- Email: [email protected]
The ship maneuverability based collision avoidance dynamic support system in close-quarters situation
Ocean Engineering 146 (2017) 486–497
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
The ship maneuverability based collision avoidance dynamic support system in close-quarters situation Xin Wang *, Zhengjiang Liu, Yao Cai Navigation College, Dalian Maritime University, Dalian 116026, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Close-quarters situation Collision avoidance support system Ship maneuverability
In this article, a ship maneuverability based collision avoidance dynamic support system in close-quarters situation is presented. The dynamic calculation model of collision avoidance parameter is employed to calculate the dynamic DCPA and TCPA in real-time when ship is maneuvering. Then the collision avoidance dynamic support system is developed by combining the mathematical model of ship maneuvering motion, the control mechanism of ship maneuvering motion and the dynamic calculation model of collision avoidance parameter. Following this approach, the proposed system is able to eliminate the insufficiency of neglect of ship maneuverability in the process of avoiding collision. Moreover, by incorporating the close-quarters situation into the proposed collision avoidance dynamic support system, simulation examples consisting three encounter scenarios of two ships in close-quarters situation are applied to demonstrate the significance and necessity of ship maneuverability in the process of collision avoidance and illustrate the merits and effectiveness of the proposed system. The simulation results show that the proposed dynamic support system is a reasonable, effective and practicable system for collision avoidance, particularly in close-quarters situation.
1. Introduction In the past decades, the collision accidents in maritime traffic engineering have drawn much attention owing to maritime safety and environmental protection (Eleftheria et al., 2016). In particularly, the collision avoidance system is recognized as an available way to prevention of the ships from collisions (Tsou et al., 2010). Through years of progress, a number of maritime collision avoidance systems or/and models have achieved many significant results (Ahn et al., 2012; Chai et al., 2017; Goerlandt and Kujala, 2014; Goerlandt and Montewka, 2015; Goerlandt et al., 2015; Hwang, 2002; Li and Pang, 2013; Tam and Bucknall, 2013; Xue et al., 2011; Zhang et al., 2015), such as autonomous collision avoidance system, decision making system etc. For example, a fuzzy collision avoidance expert system which include the experiences of experts was designed to resolve the problems of collision (Hwang, 2002); a framework for risk-informed maritime collision alert system was proposed for different navigational environments (Goerlandt et al., 2015); and a distributed anti-collision decision support formulation was studied in multi-ship encounter situations under the International Regulations for Preventing Collisions at Sea (COLREGs) (Zhang et al., 2015). Nevertheless, a shortage is still exists in above-mentioned collision avoidance systems, that is they could not forecast the Distance at Closest
Point of Approach (DCPA) and Time to the Closest Point of Approach (TCPA) if a certain action is taken by ship to avoid collision, and indicate the variation trend of DCPA and TCPA if a series of actions in a proper range is taken. In practice, the collision avoidance manoeuvres for another ships or obstacles are usually performed under the navigators’ own judgement nowadays. Hence, in order to assist the navigators to judge which and how large the action shall be taken, the collision avoidance support system which can indicate whole as well as real-time DCPA and TCPA should be studied. In addition, ships encounter in close-quarters situation is still a major and intractable issue in maritime collision avoidance. The reason lies on that there are not unified quantitative and qualitative interpretations on the term “close-quarters situation” in the world, and the exact definition is also not given in the COLREGs by far. The 1972 International Maritime Organization (IMO) conference ever considered the possibility of specifying the distance at which it would begin to apply, but after a lengthy discussion, it was decided that this distance could not be quantified. In the following decades, several literatures (Cahill, 1982; Hilgert, 1983; International Maritime Organization (IMO), 1981; Perera et al., 2011; Tam et al., 2009; Tam and Bucknall, 2010a, 2010b; Zhang and Zhao, 1991) have presented some different interpretations for the term “closequarters situation” and studied for the collision avoidance in close range
* Corresponding author. E-mail address: [email protected] (X. Wang). http://dx.doi.org/10.1016/j.oceaneng.2017.08.034 Received 27 February 2017; Received in revised form 25 June 2017; Accepted 18 August 2017 Available online 6 September 2017 0029-8018/© 2017 Elsevier Ltd. All rights reserved.
X. Wang et al.
Ocean Engineering 146 (2017) 486–497
is required to give the whistle signal prescribed in Rules and is permitted to take action to avoid collision by her manoeuvre alone, but a power driven ship must not alter course to port to avoid another power driven ship crossing from her own port side. The give-way ship is relieved of her obligation to keep out of the way. (4) When collision can not be avoided by the give-way ship alone, the stand-on ship is required to take such action as well best aid to avoid collision.
encounters or critical conditions. For example, according to the COLREGs, a collision risk assessment method and an optimal path planning algorithm for ships in close range encounters were presented (Tam and Bucknall, 2010a, 2010b), respectively. However, the lack of the study on ship maneuverability is existing in above-mentioned researches, i.e., most of the aforementioned researches ignore the impact of ship maneuverability in the process of avoiding collision. Generally speaking, ship maneuverability weighs the response capability when ship is handled by operator or interfered by external environment (Hong and Yang, 2012). Due to the characteristics of ship motion such as large inertia, time delay and nonlinear etc., the ship maneuverability is a major and important issue for the navigational safety of ships, especially in the collision avoidance of ships. The previous literature (Wang, 1991) has pointed that the distance at which a close-quarters situation applies depend upon many factors, including the ship maneuverability. In fact, as we know, the ship maneuverability plays a very important role in the process of collision avoidance in close-quarters situation on the basis of foregoing depictions. To the best knowledge of the authors, there is still no report on applying the ship maneuverability to the ships encounter in closequarters situation based on the collision avoidance system (or model) so far. Hence, the collision avoidance in close-quarters situation requires intensive analysis and should be modelled precisely. Motivated by above observation, and in order to study the influence of ship maneuverability in the process of collision avoidance in closequarters situation, a ship maneuverability based collision avoidance dynamic support system in close-quarters situation is presented in this article. The collision avoidance dynamic support system is developed by combining the mathematical model of ship maneuvering motion, the control mechanism of ship maneuvering motion and the dynamic calculation model of collision avoidance parameter. Following this approach, the proposed system is able to eliminate the insufficiency of neglect of ship maneuverability in the process of avoiding collision. Moreover, by incorporating the close-quarters situation into the proposed collision avoidance dynamic support system, simulation examples consisting three encounter scenarios of two ships in close-quarters situation are applied to demonstrate the significance and necessity of ship maneuverability in the process of collision avoidance and illustrate the merits and effectiveness of the proposed system. The simulation results show that the proposed dynamic support system is a reasonable, effective and practicable system for collision avoidance, particularly in close-quarters situation. The rest of the paper is organized as follows. Section 2 briefly describes the concept of close-quarters situation. Then the collision avoidance dynamic support system is described briefly in Section 3. In Section 4, the simulation results via three encounter scenarios of two ships in close-quarters situation are presented and analysed in detail. The paper ends with conclusion in Section 5.
