very Dangerous Zone (VDZ) and the required actions in each zone are given in Table 4. As mentioned in Section 3.3, the upper boundary for very dangerous zone of PRSD model is corresponding with value of 𝑃 𝐶𝑅 = 0.8. If the 𝑃 𝐶𝑅 of any target ship is more than 0.8 within this boundary and more than 0.8 outside this boundary. It should be ensured that the target ship does not cross this boundary or keeps a distance from it. The next step is to estimate dangerous and cautious boundaries based on value of 𝑃 𝐶𝑅. For estimation, the Least Squares method is applied again. In this case, only 𝑃 𝐶𝑅 needs to be estimated. A ship domain with series of boundary is required for approximation. Among existing ship domains, the ones of Kijima and Furukawa (2003), Dinh and Im (2016) and Hilgert (1983) consist of more than two zones. However, only the model of Kijima and Furukawa (2003) takes account into ship length and ship speed, similar with PRSD model. Kijima and

Parameters for PRSD model in two areas are shown in Table 3 Size of ship domain by PRSD model in restricted areas is determined as follows: √ −2×ln(0.8) ⎧ × 𝑣 × 0.35 × 𝐿 ⎪ 0.03𝑣2 𝑠𝑖𝑛𝜑𝑝 2 +𝑐𝑜𝑠𝜑𝑝 2 ⎪ if 0◦ ⩽ 𝜑 ⩽ 90◦ or 270◦ ⩽ 𝜑 ⩽ 360◦ 𝑝 𝑝 (18) 𝐷 𝜑 = ⎨√ ⎪ −2×ln(0.8) × 𝑣 × 0.35 × 𝐿 0.03 ⎪ ⎩ if 90◦ < 𝜑𝑝 < 270◦ Size of ship domain by PRSD model in open sea areas is determined as follows: √ −2×ln(0.8) ⎧ × 𝑣 × 1.25 × 𝐿 ⎪ 0.02𝑣2 𝑠𝑖𝑛𝜑𝑝 2 +𝑐𝑜𝑠𝜑𝑝 2 ⎪ ◦ ◦ ◦ ◦ 𝐷𝜑 = ⎨√if 0 ⩽ 𝜑𝑝 ⩽ 90 or 270 ⩽ 𝜑𝑝 ⩽ 360 (19) ⎪ −2×ln(0.8) × 𝑣 × 1.25 × 𝐿 0.02 ⎪ ⎩ if 90◦ < 𝜑𝑝 < 270◦ 9

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Radii of the watching area of Kijima and Furukawa (2003) are given by: ⎧ ⎪𝑅𝑤𝑓 = 2𝑅𝑏𝑓 + 1 ⎪𝑅𝑤𝑎 = 2𝑅𝑏𝑎 + 1 ⎨𝑅 = 2𝑅 + 1 𝑏𝑠 ⎪ 𝑤𝑠 ⎪𝑅𝑤𝑝 = 2𝑅𝑏𝑝 + 1 ⎩

(21)

where 𝑅𝑏𝑓 , 𝑅𝑏𝑎 , 𝑅𝑏𝑠 , 𝑅𝑏𝑝 : radius of blocking area of Kijima and Furukawa (2003) in fore, aft, starboard and port side, respectively; 𝑅𝑤𝑓 , 𝑅𝑤𝑎 , 𝑅𝑤𝑠 , 𝑅𝑤𝑝 : radius of watching area of Kijima and Furukawa (2003) in fore, aft, starboard and port side, respectively; 𝐿 is the own ship length; 𝑘𝐴𝐷 and 𝑘𝐷𝑇 represent gains of the advance 𝐴𝐷 and the tactical diameter 𝐷𝑇 respectively. ⎧ ⎪𝑅𝑤𝑓 = 2𝑅𝑏𝑓 + 1 ⎪𝑅𝑤𝑎 = 2𝑅𝑏𝑎 + 1 ⎨𝑅 = 2𝑅 + 1 𝑏𝑠 ⎪ 𝑤𝑠 ⎪𝑅𝑤𝑝 = 2𝑅𝑏𝑝 + 1 ⎩

The fitness function to determine 𝑃 𝐶𝑅 value for dangerous boundary is given by:

Fig. 10. Collision Assessment zone divided by three boundaries.