The distances at which the various stages begin to apply will vary considerably. For a crossing situation involving two power driven ships in the open sea, it is suggested that the outer limit of the second stage might be of the order of 5–8 miles and that the outer limit for the third stage would be about 2–3 miles. 2.2. The concept of close-quarters situation The term “close-quarters situation” is presented firstly in the paragraph (c) of Rule 8 of COLREGs states (International Maritime Organization (IMO), 2001):“If there is sufficient sea room, alteration of course alone may be the most effective action to avoid a close-quarters situation provided that it is made in good time, is substantial and does not result in another close-quarters situation”. Moreover, the paragraph (d) and (e) of Rule 19 are also refer to the close quarters situation. Nevertheless, there are not unified quantitative and qualitative interpretations on the term “close-quarters situation” in the world, and the exact definition is also not given in the COLREGs by far. For instance, in 1982, a provisional definition of “close-quarters” is presented as that area around a ship where a collision with an approaching ship could not be avoided by the action of the approached ship alone if the approaching ship made a major, sudden and unexpected course change (Cahill, 1982). In 1983, the Board of Navigation and Maritime Affairs of the German Democratic Republic determined the definition as “when two ships in danger of collision in restricted visibility, the distance travelled by two ships from both ships take all way off by crash-stop” (Hilgert, 1983). Besides, the Justice Mr. Sheen pointed that “the structure of Collision Regulations is designed to ensure that, whenever possible, ship will not reach a close-quarters situation in which there is risk of collision and in which decisions have to be taken without time for proper thought”. Moreover, the definition “when two ships are so close that action taken by a ship alone could not be such as to result in avoiding collision” was presented for “close-quarter situation” (Zhang and Zhao, 1991). However, the aforementioned definitions (or conceptions) are imprecise due to various reasons. Particularly, the literature (Wang, 1991) pointed that the close-quarters situation is the distance between two encounter ships from when two ships are so close that action taken by a ship alone could not be such as to result in passing at a safe distance to the collision can not be avoided by the most effective action of a ship alone. This definition (or conception) has been accepted by most experts and it is employed in this paper. The distance at which a close-quarters situation first applies has not been defined in miles, and is not likely to be, as it will depend upon a number of factors (Cockcroft and Lameijer, 2011). In restricted visibility, in the open sea, a close quarters situation is generally considered to begin to apply at a distance of at least 2 n miles in any direction forward of the beam as this is the typical range of audibility for the whistle of a large ship in still conditions (see Annex III(1) (c) of COLREGs). The smaller distances, probably of the order of 1 n mile, would probably be accepted for ships in sight of one another. Furthermore, according to the Paragraph (c) of Rule 8, an alteration of course is generally more effective than an alteration of speed since it is quick to take effect and is easily observed, both visually and on radar from the other ship. Therefore, an alteration of course alone may be the most effective action to avoid a close-quarters situation when there is
2. Close-quarters situation 2.1. The four stages in collision avoidance procedure 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 is required to keep out of the way by the COLREGs. In general, there may be four stages relating to the permitted or required action for each ship (Cockcroft and Lameijer, 2011): (1) At long range, before risk of collision exists, both ships are free to take any action. (2) 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. (3) When it becomes apparent that the give-way ship is not taking appropriate action in compliance with the Rules, the stand-on ship 487
X. Wang et al.
Ocean Engineering 146 (2017) 486–497
sufficient sea room, and the alteration must be made in good time, substantial, and should not result in another close-quarters situation. In view of above observation, the mainly action taken to avoid collision is alteration of course in this article.
e ¼ ϕr ϕ, where ϕr is the desired heading angle, as expressed by the following equation (Le et al., 2004):
3. The collision avoidance dynamic support system
where Kp, Kd and Ki are designed parameters of PID controller and they can be estimated by following equation (Yang et al., 1999):
δE ¼ Kp e þ Kd e_ þ Ki ∫ edt
3.1. The mathematical model of ship maneuvering motion
Kp
The mathematical model relating the rudder angle δ to the ship's heading angle ϕ is expressed as follows (Yang and Ren, 2003):
€ þ K H ϕ_ ¼ K δ ϕ T T
where α, β are real-valued constants. Moreover, the ship's rudder actuator dynamics is expressed as follows:
(3)
where TE is time constant of a steering gear, KE is the steering quality index, δE is the order angle of the steering gear. The mathematical model of velocity of ship maneuvering can be expressed as (Yoshimura and Nomoto, 1978):
(4)
where V is velocity of ship, n is propeller revolution, ann, anv are thrust coefficients, avv, arr and aδδ are damping coefficients, expressed as:
0 lp m0 þ cm m0y 1 0 1 Xvv 1 ; arr ¼ L⋅a0rr ¼ ; aδδ ¼ ⋅a0δδ avv ¼ avv ¼ ⋅ 0 L L m þ m0x L m0 þ m0x 1 X0 AR ; ann ¼ L⋅a0nn ¼ ⋅ 0 δδ 0 ⋅ L m þ mx L⋅d 3 2 2C1 D D 2C2 D D 0 ⋅ ⋅ ; a ¼ L⋅ 0 ⋅ ¼ a ¼ ⋅ nv nv L d L d m þ m0x m0 þ m0x
XT0 ¼ R0 sinðϕ0 þ α0 Þ;
t
XO ðtÞ ¼ ∫ 0 V sin ϕdt;
0 Xvv ¼ 2Rt ρLdV 2 ;
¼ fα ðλÞ⋅ðUReo =VÞ
2
(8)
YT0 ¼ R0 cosðϕ0 þ α0 Þ
(9)
t
YO ðtÞ ¼ ∫ 0 V cos ϕdt
(10)
where V and ϕ are the instantaneous velocity and the instantaneous course of own ship, which are calculated by Eqs. (1), (4) and (7). Furthermore, the relative displacement from the target ship to the own ship is expressed as follows
where L is the length of ship, d is the draft of a ship, D is the diameter of ship's propeller, AR is the rudder area, lp is the distance between pivoting point and center of gravity of ship, cm is equal to block coefficient of hull cb, C1 and C2 are parameters of non-dimensional effective thrust, m0 , m0 x and m0 y are non-dimensional of ship's mass m, longitudinal added mass mx and lateral added mass my, respectively. Here, m0 , m0 x, m0 y, X0 vv and X0 δδ are expressed as follows:
m0x;y ¼ 2mx;y ρL2 d;
kn3 T 10K
where ϕ0 is the initial course of own ship. At the time of t after the alteration of course, the positive of the own ship is
(5)
m0 ¼ 2m ρL2 d;
Ki
In general, the DCPA and TCPA between two meeting ships are used to determine whether the risk of collision is exist and/or assess the risk level of collision of two ships. The dynamic mathematical model of ship collision avoidance is shown as follows (Yang, 1995): Fig. 1 shows the space-fixed coordinate system XOY, where X, Y axes point towards the East and North of the earth, respectively. In the coordinate system XOY, the velocity and the heading angle of the own ship are VO and ϕO, and point O is the midship of the own ship. Similarly, VT, ϕT and T denote the velocity, the heading angle, and the point of midship of the target ship. α is the relative bearing angle of the target ship T relative to the own ship O. R is defined as the relative distance between the own ship O and the target ship T. Therefore, the relative velocity VR and the relative heading angle ϕR is obtained by geometrograph. In the meantime, the distance of the perpendicular line OA is the DCPA between the own ship and the target ship, and the relative velocity VR is divided by the distance of line TA is the TCPA. Supposing that the initial position of the midship of own ship is located at the point of origin O (0,0), the initial relative distance between own ship and target ship and the initial relative bearing of target ship relative to own ship are R0 and α0, respectively. Then the initial positive of the target ship is
(2)
2 V_ þ avv V 2 þ arr ϕ_ þ aδδ V 2 δ2 ¼ ann n2 þ anv nV
1:8Tkn 1 ; K
3.3. The dynamic calculation model of collision avoidance parameter
where K is steering quality index, T is steering quality time constant. An _ can be experiment known as the “spiral test” has shown that HðϕÞ approximated by
1 KE δ_ ¼ δ þ δE TE TE
Kd
where kn < kw is designed parameter, kw is the frequency of steering gear.
(1)
3 H ϕ_ ¼ αϕ_ þ βϕ_
Tkn2 ; K
(7)
0 Xδδ
(6)
where ρ is the density of water, Rt is the total drag of a ship at steady straight-running, UReo is the effective rudder inflow velocity at steady straight-running and fα(λ) is the gradient of rudder normal force against attack angle in open water. 3.2. The control mechanism of ship maneuvering motion In the PID controller based control system, rudder deflection angle δE is controlled by PID control law based on the error of the heading angle
Fig. 1. The space-fixed coordinate system XOY. 488
X. Wang et al.
Ocean Engineering 146 (2017) 486–497
t
ΔX ¼ XT0 þ ∫ 0 ðVT sin ϕT V sin ϕÞdt;
Table 1 Principal particulars and maneuverability characteristics of the sample ship.