⎧ 2 ⎪𝛥𝑏𝑓 = (𝐷𝑓 − 𝑅𝑏𝑓 ) 2 ⎪𝛥𝑏𝑎 = (𝐷𝑎 − 𝑅𝑏𝑎 ) ⎨𝛥𝑏 = (𝐷 − 𝑅 )2 𝑠 𝑏𝑠 ⎪ 𝑠 ⎪𝛥𝑏𝑝 = (𝐷𝑝 − 𝑅𝑏𝑝 )2 ⎩

Table 4 New proposal of Collision Assessment zone around a ship. Collision Assessment zone

Definition

Action requirement

Safe zone

At long range, before risk of collision exists

Free to take any action

Cautious zone

Collision risk first begins to exist

Maintain a proper look-out, take all the necessary precautions and make a full appraisal of the situation and of the risk of collision

Dangerous zone

Collision risk becomes apparent

Adherence to COLREGs and take preparations to avoid any kind of accident and sail through such areas

Very dangerous zone

Collision risk becomes critical

Take actions to keep clear of other vessels

(22)

𝛥𝑏 =

4 ∑∑

𝛥𝑏𝑖

(23)

(24)

𝑣 𝑖=1

where 𝛥𝑏𝑓 , 𝛥𝑏𝑎 , 𝛥𝑏𝑠 , 𝛥𝑏𝑝 : squared distance difference at fore, aft, starboard and port side, respectively; 𝛥𝑏: sum of squared distance differences with the blocking area of Kijima and Furukawa (2003). The fitness function to determine 𝑃 𝐶𝑅 value for cautious boundary is given by: ⎧ 2 ⎪𝛥𝑤𝑓 = (𝐷𝑓 − 𝑅𝑤𝑓 ) ⎪𝛥𝑤𝑎 = (𝐷𝑎 − 𝑅𝑤𝑎 )2 ⎨𝛥𝑤 = (𝐷 − 𝑅 )2 𝑠 𝑤𝑠 ⎪ 𝑠 ⎪𝛥𝑤𝑝 = (𝐷𝑝 − 𝑅𝑤𝑝 )2 ⎩

Furukawa (2003) decided to utilize ‘‘Safety blocking area’’ and ‘‘Blocking area with space’’ of Arimura et al. (1994) to evaluate collision risk with other ships. Hereafter, those domains are simply called as Blocking area and Watching area, respectively. They also declared the general action for ship in these two areas. According to Kijima and Furukawa (2003), existing ship in the Watching area is regarded as target ship which should be watched out for by own ship. When the Watching area of the target ship invades the Blocking area, own ship should decide whether she will change or keep her course. These areas are based on traffic regulation and expressed as a relation of elements such as tactical performance, speed and ship length. Their validity was examined by numerical simulations. Therefore, Kijima and Furukawa (2003) model with blocking area and watching area is subject domain for approximation of dangerous boundary and cautious boundary of PRSD model, respectively. Since the model of Kijima and Furukawa (2003) is for restricted areas, the set value of influence parameters for this area (𝜎 = 0.35, 𝜆 = 0.03) is adopted. Then, two fitness functions are defined. Radii of the blocking area of Kijima and Furukawa (2003) are given by: √ ⎧𝑅 = (1 + 1.34 𝑘2 + 0.25𝑘2 ) × 𝐿 𝑏𝑓 𝐷𝑇 ⎪ √ 𝐴𝐷 ⎪𝑅 = (1 + 0.67 𝑘2 + 0.25𝑘2 ) × 𝐿 (20) 𝐴𝐷 𝐷𝑇 ⎨ 𝑏𝑎 ⎪𝑅𝑏𝑠 = (0.2 + 𝑘𝐷𝑇 ) × 𝐿 ⎪ ⎩𝑅𝑏𝑝 = (0.2 + 0.75𝑘𝐷𝑇 ) × 𝐿

𝛥𝑤 =

4 ∑∑

𝛥𝑤𝑖

(25)