ΔY
t
¼ YT0 þ ∫ 0 ðVT cos ϕT V cos ϕÞdt
Length (m) Breadth (m) Draft (m) Block coefficient Displacement (ton)
(11)
Therefore, the relative distance R(t) between the own ship and the target ship is
RðtÞ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΔX 2 þ ΔY 2
ΔX þ Δα ΔY
(13)
ΔVY ¼ VT cos ϕT V cos ϕ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΔV 2x þ ΔV 2y
(14)
(15)
And the relative course ϕR (t) is
ϕR ðtÞ ¼ arctan
ΔVX þ ΔϕR ΔVY
(16)
8 if ΔVX 0; ΔVY 0 < 0 where ΔϕR ¼ 360 if ΔVX < 0; ΔVY 0 . : 180 others Then, as indicated in Fig. 1, the DCPA between the own ship and the target ship is expressed as
DCPA ¼ RðtÞsin½ϕR ðtÞ αðtÞ π
1.4 104 101.5 1.6 103 1.4 102 5.9 104
4. Simulation examples in close-quarters situation In order to certify the effectiveness and advantages of the presented collision avoidance dynamic support system, three encounter scenarios of two ships involved in close-quarters situation are provided for simulation, i.e. head-on situation, crossing situation and overtaking situation. The own ship and the target ship have the same principal particulars and maneuverability characteristics, as shown in Table 1. Moreover, the PID control mechanism parameters are kn ¼ 0.1, Kp ¼ 10, Kd ¼ 150 and Ki ¼ 0.001.
(17)
And the TCPA is expressed as
TCPA ¼
avv arr aδδ ann anv
(i) For the mathematical model of ship maneuvering motion, all above-mentioned literatures only take into account the mathematical model of course alteration. In this paper, no only the mathematical model relating the rudder angle to the ship's heading angle is considered (see Eqs. (1)–(3)), but also the mathematical model of ship's velocity when ship is maneuvering (see Eq. (4)); (ii) For the calculation model of collision avoidance parameter, all above-mentioned literatures are studied based on static calculation model, e.g. traditional geometrograph. In this paper, the dynamic calculation model of collision avoidance parameter is applied to indicate the real-time DCPA and TCPA when ship is maneuvering. As a result, it is convenient for navigators to forecast and analyze the variation trend of DCPA and TCPA if a certain action is taken to avoid collision, so that navigators can make a proper and correct decision.
Hence, the relative velocity VR (t) between the own ship and the target ship is
VR ðtÞ ¼
0.48 216.5 1 2.5 120
Remark 1. In recent years, although several ship collision avoidance systems are studied by considering partial mathematical model of ship maneuvering motion, the ship maneuverability considered in these systems is insufficiently. Moreover, the main differences between abovementioned literatures and this paper are listed as follows:
8 if ΔX 0; ΔY 0 < 0 where Δα ¼ 360 if ΔX < 0; ΔY 0 . : 180 others Similarly, the components of the relative velocity VR (t) along X-axis and Y-axis is expressed as follows
ΔVX ¼ VT sin ϕT V sin ϕ;
K T KE TE n (r/min)
(12)
And the relative bearing α(t) is
αðtÞ ¼ arctan
126.0 20.8 8.0 0.681 14,278
RðtÞcos½ϕR ðtÞ αðtÞ π VR ðtÞ
(18)
3.4. The collision avoidance dynamic support system 4.1. Scenario 1: head-on situation On the basis of the mathematical model of ship maneuvering motion, the control mechanism of ship maneuvering motion and the dynamic calculation model of collision avoidance parameter, the ship maneuverability based collision avoidance dynamic support system is developed in this section. The flow chart of collision avoidance dynamic support system is shown in Fig. 2, where ϕa is the alteration angle of course of the own ship or/and target ship.
In the initial status, the initial heading angle of the own ship and target ship are 0 and 180 , respectively. The initial velocity of the own ship and the target ship are same, namely 13 kn. The initial relative bearing angle of the target ship relative to the own ship is 358 , and the initial relative distance is 1 n mile. According to Section 2 of this article and the paragraph (a) of Rule 14 of COLREGs, it is obvious that the own
Fig. 2. The flow chart of collision avoidance dynamic support system. 489
X. Wang et al.
Ocean Engineering 146 (2017) 486–497
ship and target ship are reaching the close-quarters situation and meeting on the nearly reciprocal course so as to involve risk of collision, each shall alter her course to starboard so that each shall pass on the port side of the other. In order to illustrate the effectiveness of the collision avoidance dynamic support system clearly, it is supposed that the action to avoid collision is taken by the own ship or target ship alone in this section. Either own ship or target ship alter her course to starboard in a certain degree, e.g. the own ship alter her course to 060 . The simulation results are presented in Figs. 3–13. The trajectories of two ships and the target ship relative to own ship at time t ¼ 50s, 100s, 200s and 300s are shown in Figs. 3 and 4 respectively. Fig. 5 shows the curves of the expected DCPA, the dynamic DCPA and the actual DCPA between the own ship and target ship, respectively. It can be seen from Fig. 5 that the expected DCPA is 0.53 n miles by using traditional geometrograph, while actual DCPA is 0.3772 n miles by using collision avoidance dynamic support system. It is obvious that actual DCPA which is calculated by collision avoidance dynamic support system is much less than the expected DCPA which is calculated by traditional geometrograph. The main reason is the traditional geometrograph ignores the maneuvering characteristics of ship's motion and control, such as inertia and nonlinear. Fig. 6 gives the curves of the expected TCPA and the actual TCPA between the own ship and target ship, respectively. From Fig. 6, it can be seen that expected TCPA is 135s by using traditional geometrograph, while actual TCPA is 151s by using collision avoidance dynamic support system. Fig. 7 shows the curve of the relative distance of two ships. The curves of the relative bearing angle, the relative heading angle and the relative velocity of the target ship relative to the own ship are shown in Figs. 8–10, respectively. Fig. 11 shows the good tracking performance for the own ship's heading angle relative to the desired heading angle is achieved. Figs. 12 and 13 gives the curves of the velocity and rudder angle of the own ship, respectively. Either own ship or target ship alter her course to starboard in a proper range, e.g. the range of the own ship's alteration angle of course is from 1 to 60 . The simulation results are presented in Figs. 14–16. The curves of the expected DCPA and the actual DCPA with increasing alteration angle of course are shown in Fig. 14. Fig. 15 gives the curves of the expected TCPA and the actual TCPA, respectively. And the curve of the ratio of actual DCPA to expected DCPA is given in Fig. 16.
Fig. 4. Relative trajectories of target ship.
4.2. Scenario 2: crossing situation In the initial status, the initial heading angle of the own ship and target ship are 030 and 290 , respectively. The initial velocity of the own ship and the target ship are same, namely 13 kn. The initial relative
Fig. 5. The DCPA between two ships.
Fig. 3. Trajectories of two ships.
Fig. 6. The TCPA between two ships. 490
X. Wang et al.
Ocean Engineering 146 (2017) 486–497
Fig. 7. Relative distance between two ships.
Fig. 10. Relative velocity of target ship to own ship.
Fig. 8. The relative bearing angle of target ship to own ship.
Fig. 11. The heading angle of own ship.
Fig. 9. Relative heading angle of target ship to own ship.
Fig. 12. The velocity of own ship. 491
X. Wang et al.
Ocean Engineering 146 (2017) 486–497
Fig. 13. The rudder angle of own ship.
Fig. 16. The ratio of actual DCPA to expected DCPA.
bearing angle of the target ship relative to the own ship is 40 , and the initial relative distance is 1 n mile. According to Section 2 of this article and the paragraph (a) of Rule 15, it is obvious that the own ship and target ship are reaching the close-quarters situation and crossing so as to involve risk of collision, the own ship shall keep out of the way of the target ship and avoid crossing ahead of the target ship. If own ship alter her course to starboard in a certain degree, e.g. the own ship alter her course to 060 , the alteration angle of course is 30 . The simulation results are shown in Figs. 17–23. The trajectories of two ships and the target ship relative to own ship at time t ¼ 50s, 100s, 200s and 300s are shown in Figs. 17 and 18, respectively. Fig. 19 shows the curves of the expected DCPA, the dynamic DCPA and the actual DCPA between the own ship and target ship, respectively. It can be observed from Fig. 19 that the expected DCPA is 0.26 n miles by using traditional geometrograph, while actual DCPA is 0.24 n miles by using collision avoidance dynamic support system. Furthermore, it also can be seen from Fig. 19 that there is a short period that the dynamic DCPA is larger than the actual DCPA, the main reason is the heading angle and the velocity of the own ship are varying (as shown in Figs. 22 and 23) during the collision avoidance maneuver. This phenomenon should be noted since the officer may take the largest value of DCPA as the actual DCPA during the collision avoidance maneuvering. Fig. 20 gives the curves of the expected TCPA and the actual TCPA, respectively. From Fig. 20, it can be
Fig. 14. The varied trend of expected and actual DCPA.
Fig. 15. The varied trend of expected and actual TCPA.
Fig. 17. Trajectories of two ships. 492
X. Wang et al.
Ocean Engineering 146 (2017) 486–497
Fig. 21. Relative distance between two ships.
Fig. 18. Relative trajectories of target ship.
seen that expected TCPA is 146s, while actual TCPA is 156s. The curve of the relative distance of two ships is shown in Fig. 21. If own ship alter her course to starboard in a proper range, e.g. the range of the own ship's alteration angle of course is from 1 to 60 . The simulation results are shown in Figs. 24–26. The curves of the expected DCPA and the actual DCPA with increasing alteration angle of course are shown in Figs. 24 and 25 gives the curves of the expected TCPA and the actual TCPA, respectively. And the curve of the ratio of actual DCPA to expected DCPA is given in Fig. 26. 4.3. Scenario 3: overtaking situation In the initial status, the initial heading angle of the own ship and target ship are 000 and 020 , respectively. The initial velocity of the own ship is 13 kn, while initial velocity of the target ship is 8.5 kn. The initial relative bearing angle of the target ship relative to the own ship is 330 , and the initial relative distance is 1 n mile. According to Section 2 of this article and the paragraph (b) of Rule 13, it is obvious that the own ship and target ship are reaching the close-quarters situation and the own ship is coming up with the target ship from a direction more than 22.5 abaft her beam so as to involve risk of collision, the own ship which is overtaking target ship shall keep out of the way of the target ship which is being overtaken.
Fig. 19. The DCPA between two ships.
Fig. 20. The TCPA between two ships.
Fig. 22. The heading angle of own ship. 493
X. Wang et al.
Ocean Engineering 146 (2017) 486–497
Fig. 23. The velocity of own ship.
Fig. 26. The ratio of actual DCPA to expected DCPA.
If own ship alter her course to starboard in a certain degree, e.g. the own ship alter her course to 020 , the alteration angle of course is 20 . The simulation results are shown in Figs. 27–31. The trajectories of two ships and the target ship relative to own ship at time t ¼ 100s, 300s, 500s, 700s and 900s are shown in Figs. 27 and 28, respectively. Fig. 29 shows the curves of the expected DCPA, the dynamic DCPA and the actual DCPA between the own ship and target ship, respectively. It can be observed from Fig. 29 that the expected DCPA is 0.77 n miles by using traditional geometrograph, while actual DCPA is 0.74 n miles by using collision avoidance dynamic support system. Fig. 30 gives the curves of the expected TCPA and the actual TCPA, respectively. From Fig. 30, it can be seen that expected TCPA is 513s, while actual TCPA is 535s. The curve of the relative distance of two ships is shown in Fig. 31. If own ship alter her course to starboard in a proper range, e.g. the range of the own ship's alteration angle of course is from 1 to 60 . The simulation results are shown in Figs. 32–34. The curves of the expected DCPA and the actual DCPA with increasing alteration angle of course are shown in Fig. 32. It can be seen from Fig. 32 that the expected DCPA and actual DCPA are constant if the alteration angle of course is more than a certain degree, such as 34 in this scenario, the main reason is the initial status of encounter is the Closest Point of Approach (CPA) of two ships, i.e. the initial relative distance is the DCPA. Fig. 33 gives the curves of the expected TCPA and the actual TCPA, respectively. And the curve of the ratio of actual DCPA to expected DCPA is given in Fig. 34.
Fig. 24. The varied trend of expected and actual DCPA.
4.4. Discussions and analysis The ratio of actual DCPA to expected DCPA of ships meeting in closequarters situation for head-on situation, crossing situation and overtaking situation are shown in Figs. 16, 26 and 34, respectively. According to Figs. 16, 26 and 34, it is obvious that the curves of the ratio of actual DCPA to expected DCPA vary with alteration angle of course for each scenario are vary different. For example, for head-on situation, it can be observed from Fig. 16 that the ratio of actual DCPA to expected DCPA is decrease with increasing alteration angle of course apparently, i.e. the ratio is decrease from 98% to 77% with alteration angle of course of own ship increase from 1 to 60 . For crossing situation, it can be observed from Fig. 26 that the ratio is decrease smoothly with increasing alteration angle of course overall, i.e. the ratio is decrease from 95% to 88% with alteration angle of course of own ship increasing from 1 to 60 . And it can be observed from Fig. 34 that the ratio of actual DCPA to expected DCPA is very high and fluctuating between 95% and 98% with increasing alteration angle of course for overtaking situation. Furthermore, no matter in term of COLREGs or in practice, a
Fig. 25. The varied trend of expected and actual TCPA. 494
X. Wang et al.
Ocean Engineering 146 (2017) 486–497
Fig. 30. The TCPA between two ships. Fig. 27. Trajectories of two ships.
Fig. 31. Relative distance between two ships. Fig. 28. Relative trajectories of target ship.
Fig. 29. The DCPA between two ships.
Fig. 32. The varied trend of expected and actual DCPA. 495
X. Wang et al.
Ocean Engineering 146 (2017) 486–497
Meanwhile, based on the collision avoidance dynamic support system, the quantitative results of ship maneuverability effect on the action which is taken to avoid collision in close-quarters situation are revealed in this study. 5. Conclusions In this article, a ship maneuverability based collision avoidance dynamic support system in close-quarters situation is presented. The dynamic calculation model of collision avoidance parameter is employed to calculate the dynamic DCPA and TCPA in real-time when ship is maneuvering. Then the collision avoidance dynamic support system is developed by combining the mathematical model of ship maneuvering motion, the control mechanism of ship maneuvering motion as well as the dynamic calculation model of collision avoidance parameter. Following this approach, the proposed system is able to eliminate the insufficiency of neglect of ship maneuverability in the process of avoiding collision. Moreover, by incorporating the close-quarters situation into the proposed collision avoidance dynamic support system, simulation examples consisting three encounter scenarios of two ships in closequarters situation are applied to demonstrate the significance and necessity of ship maneuverability in the process of collision avoidance and illustrate the merits and effectiveness of the proposed system. The simulation results show that the proposed dynamic support system is a reasonable, effective and practicable system for collision avoidance, particularly in close-quarters situation. Furthermore, since the different velocity ratio of two meeting ships is another factor in the process of collision avoidance, our future research will be concentrated on the influence of different velocity ratio of two meeting ships on the process of collision avoidance.
Fig. 33. The varied trend of expected and actual TCPA.
Acknowledgements This work is supported in part by National Natural Science Foundation of China (Grant 51309041, 51179019 and 61374114), and the Fundamental Research Program for Key Laboratory of the Education Department of Liaoning Province (Grant. LZ2015006). References Fig. 34. The ratio of actual DCPA to expected DCPA.