(26)

𝑣 𝑖=1

where 𝛥𝑤𝑓 , 𝛥𝑤𝑎 , 𝛥𝑤𝑠 , 𝛥𝑤𝑝 : squared distance difference at fore, aft, starboard and port side, respectively; 𝛥𝑤: sum of squared distance differences with the watching area of Kijima and Furukawa (2003). The estimations for dangerous boundary and cautious boundary are conducted separately. The former is estimated by comparison with blocking area while the latter is estimated by comparison with the watching area of Kijima and Furukawa (2003). At this time, only 𝑃 𝐶𝑅 is involved in the fitness functions of the approximation. The fitness functions are defined by summing up the squared distance difference between four radii of PRSD model and (Kijima and Furukawa, 2003) model (𝛥𝑏 to estimate PCR value for dangerous boundary by Eq. (24), 𝛥𝑤 to estimate PCR value for cautious boundary by Eq. (26). The aim is to find out corresponding values of 𝑃 𝐶𝑅 with minimum 𝛥𝑏 and 𝛥𝑤. Implementing the same method as for restricted areas, the boundaries of dangerous zone and cautious zone of PRSD model according to values of 𝑃 𝐶𝑅 are figured out. The results of 𝑃 𝐶𝑅 value for boundaries are shown in Table 5. To verify the size of these boundaries, the same comparison with past ship domains in restricted areas as in Section 3.1 is carried out and presented Fig. 11. 10

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N. Im and T.N. Luong Table 5 𝑃 𝐶𝑅 values for three boundaries. Boundary

𝑃 𝐶𝑅

Cautious boundary Dangerous boundary Very dangerous boundary

0.01 0.4 0.8

Fig. 12. Ship domains in open sea areas in case 𝑣 = 15𝑘𝑛, 𝐿𝑂𝐴 = 200 m.

Table 6 Criterion of Collision Assessment zone around a ship. Collision Assessment 𝑃 𝐶𝑅 zone Fig. 11. Ship domains in restricted areas in case 𝑣 = 12𝑘𝑛.

It is clearly seen that the cautious boundary of PRSD model is almost the same as boundary of Kijima and Furukawa (2003) watching area. The comparison shows that this boundary could be used as boundary for ship or obstacle moving into cautious zone should be watched out for by own ship. In head-on situation, ship domain in Coldwell (1983) on the starboard side is assumed larger than that on the port side base on the general preference of navigators to pass on the port side instead of the starboard side. It is close to dangerous boundary of PRSD model in fore and starboard side. The dangerous boundary of PRSD is also quite similar with the blocking area in Kijima and Furukawa (2003), only slightly smaller on aft and starboard sides. When other ship or obstacle moving across the dangerous boundary, own ship should decide whether to act for collision avoidance. In order to see if these boundaries of PRSD model are suitable in open sea areas, we can compare it after applying 𝑃 𝐶𝑅 = 0.01 and 𝑃 𝐶𝑅 = 0.4 with other ship domains (as can be shown in Fig. 12). The cautious boundary significantly greater than other ship domains in open sea areas. In open sea areas, ships usually navigate at high speed, so they should be given sufficient sea-room for watching others. There are again good matches between the dangerous boundary of PRSD model and (Pietrzykowski and Urias, 2009) maximum ship domain in Pietrzykowski and Urias (2009) and ship domain of Goodwin (1975), particularly on the fore and aft side, although these two ship domains show a larger increase on the port and starboard sides. It may be concluded that the cautious and dangerous boundaries in both restricted areas and open sea areas are reasonable when correspondingly applying 𝑃 𝐶𝑅 = 0.01 and 𝑃 𝐶𝑅 = 0.4. Based on these values, Collision Assessment zone around a ship is divided into four zones by three boundaries can be defined in Table 6. The summary of the proposed PRSD model’s contribution, when compared with past ship domains, is given in Table 7. Similar with previous ship domains, the PRSD model also proposes an area where other ships need to avoid entering. The size of ship domain by PRSD model is dynamically changed due to the changing in speed during