Ahn, J.H., Rhee, K.P., You, Y.J., 2012. A study on the collision avoidance of a ship using neural networks and fuzzy logic. Appl. Ocean Res. 37, 162–173. Cahill, R.A., 1982. The avoidance of close quarters in clear weather. J. Navig. 35 (1), 151–159. Chai, T., Weng, J.X., Xiong, D.Q., 2017. Development of a quantitative risk assessment model for ship collisions in fairways. Saf. Sci. 91, 71–83. Cockcroft, A.N., Lameijer, J.N.F., 2011. A Guide to the Collision Avoidance Rules: International Regulations for Preventing Collisions at Sea, seventh ed. Butterworth–Heinemann. Eleftheria, E., Apostolos, P., Markos, V., 2016. Statistical analysis of ship accidents and review of safety level. Saf. Sci. 85, 282–292. Goerlandt, F., Kujala, P., 2014. On the reliability and validity of ship–ship collision risk analysis in light of different perspectives on risk. Saf. Sci. 62, 348–365. Goerlandt, F., Montewka, J., 2015. A framework for risk analysis of maritime transportation systems: a case study for oil spill from tankers in a ship–ship collision. Saf. Sci. 76, 42–66. Goerlandt, F., Montewka, J., Kuzmin, V., Kujala, P., 2015. A risk-informed ship collision alert system: framework and application. Saf. Sci. 77, 182–204. Hilgert, H., 1983. Defining the close-quarters situation at sea. J. Navig. 36 (3), 454–461. Hong, B.G., Yang, L.J., 2012. Ship Handing. Dalian Maritime University Press, Dalian. Hwang, C.N., 2002. The integrated design of fuzzy collision-avoidance and H∞Autopilots on ships. J. Navig. 55 (1), 117–136. International Maritime Organization (IMO), 1981. Definition of the Term “Close-quarters Situation”. Note by the Government of the German Democratic Republic, NAV 26/4/ 2. International Maritime Organization (IMO), 2001. International Regulations for Preventing Collisions at Sea (COLREGs) 1972. Le, M.D., Nguyen, S.H., Nguyen, L.A., 2004. Study on a new and effective fuzzy PID ship autopilot. Artif. Life Robot. 8, 197–201. Li, B., Pang, F.W., 2013. An approach of vessel collision risk assessment based on the D-S evidence theory. Ocean. Eng. 74, 16–21. Perera, L.P., Carvalho, J.P., Soares, C.G., 2011. Fuzzy logic based decision making system for collision avoidance of ocean navigation under critical collision conditions. J. Mar. Sci. Technol. 16, 84–99.
substantial alteration of course is the most common and effective action when a ship involved in a close-quarters situation. Significant differences are also observed for a certain substantial alteration angle of course in each scenario (as shown in Figs. 16, 26 and 34). For instance, when the alteration angle of course is 60 , the ratio of actual DCPA to expected DCPA in head-on situation is 77% which is nearly three quarters, it means that the difference between actual DCPA and expected DCPA is very large and visible. At the same time, the ratio of actual DCPA to expected DCPA is 88% in crossing situation, it means that the difference between actual DCPA and expected DCPA is also large. Meanwhile, the ratio of actual DCPA to expected DCPA is 96% in overtaking situation, it means that the difference between actual DCPA and expected DCPA is small. On the basis of above analyses, it is obvious that the difference between the actual DCPA and expected DCPA of meeting ships are universal exist in the close-quarters situation due to ship maneuverability. Moreover, it can be seen that the maximum difference between actual DCPA and expected DCPA and the lowest ratio of actual DCPA to expected DCPA are indicated in the head-on situation on the whole, followed by the crossing situation, and the last is the overtaking situation. As a result, it is easily to find out that the main scenario which is influenced by ship maneuverability during collision avoidance is head-on situation. The main reason is that the absolute value of course difference and relative velocity of two meeting ships are maximum in head-on situation by comparing with the crossing situation and overtaking situation. 496
X. Wang et al.
Ocean Engineering 146 (2017) 486–497 Yang, Y.S., 1995. Study on mathematical model for ship collision avoidance dynamic system. J. Dalian Marit. Univ. 21 (1), 30–36. Yang, Y.S., Yu, X.L., Jia, X.L., 1999. Robust PID autopilot for ships. J. Dalian Marit. Univ. 25 (4), 11–15. Yang, Y.S., Ren, J.S., 2003. Adaptive fuzzy robust tracking controller design via small gain approach and its application. IEEE Trans. Fuzzy Syst. 11 (6), 783–795. Yoshimura, Y., Nomoto, K., 1978. Modeling of manoeuvring behaviour of ships with a propeller idling, boosting and reversing. J. Soc. Nav. Archit. Jpn. 144, 57–69 (in Japanese). Zhang, Y.K., Zhao, L.D., 1991. The concept and model of close-quarters situation. J. Dalian Marit. Univ. 17 (1), 7–13. Zhang, J.F., Zhang, D., Yan, X.P., Haugen, S., Soares, C.G., 2015. A distributed anticollision decision support formulation in multi-ship encounter situations under COLREGs. Ocean. Eng. 105, 336–348.
Tam, C.K., Bucknall, R., Greig, A., 2009. Review of collision avoidance and path planning methods for ships in close range encounters. J. Navig. 62, 455–476. Tam, C.K., Bucknall, R., 2010a. Collision risk assessment for ships. J. Mar. Sci. Technol. 15, 257–270. Tam, C.K., Bucknall, R., 2010b. Path-planning algorithm for ships in close-range encounters. J. Mar. Sci. Technol. 15, 395–407. Tam, C.K., Bucknall, R., 2013. Cooperative path planning algorithm for marine surface vessels. Ocean Eng. 57, 25–33. Tsou, M.C., Kao, S.L., Su, C.M., 2010. Decision support from genetic algorithms for ship collision avoidance route planning and alerts. J. Navig. 63 (1), 167–182. Wang, F.C., 1991. A discussion on the explanation of “close-quarters situation”. J. Dalian Mar. Coll. 17 (1), 1–6. Xue, Y.Z., Clelland, D., Lee, B.S., Han, D.F., 2011. Automatic simulation of ship navigation. Ocean. Eng. 38 (17–18), 2290–2305.
497
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
The ship maneuverability based collision avoidance dynamic support system in close-quarters situation Xin Wang *, Zhengjiang Liu, Yao Cai Navigation College, Dalian Maritime University, Dalian 116026, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Close-quarters situation Collision avoidance support system Ship maneuverability
In this article, a ship maneuverability based collision avoidance dynamic support system in close-quarters situation is presented. The dynamic calculation model of collision avoidance parameter is employed to calculate the dynamic DCPA and TCPA in real-time when ship is maneuvering. Then the collision avoidance dynamic support system is developed by combining the mathematical model of ship maneuvering motion, the control mechanism of ship maneuvering motion and the dynamic calculation model of collision avoidance parameter. Following this approach, the proposed system is able to eliminate the insufficiency of neglect of ship maneuverability in the process of avoiding collision. Moreover, by incorporating the close-quarters situation into the proposed collision avoidance dynamic support system, simulation examples consisting three encounter scenarios of two ships in close-quarters situation are applied to demonstrate the significance and necessity of ship maneuverability in the process of collision avoidance and illustrate the merits and effectiveness of the proposed system. The simulation results show that the proposed dynamic support system is a reasonable, effective and practicable system for collision avoidance, particularly in close-quarters situation.