Evaluation

Very dangerous zone

0.8 ≤ 𝑃 𝐶𝑅 < 1

Other ships are never allowed to be entered, because this area is very close to own ship, target ship surpassed this area of own ship domain

Dangerous zone

0.4 ≤ 𝑃 𝐶𝑅 < 0.8

Other ships probably enter the very dangerous area and should avoid this area as soon as possible

Cautious zone

0.01 ≤ 𝑃 𝐶𝑅 < 0.4

Other ships are inside boundary of watching area and should pay attention to the upcoming situation

Safe zone

0 < 𝑃 𝐶𝑅 < 0.01

Other ships are out of boundary of watching area

navigation. It depends on the value of parameters as well, leading to the ability to apply in both restricted areas and open sea areas just by adopting different set of value of parameters. Another new function of PRSD model is to calculate continuous and dynamic potential collision risk level of any point in area around the ship. It has been solved the quantitative quantification problem of existing deterministic domain. The developed series of boundaries that divided Collision Assessment zone into four zones combination with criterion for risk evaluation seems to be powerful in determining the navigational level of collision risk. The feature makes the model applicable in assessing the risk of collisions. Furthermore, when two ships are approaching each other, their domains can be overlapped, and the visualization can help to predict the potential collision area, that will be represented in Section 5. 5. Application of the proposed PRSD In this section, the encounter situations of real accidents are carried out based on AIS data to demonstrate the validity and superiority of the presented PRSD model. The results are illustrated in multiple figures, representing snapshots of situations. A two-dimensional Cartesian coordinate system presents the distance in nautical miles (nm); the vertical axis in the positive direction shows North 0◦ , and the horizontal axis in the positive direction is 90◦ . 11

Ocean Engineering 194 (2019) 106610

N. Im and T.N. Luong Table 7 A summary of new features in PRSD model.

Table 8 Positions and motion parameters of two ships in scenario 1.

Function

Fujii

Goodwin

Wang

Pietrzykowski

Kijima

PRSD

Ship domain Dynamic size All sea areas supported Series of boundaries Overlapped domain Real-time potential collision risk calculation Risk evaluation criteria by zones





 

  

 

  

𝑂 𝑂

𝑂

𝑂 𝑂 𝑂

𝑂 𝑂 𝑂

𝑂 𝑂 𝑂

𝑂 𝑂



𝑂 𝑂 𝑂

  

𝑂

𝑂

𝑂

𝑂

𝑂





𝑂

𝑂

t (s)

0 60 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200 1260

Distance (nm)

7.82 7.40 6.90 6.49 6.02 5.57 5.10 4.60 4.18 3.80 3.38 2.95 2.54 2.15 1.76 1.48 1.18 0.93 0.71 0.49 0.27 Collision

Ship A

Ship B

Heading (degree)

Speed (kn)

Heading (degree)

Speed (kn)

84.0 83.8 84.1 84.0 84.8 84.5 84.2 83.6 84.6 84.4 84.0 83.2 83.0 83.5 83.7 84.1 83.2 84.0 83.8 84.0 83.7 83.5

15.6 15.4 15.2 15.7 15.3 15.1 14.9 14.6 14.3 14.0 13.8 13.6 13.5 12.4 11.6 10.2 9.6 8.4 7.6 7.5 6.6 5.3

229.5 231.3 230.2 229.5 229.9 231.6 230.5 229.0 231.2 233.3 232.4 229.3 231.7 236.7 249.0 247.2 233.6 255.8 235.8 230.8 230.0 218.8

12.4 12.6 12.4 12.8 12.1 11.9 11.7 11.5 11.2 11.0 10.4 10.4 10.3 9.4 8.8 8.3 7.4 6.2 5.8 5.5 5.2 4.4

maintained their courses and started to reduce their speeds to avoid this area. At time step 540, 𝑃 𝐶𝑅 of ship B (seen from ship A) was 0.01, it means that ship B was in the cautious boundary of ship A. Ship A should start to watch if ship B has any action. Distance between two ships at this moment was 3.8 nm. However, ship B did not alter her course and two ships were still approaching each other. Arguably, both ships’ courses were continuously consistent with small fluctuations. Dangerous and very dangerous boundaries were overlapped at time step 660 (Fig. 14c) and 840 (as Fig. 14d), respectively. It means that the potential collision area was becoming closer to two ships and the collision probability was also greater. From time step 660 to 840, ship B tried to turn to starboard. However, due to the bad weather conditions on the route, the ability to maneuver of ship B became restricted and it was difficult to maneuver the ship as desired. The high winds and waves made the heading of ship B fluctuated. At time step 840 and in the range of 1.8 nm, it would be still enough time for both ships to avoid collisions. However, collision finally occurred due to no effective actions taken from two ships (as Fig. 14e). Size of a single ship domain is changing due to the change in speeds (represented by fore radii of three boundaries in Fig. 15). Due to the different between length and speed, ship A had a larger domain than ship B. From start to time step 180, both sizes of two ship domains were almost keeping the same because the speeds were not changed. At time step 180, two ships started to reduce their own speeds but did not change their own courses. At same moment, the cautious boundaries started to be overlapped. When two ships were approaching each other, the distance became closer, dangerous boundaries were overlapped, it would be safe to combine of changing courses and speeds by two ships. Ship A collided with ship B had none of them change their courses despite decreasing speed. When two ships were navigating at their full speeds, the size of cautious boundaries of ship A and B are around 4.2 nm and 2.4 nm, respectively. During the period (0–540 s), the distance between two ships are greater than the cautious boundaries of both ships (minimum 3.8 nm at time step 540). This distance can be considered as safe distance in open sea area. After that, the distance continued to reduce. From time step 540 to 840, it was smaller than the cautious boundary of ship A but still larger than the one of ship B. It means that ship B entered the cautious zone of ship A sooner. It can be explained that ship A is bigger and sailing at higher speed, so her domain should be larger than ship B’s. At time step 960, when

Fig. 13. Location (red mark on the map) and ships trajectories of the scenario 1.

5.1. Scenario 1: Teaan accident A simulation of collision in west coastal waters of South Korea is carried out. The collision happened at 7: 10A.M, 19th January 2012. The position of collision was at 36.99◦ 𝑁, 126.10◦ 𝐸 and has been plotted as red mark in Fig. 13. The parameters for open sea areas are used in these simulations: 𝜎 = 1.25 and 𝜆 = 0.02. Fig. 13 also displays the two ships, labeled as ship A and ship B, both the vessels are moving with different speeds and courses. Ship A is believed to be moving in the east–northeast direction while the ship B is moving towards the south-west direction. A detail of distance between two ships, ships’ positions and motion parameters of each ship during navigation is given in Table 8. An important issue should be mentioned here. When two ships are moving closer to each other, their respective ship domains will have an overlap. The overlapping area, which can be understood as the potential collision area is the collision position if the two ships do not take measures to avoid collision. This area will vary at different moments, from cautious boundary to very dangerous boundary. To a certain extent, the changes of the overlapping area also demonstrate the degree of the collision probability in a moment of encounter situation. The more inside that the overlapping boundaries are, the bigger that the risk of collision is between the two ships and the greater the collision probability. Collision Assessment zones of two ships during navigation using PRSD model is illustrated in Fig. 14. Two ships would collide if none of them change their courses. Theoretically, making decision in crossing encounter situations should be easy: only one of the ships is supposed to maneuver and the direction of a turn is specified by COLREGS. Since ship A is on starboard, it is ship B that should maneuver, if possible, to starboard. As Fig. 14b informed us, at time step 240, two ships’ cautious boundaries were overlapped and distance between them was around 6 nm. This overlapping area would be likely the collision area. The 𝑃 𝐶𝑅 value of any point in this area (seen from both ships) would be greater than 0.01. Since then, two ships only 12

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Fig. 14. (a) When cautious boundaries are not overlapped (time step 0); (b) When cautious boundaries are overlapped (time step 240); (c) When dangerous boundaries are overlapped (time step 660); (d) When very dangerous boundaries are overlapped (time step 840); (e) Collision happened (time step 1260).

the distance was 1.18 nm, same as dangerous boundary of ship A, it showed that ship B was going inside the dangerous zone of ship A. This distance is kindly close and danger in this situation. When two ships were coming closer (0.49 nm at time step 1140), it was also the

boundary of very dangerous zone of ship A. After this boundary was trespassed, collision happened. Fig. 16 shows the 𝑃 𝐶𝑅 value of two ships during approaching each other. Ship B entered cautious zone, dangerous zone and very 13

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Fig. 15. Dynamic 𝐷𝑓 of 2 ships in real time.