1. Introduction In the past decades, the collision accidents in maritime traffic engineering have drawn much attention owing to maritime safety and environmental protection (Eleftheria et al., 2016). In particularly, the collision avoidance system is recognized as an available way to prevention of the ships from collisions (Tsou et al., 2010). Through years of progress, a number of maritime collision avoidance systems or/and models have achieved many significant results (Ahn et al., 2012; Chai et al., 2017; Goerlandt and Kujala, 2014; Goerlandt and Montewka, 2015; Goerlandt et al., 2015; Hwang, 2002; Li and Pang, 2013; Tam and Bucknall, 2013; Xue et al., 2011; Zhang et al., 2015), such as autonomous collision avoidance system, decision making system etc. For example, a fuzzy collision avoidance expert system which include the experiences of experts was designed to resolve the problems of collision (Hwang, 2002); a framework for risk-informed maritime collision alert system was proposed for different navigational environments (Goerlandt et al., 2015); and a distributed anti-collision decision support formulation was studied in multi-ship encounter situations under the International Regulations for Preventing Collisions at Sea (COLREGs) (Zhang et al., 2015). Nevertheless, a shortage is still exists in above-mentioned collision avoidance systems, that is they could not forecast the Distance at Closest
Point of Approach (DCPA) and Time to the Closest Point of Approach (TCPA) if a certain action is taken by ship to avoid collision, and indicate the variation trend of DCPA and TCPA if a series of actions in a proper range is taken. In practice, the collision avoidance manoeuvres for another ships or obstacles are usually performed under the navigators’ own judgement nowadays. Hence, in order to assist the navigators to judge which and how large the action shall be taken, the collision avoidance support system which can indicate whole as well as real-time DCPA and TCPA should be studied. In addition, ships encounter in close-quarters situation is still a major and intractable issue in maritime collision avoidance. The reason lies on that there are not unified quantitative and qualitative interpretations on the term “close-quarters situation” in the world, and the exact definition is also not given in the COLREGs by far. The 1972 International Maritime Organization (IMO) conference ever considered the possibility of specifying the distance at which it would begin to apply, but after a lengthy discussion, it was decided that this distance could not be quantified. In the following decades, several literatures (Cahill, 1982; Hilgert, 1983; International Maritime Organization (IMO), 1981; Perera et al., 2011; Tam et al., 2009; Tam and Bucknall, 2010a, 2010b; Zhang and Zhao, 1991) have presented some different interpretations for the term “closequarters situation” and studied for the collision avoidance in close range
* Corresponding author. E-mail address: [email protected] (X. Wang). http://dx.doi.org/10.1016/j.oceaneng.2017.08.034 Received 27 February 2017; Received in revised form 25 June 2017; Accepted 18 August 2017 Available online 6 September 2017 0029-8018/© 2017 Elsevier Ltd. All rights reserved.
X. Wang et al.
Ocean Engineering 146 (2017) 486–497
is required to give the whistle signal prescribed in Rules and is permitted to take action to avoid collision by her manoeuvre alone, but a power driven ship must not alter course to port to avoid another power driven ship crossing from her own port side. The give-way ship is relieved of her obligation to keep out of the way. (4) When collision can not be avoided by the give-way ship alone, the stand-on ship is required to take such action as well best aid to avoid collision.
encounters or critical conditions. For example, according to the COLREGs, a collision risk assessment method and an optimal path planning algorithm for ships in close range encounters were presented (Tam and Bucknall, 2010a, 2010b), respectively. However, the lack of the study on ship maneuverability is existing in above-mentioned researches, i.e., most of the aforementioned researches ignore the impact of ship maneuverability in the process of avoiding collision. Generally speaking, ship maneuverability weighs the response capability when ship is handled by operator or interfered by external environment (Hong and Yang, 2012). Due to the characteristics of ship motion such as large inertia, time delay and nonlinear etc., the ship maneuverability is a major and important issue for the navigational safety of ships, especially in the collision avoidance of ships. The previous literature (Wang, 1991) has pointed that the distance at which a close-quarters situation applies depend upon many factors, including the ship maneuverability. In fact, as we know, the ship maneuverability plays a very important role in the process of collision avoidance in close-quarters situation on the basis of foregoing depictions. To the best knowledge of the authors, there is still no report on applying the ship maneuverability to the ships encounter in closequarters situation based on the collision avoidance system (or model) so far. Hence, the collision avoidance in close-quarters situation requires intensive analysis and should be modelled precisely. Motivated by above observation, and in order to study the influence of ship maneuverability in the process of collision avoidance in closequarters situation, a ship maneuverability based collision avoidance dynamic support system in close-quarters situation is presented in this article. The collision avoidance dynamic support system is developed by combining the mathematical model of ship maneuvering motion, the control mechanism of ship maneuvering motion and the dynamic calculation model of collision avoidance parameter. Following this approach, the proposed system is able to eliminate the insufficiency of neglect of ship maneuverability in the process of avoiding collision. Moreover, by incorporating the close-quarters situation into the proposed collision avoidance dynamic support system, simulation examples consisting three encounter scenarios of two ships in close-quarters situation are applied to demonstrate the significance and necessity of ship maneuverability in the process of collision avoidance and illustrate the merits and effectiveness of the proposed system. The simulation results show that the proposed dynamic support system is a reasonable, effective and practicable system for collision avoidance, particularly in close-quarters situation. The rest of the paper is organized as follows. Section 2 briefly describes the concept of close-quarters situation. Then the collision avoidance dynamic support system is described briefly in Section 3. In Section 4, the simulation results via three encounter scenarios of two ships in close-quarters situation are presented and analysed in detail. The paper ends with conclusion in Section 5.
The distances at which the various stages begin to apply will vary considerably. For a crossing situation involving two power driven ships in the open sea, it is suggested that the outer limit of the second stage might be of the order of 5–8 miles and that the outer limit for the third stage would be about 2–3 miles. 2.2. The concept of close-quarters situation The term “close-quarters situation” is presented firstly in the paragraph (c) of Rule 8 of COLREGs states (International Maritime Organization (IMO), 2001):“If there is sufficient sea room, alteration of course alone may be the most effective action to avoid a close-quarters situation provided that it is made in good time, is substantial and does not result in another close-quarters situation”. Moreover, the paragraph (d) and (e) of Rule 19 are also refer to the close quarters situation. Nevertheless, there are not unified quantitative and qualitative interpretations on the term “close-quarters situation” in the world, and the exact definition is also not given in the COLREGs by far. For instance, in 1982, a provisional definition of “close-quarters” is presented as that area around a ship where a collision with an approaching ship could not be avoided by the action of the approached ship alone if the approaching ship made a major, sudden and unexpected course change (Cahill, 1982). In 1983, the Board of Navigation and Maritime Affairs of the German Democratic Republic determined the definition as “when two ships in danger of collision in restricted visibility, the distance travelled by two ships from both ships take all way off by crash-stop” (Hilgert, 1983). Besides, the Justice Mr. Sheen pointed that “the structure of Collision Regulations is designed to ensure that, whenever possible, ship will not reach a close-quarters situation in which there is risk of collision and in which decisions have to be taken without time for proper thought”. Moreover, the definition “when two ships are so close that action taken by a ship alone could not be such as to result in avoiding collision” was presented for “close-quarter situation” (Zhang and Zhao, 1991). However, the aforementioned definitions (or conceptions) are imprecise due to various reasons. Particularly, the literature (Wang, 1991) pointed that the close-quarters situation is the distance between two encounter ships from when two ships are so close that action taken by a ship alone could not be such as to result in passing at a safe distance to the collision can not be avoided by the most effective action of a ship alone. This definition (or conception) has been accepted by most experts and it is employed in this paper. The distance at which a close-quarters situation first applies has not been defined in miles, and is not likely to be, as it will depend upon a number of factors (Cockcroft and Lameijer, 2011). In restricted visibility, in the open sea, a close quarters situation is generally considered to begin to apply at a distance of at least 2 n miles in any direction forward of the beam as this is the typical range of audibility for the whistle of a large ship in still conditions (see Annex III(1) (c) of COLREGs). The smaller distances, probably of the order of 1 n mile, would probably be accepted for ships in sight of one another. Furthermore, according to the Paragraph (c) of Rule 8, an alteration of course is generally more effective than an alteration of speed since it is quick to take effect and is easily observed, both visually and on radar from the other ship. Therefore, an alteration of course alone may be the most effective action to avoid a close-quarters situation when there is
2. Close-quarters situation 2.1. The four stages in collision avoidance procedure 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 is required to keep out of the way by the COLREGs. In general, there may be four stages relating to the permitted or required action for each ship (Cockcroft and Lameijer, 2011): (1) At long range, before risk of collision exists, both ships are free to take any action. (2) 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. (3) When it becomes apparent that the give-way ship is not taking appropriate action in compliance with the Rules, the stand-on ship 487
X. Wang et al.
Ocean Engineering 146 (2017) 486–497
sufficient sea room, and the alteration must be made in good time, substantial, and should not result in another close-quarters situation. In view of above observation, the mainly action taken to avoid collision is alteration of course in this article.