Fig. 17. Location (red mark on the map) and ships trajectories of the scenario 2.

Table 9 Positions and motion parameters of two ships in scenario 2. t (s)

0 60 120 180 240 300 360 420 480 540 600 660 720 780

Fig. 16. Dynamic 𝑃 𝐶𝑅 indexes of 2 ships in real time.

dangerous zone of ship A earlier. It means that 𝑃 𝐶𝑅 of ship B reached 0.01, 0.4 and 0.8 sooner than ship A did. The reason is that ship A is longer and was navigating with higher speed, so the Collision Assessment zone around ship A was larger than the one of ship B. Therefore, ship B crossed each boundary of ship A sooner. When ship B was seen crossing into cautious zone from ship A (time step 540), it should be watched if 𝑃 𝐶𝑅 was increasing. Ship B continued to keep direction and surpassed into dangerous zone (𝑃 𝐶𝑅 reached 0.4) from time step 540 to 960. Navigators should have action to avoid collision. However, both ships had no action and very dangerous zones of two ships were invaded. 𝑃 𝐶𝑅s of two ships reached almost 1 when collision.

Distance (nm)

0.76 0.66 0.56 0.44 0.34 0.23 0.19 0.16 0.19 0.18 0.17 0.11 0.06 0

Ship A

Ship B

Heading (degree)

Speed (kn)

Heading (degree)

Speed (kn)

105.6 104.2 102.7 101.5 98.1 93.1 95.2 92.3 89.2 86.1 88.4 94.3 91.5 85.1

12.8 12.7 12.9 12.8 12.8 12.7 12.5 12.3 12.2 12.6 12.2 11.6 11.5 10.3

88.6 87.7 89.5 89.7 90.4 92.6 94.6 97.5 95 85.1 75.4 75.8 75.7 74.7

12.5 12.3 12.6 12.6 12.4 12.5 12.7 12.5 12.2 12.1 12 11.9 11.8 11.7

two ships were too close (0.34 nm at time step 240 and dangerous boundaries were overlapped, the distance from two ships to potential collision area were around 0.18 nm), they decided to change courses. But the appearance of a landmass probably made it hard to find a collision avoidance maneuver to starboard and the navigator may be tempted to turn to port again, leading to overlap of very dangerous boundaries. One thing should be added here is that water of collision is a high traffic area with a lot of small fishing boats around. After ship B turned to starboard, there were a couple of fishing boats in front, sailing towards ship B. Therefore, ship B had to turn back to port side again to avoid a landmark and other boats. Both ships remained their own courses and speeds until collision. Different with accident 1, when two ship were coming closer to each other, they both attempted to change their own course but keep own speeds. Therefore, the size of ship domains was changed slightly during navigation (as Fig. 19). Two ships were navigating in high-density and busy water. The distance between two ships was reducing, smaller than cautious boundaries (around 0.6 nm). According to the situation of area and the size of both ships, this distance could be enough for minimum passing distance. However, when the distance was smaller than dangerous boundaries (around 0.26 nm), it was dangerous for both ships. At time step 660, when the distance was smaller than very dangerous boundaries (around 0.12 nm), both ships were in critical situation. As we can see from Fig. 20, both ships entered cautious zones of each other (𝑃 𝐶𝑅 values of two ships reached around 0.01) at time step 240 and the distance was 0.34 nm. Since two ships were approaching from side, the required action at this moment was changing direction.