e ¼ ϕr ϕ, where ϕr is the desired heading angle, as expressed by the following equation (Le et al., 2004):
3. The collision avoidance dynamic support system
where Kp, Kd and Ki are designed parameters of PID controller and they can be estimated by following equation (Yang et al., 1999):
δE ¼ Kp e þ Kd e_ þ Ki ∫ edt
3.1. The mathematical model of ship maneuvering motion
Kp
The mathematical model relating the rudder angle δ to the ship's heading angle ϕ is expressed as follows (Yang and Ren, 2003):
€ þ K H ϕ_ ¼ K δ ϕ T T
where α, β are real-valued constants. Moreover, the ship's rudder actuator dynamics is expressed as follows:
(3)
where TE is time constant of a steering gear, KE is the steering quality index, δE is the order angle of the steering gear. The mathematical model of velocity of ship maneuvering can be expressed as (Yoshimura and Nomoto, 1978):
(4)
where V is velocity of ship, n is propeller revolution, ann, anv are thrust coefficients, avv, arr and aδδ are damping coefficients, expressed as:
0 lp m0 þ cm m0y 1 0 1 Xvv 1 ; arr ¼ L⋅a0rr ¼ ; aδδ ¼ ⋅a0δδ avv ¼ avv ¼ ⋅ 0 L L m þ m0x L m0 þ m0x 1 X0 AR ; ann ¼ L⋅a0nn ¼ ⋅ 0 δδ 0 ⋅ L m þ mx L⋅d 3 2 2C1 D D 2C2 D D 0 ⋅ ⋅ ; a ¼ L⋅ 0 ⋅ ¼ a ¼ ⋅ nv nv L d L d m þ m0x m0 þ m0x
XT0 ¼ R0 sinðϕ0 þ α0 Þ;
t
XO ðtÞ ¼ ∫ 0 V sin ϕdt;
0 Xvv ¼ 2Rt ρLdV 2 ;
¼ fα ðλÞ⋅ðUReo =VÞ
2
(8)
YT0 ¼ R0 cosðϕ0 þ α0 Þ
(9)
t
YO ðtÞ ¼ ∫ 0 V cos ϕdt
(10)
where V and ϕ are the instantaneous velocity and the instantaneous course of own ship, which are calculated by Eqs. (1), (4) and (7). Furthermore, the relative displacement from the target ship to the own ship is expressed as follows
where L is the length of ship, d is the draft of a ship, D is the diameter of ship's propeller, AR is the rudder area, lp is the distance between pivoting point and center of gravity of ship, cm is equal to block coefficient of hull cb, C1 and C2 are parameters of non-dimensional effective thrust, m0 , m0 x and m0 y are non-dimensional of ship's mass m, longitudinal added mass mx and lateral added mass my, respectively. Here, m0 , m0 x, m0 y, X0 vv and X0 δδ are expressed as follows:
m0x;y ¼ 2mx;y ρL2 d;
kn3 T 10K
where ϕ0 is the initial course of own ship. At the time of t after the alteration of course, the positive of the own ship is
(5)
m0 ¼ 2m ρL2 d;
Ki
In general, the DCPA and TCPA between two meeting ships are used to determine whether the risk of collision is exist and/or assess the risk level of collision of two ships. The dynamic mathematical model of ship collision avoidance is shown as follows (Yang, 1995): Fig. 1 shows the space-fixed coordinate system XOY, where X, Y axes point towards the East and North of the earth, respectively. In the coordinate system XOY, the velocity and the heading angle of the own ship are VO and ϕO, and point O is the midship of the own ship. Similarly, VT, ϕT and T denote the velocity, the heading angle, and the point of midship of the target ship. α is the relative bearing angle of the target ship T relative to the own ship O. R is defined as the relative distance between the own ship O and the target ship T. Therefore, the relative velocity VR and the relative heading angle ϕR is obtained by geometrograph. In the meantime, the distance of the perpendicular line OA is the DCPA between the own ship and the target ship, and the relative velocity VR is divided by the distance of line TA is the TCPA. Supposing that the initial position of the midship of own ship is located at the point of origin O (0,0), the initial relative distance between own ship and target ship and the initial relative bearing of target ship relative to own ship are R0 and α0, respectively. Then the initial positive of the target ship is
(2)
2 V_ þ avv V 2 þ arr ϕ_ þ aδδ V 2 δ2 ¼ ann n2 þ anv nV
1:8Tkn 1 ; K
3.3. The dynamic calculation model of collision avoidance parameter
where K is steering quality index, T is steering quality time constant. An _ can be experiment known as the “spiral test” has shown that HðϕÞ approximated by
1 KE δ_ ¼ δ þ δE TE TE
Kd
where kn < kw is designed parameter, kw is the frequency of steering gear.
(1)
3 H ϕ_ ¼ αϕ_ þ βϕ_
Tkn2 ; K
(7)
0 Xδδ
(6)
where ρ is the density of water, Rt is the total drag of a ship at steady straight-running, UReo is the effective rudder inflow velocity at steady straight-running and fα(λ) is the gradient of rudder normal force against attack angle in open water. 3.2. The control mechanism of ship maneuvering motion In the PID controller based control system, rudder deflection angle δE is controlled by PID control law based on the error of the heading angle
Fig. 1. The space-fixed coordinate system XOY. 488
X. Wang et al.
Ocean Engineering 146 (2017) 486–497
t
ΔX ¼ XT0 þ ∫ 0 ðVT sin ϕT V sin ϕÞdt;
Table 1 Principal particulars and maneuverability characteristics of the sample ship.
ΔY
t
¼ YT0 þ ∫ 0 ðVT cos ϕT V cos ϕÞdt
Length (m) Breadth (m) Draft (m) Block coefficient Displacement (ton)
(11)
Therefore, the relative distance R(t) between the own ship and the target ship is
RðtÞ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΔX 2 þ ΔY 2
ΔX þ Δα ΔY
(13)
ΔVY ¼ VT cos ϕT V cos ϕ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΔV 2x þ ΔV 2y
(14)
(15)
And the relative course ϕR (t) is
ϕR ðtÞ ¼ arctan
ΔVX þ ΔϕR ΔVY
(16)
8 if ΔVX 0; ΔVY 0 < 0 where ΔϕR ¼ 360 if ΔVX < 0; ΔVY 0 . : 180 others Then, as indicated in Fig. 1, the DCPA between the own ship and the target ship is expressed as
DCPA ¼ RðtÞsin½ϕR ðtÞ αðtÞ π
1.4 104 101.5 1.6 103 1.4 102 5.9 104
4. Simulation examples in close-quarters situation In order to certify the effectiveness and advantages of the presented collision avoidance dynamic support system, three encounter scenarios of two ships involved in close-quarters situation are provided for simulation, i.e. head-on situation, crossing situation and overtaking situation. The own ship and the target ship have the same principal particulars and maneuverability characteristics, as shown in Table 1. Moreover, the PID control mechanism parameters are kn ¼ 0.1, Kp ¼ 10, Kd ¼ 150 and Ki ¼ 0.001.
(17)
And the TCPA is expressed as
TCPA ¼
avv arr aδδ ann anv
(i) For the mathematical model of ship maneuvering motion, all above-mentioned literatures only take into account the mathematical model of course alteration. In this paper, no only the mathematical model relating the rudder angle to the ship's heading angle is considered (see Eqs. (1)–(3)), but also the mathematical model of ship's velocity when ship is maneuvering (see Eq. (4)); (ii) For the calculation model of collision avoidance parameter, all above-mentioned literatures are studied based on static calculation model, e.g. traditional geometrograph. In this paper, the dynamic calculation model of collision avoidance parameter is applied to indicate the real-time DCPA and TCPA when ship is maneuvering. As a result, it is convenient for navigators to forecast and analyze the variation trend of DCPA and TCPA if a certain action is taken to avoid collision, so that navigators can make a proper and correct decision.