5.2. Scenario 2: Jindo accident Another accident in south coastal waters of South Korea is carried out to verify the usefulness of PRSD model in restricted areas. The collision happened at 8: 10A.M, 10th August 2013. The position of collision was at 34.35◦ 𝑁, 126.09◦ 𝐸 and has been plotted as red mark in Fig. 17. The parameters for restricted areas were used in these simulations: 𝜎 = 0.35 and 𝜆 = 0.03 due to the high-density traffic. Trajectories of two ships in this accident is displayed in Fig. 17. Ship A is moving in the east–southeast direction while the ship B is moving in the east–northeast direction. Table 9 shows distance between two ships, positions and motion parameters at every time step. Fig. 18 displays Collision Assessment zones of two ships during navigation. It would be safe if ship A altered to starboard according to COLREGs when the range was around 0.75 nm. Lateral radii of ship A and ship B were around 0.28 nm and 0.24 nm, respectively. The cautious boundaries were overlapped from the side where the potential collision area is likely. However, ship A chose to keep her course. When 14

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Fig. 18. (a) When cautious boundaries are not overlapped (time step 0); (b) When cautious boundaries are overlapped (time step 240); (c) When dangerous boundaries are overlapped (time step 660); (d) When very dangerous boundaries are overlapped (time step 600); (e) Collision happened (time step 780).

However, when ship B came inside the dangerous zone of ship A (at

from landmarks, it maneuvered to port again and did not reduce speed

time step 420), ship B had action to avoid collision by changing to

then dangerous zones were invade again, following by very dangerous

starboard. Then they came back to cautious zones (𝑃 𝐶𝑅s decreases to

zone (𝑃 𝐶𝑅s continued to increase again over 0.4 and 0.8, respectively)

less than 0.4). However, the fact that ship B also tried to steer away

until collision happened. 15

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to 𝐷𝐶𝑃 𝐴 and 𝑇 𝐶𝑃 𝐴, offering a picture of risk zone around a ship. Ship length, speed as well as navigation situation on the size of ship domain by PRSD model are considered. Then, the size of PRSD model in restricted and open sea areas are obtained by parameters determination based on the comparison with existing ship domains through approximation method. Finally, a Collision Assessment zone around the ship can be divided into four zones by three boundaries with different levels of risk. Such zones can recommend action for maneuver and make the navigator’s work easier. To illustrate the effectiveness of the proposed PRSD model, two real accidents within the coast of South Korea are simulated by AIS data. By using the criterion in accordance with the Rules, collisions may be avoided with a high probability. The results show that PRSD model is effective and efficient to detect and evaluate risk of collision in real-time, has a good adaptability to any water type and can be applied for risk assessment of ship collision accidents as a decision support tool for maritime safety administration to improve navigation safety. PRSD model can also be used as reference for the evaluation and supervision of VTS centers in vessel traffic management.

Fig. 19. Dynamic 𝐷𝑓 of 2 ships in real time.

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Fig. 20. Dynamic 𝑃 𝐶𝑅 indexes of 2 ships in real time.

5.3. Discussion Collision candidate is encountered ships which are in a situation that they would collide with each other if no evasive maneuver was performed. It indicates ships with geometrical possibility of collision (Chen et al., 2017). According to the definition, the ship collision avoidance is defined as prediction and avoidance. Prediction is to forecast the target ship when and where will stay the same point or have collision risk with own ship on the sea. Avoidance is that the action been taken by both ships, so that the two ship are not simultaneously occupy the same point or avoid encountering the situation of risk of collision (Liu and Xiao, 2014). Through the above simulations, some following conclusions can be proved: 1. dynamic extended zones around a ship can be obtained by PRSD model, delineate the area required for safe, collision escape maneuvers; 2. overlapping area of ship domains can predict the potential collision area, showing at which sub-zone in Collision Assessment zone of ship this area is, and the target ship with dynamic PCR value will help the own ship quickly evaluate the situation and have plan for collision evasive maneuvers; In the future, we will conduct a more comprehensive analysis to making PRSD model suitable for a variety of complex environments with multiple ships. 6. Conclusion This paper proposes a novel ship domain, named Potential Risk Ship Domain (PRSD), for real-time potential collision risk assessment. First, kernel-density function is applied to construct ship domain according 16

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