Hence, the relative velocity VR (t) between the own ship and the target ship is
VR ðtÞ ¼
0.48 216.5 1 2.5 120
Remark 1. In recent years, although several ship collision avoidance systems are studied by considering partial mathematical model of ship maneuvering motion, the ship maneuverability considered in these systems is insufficiently. Moreover, the main differences between abovementioned literatures and this paper are listed as follows:
8 if ΔX 0; ΔY 0 < 0 where Δα ¼ 360 if ΔX < 0; ΔY 0 . : 180 others Similarly, the components of the relative velocity VR (t) along X-axis and Y-axis is expressed as follows
ΔVX ¼ VT sin ϕT V sin ϕ;
K T KE TE n (r/min)
(12)
And the relative bearing α(t) is
αðtÞ ¼ arctan
126.0 20.8 8.0 0.681 14,278
RðtÞcos½ϕR ðtÞ αðtÞ π VR ðtÞ
(18)
3.4. The collision avoidance dynamic support system 4.1. Scenario 1: head-on situation On the basis of the mathematical model of ship maneuvering motion, the control mechanism of ship maneuvering motion and the dynamic calculation model of collision avoidance parameter, the ship maneuverability based collision avoidance dynamic support system is developed in this section. The flow chart of collision avoidance dynamic support system is shown in Fig. 2, where ϕa is the alteration angle of course of the own ship or/and target ship.
In the initial status, the initial heading angle of the own ship and target ship are 0 and 180 , respectively. The initial velocity of the own ship and the target ship are same, namely 13 kn. The initial relative bearing angle of the target ship relative to the own ship is 358 , and the initial relative distance is 1 n mile. According to Section 2 of this article and the paragraph (a) of Rule 14 of COLREGs, it is obvious that the own
Fig. 2. The flow chart of collision avoidance dynamic support system. 489
X. Wang et al.
Ocean Engineering 146 (2017) 486–497
ship and target ship are reaching the close-quarters situation and meeting on the nearly reciprocal course so as to involve risk of collision, each shall alter her course to starboard so that each shall pass on the port side of the other. In order to illustrate the effectiveness of the collision avoidance dynamic support system clearly, it is supposed that the action to avoid collision is taken by the own ship or target ship alone in this section. Either own ship or target ship alter her course to starboard in a certain degree, e.g. the own ship alter her course to 060 . The simulation results are presented in Figs. 3–13. The trajectories of two ships and the target ship relative to own ship at time t ¼ 50s, 100s, 200s and 300s are shown in Figs. 3 and 4 respectively. Fig. 5 shows the curves of the expected DCPA, the dynamic DCPA and the actual DCPA between the own ship and target ship, respectively. It can be seen from Fig. 5 that the expected DCPA is 0.53 n miles by using traditional geometrograph, while actual DCPA is 0.3772 n miles by using collision avoidance dynamic support system. It is obvious that actual DCPA which is calculated by collision avoidance dynamic support system is much less than the expected DCPA which is calculated by traditional geometrograph. The main reason is the traditional geometrograph ignores the maneuvering characteristics of ship's motion and control, such as inertia and nonlinear. Fig. 6 gives the curves of the expected TCPA and the actual TCPA between the own ship and target ship, respectively. From Fig. 6, it can be seen that expected TCPA is 135s by using traditional geometrograph, while actual TCPA is 151s by using collision avoidance dynamic support system. Fig. 7 shows the curve of the relative distance of two ships. The curves of the relative bearing angle, the relative heading angle and the relative velocity of the target ship relative to the own ship are shown in Figs. 8–10, respectively. Fig. 11 shows the good tracking performance for the own ship's heading angle relative to the desired heading angle is achieved. Figs. 12 and 13 gives the curves of the velocity and rudder angle of the own ship, respectively. Either own ship or target ship alter her course to starboard in a proper range, e.g. the range of the own ship's alteration angle of course is from 1 to 60 . The simulation results are presented in Figs. 14–16. The curves of the expected DCPA and the actual DCPA with increasing alteration angle of course are shown in Fig. 14. Fig. 15 gives the curves of the expected TCPA and the actual TCPA, respectively. And the curve of the ratio of actual DCPA to expected DCPA is given in Fig. 16.
Fig. 4. Relative trajectories of target ship.
4.2. Scenario 2: crossing situation In the initial status, the initial heading angle of the own ship and target ship are 030 and 290 , respectively. The initial velocity of the own ship and the target ship are same, namely 13 kn. The initial relative
Fig. 5. The DCPA between two ships.
Fig. 3. Trajectories of two ships.
Fig. 6. The TCPA between two ships. 490
X. Wang et al.
Ocean Engineering 146 (2017) 486–497
Fig. 7. Relative distance between two ships.
Fig. 10. Relative velocity of target ship to own ship.
Fig. 8. The relative bearing angle of target ship to own ship.
Fig. 11. The heading angle of own ship.
Fig. 9. Relative heading angle of target ship to own ship.
Fig. 12. The velocity of own ship. 491
X. Wang et al.
Ocean Engineering 146 (2017) 486–497
Fig. 13. The rudder angle of own ship.
Fig. 16. The ratio of actual DCPA to expected DCPA.
bearing angle of the target ship relative to the own ship is 40 , and the initial relative distance is 1 n mile. According to Section 2 of this article and the paragraph (a) of Rule 15, it is obvious that the own ship and target ship are reaching the close-quarters situation and crossing so as to involve risk of collision, the own ship shall keep out of the way of the target ship and avoid crossing ahead of the target ship. If own ship alter her course to starboard in a certain degree, e.g. the own ship alter her course to 060 , the alteration angle of course is 30 . The simulation results are shown in Figs. 17–23. The trajectories of two ships and the target ship relative to own ship at time t ¼ 50s, 100s, 200s and 300s are shown in Figs. 17 and 18, respectively. Fig. 19 shows the curves of the expected DCPA, the dynamic DCPA and the actual DCPA between the own ship and target ship, respectively. It can be observed from Fig. 19 that the expected DCPA is 0.26 n miles by using traditional geometrograph, while actual DCPA is 0.24 n miles by using collision avoidance dynamic support system. Furthermore, it also can be seen from Fig. 19 that there is a short period that the dynamic DCPA is larger than the actual DCPA, the main reason is the heading angle and the velocity of the own ship are varying (as shown in Figs. 22 and 23) during the collision avoidance maneuver. This phenomenon should be noted since the officer may take the largest value of DCPA as the actual DCPA during the collision avoidance maneuvering. Fig. 20 gives the curves of the expected TCPA and the actual TCPA, respectively. From Fig. 20, it can be
Fig. 14. The varied trend of expected and actual DCPA.
Fig. 15. The varied trend of expected and actual TCPA.
Fig. 17. Trajectories of two ships. 492
X. Wang et al.
Ocean Engineering 146 (2017) 486–497
Fig. 21. Relative distance between two ships.
Fig. 18. Relative trajectories of target ship.
seen that expected TCPA is 146s, while actual TCPA is 156s. The curve of the relative distance of two ships is shown in Fig. 21. If own ship alter her course to starboard in a proper range, e.g. the range of the own ship's alteration angle of course is from 1 to 60 . The simulation results are shown in Figs. 24–26. The curves of the expected DCPA and the actual DCPA with increasing alteration angle of course are shown in Figs. 24 and 25 gives the curves of the expected TCPA and the actual TCPA, respectively. And the curve of the ratio of actual DCPA to expected DCPA is given in Fig. 26. 4.3. Scenario 3: overtaking situation In the initial status, the initial heading angle of the own ship and target ship are 000 and 020 , respectively. The initial velocity of the own ship is 13 kn, while initial velocity of the target ship is 8.5 kn. The initial relative bearing angle of the target ship relative to the own ship is 330 , and the initial relative distance is 1 n mile. According to Section 2 of this article and the paragraph (b) of Rule 13, it is obvious that the own ship and target ship are reaching the close-quarters situation and the own ship is coming up with the target ship from a direction more than 22.5 abaft her beam so as to involve risk of collision, the own ship which is overtaking target ship shall keep out of the way of the target ship which is being overtaken.
Fig. 19. The DCPA between two ships.
Fig. 20. The TCPA between two ships.
Fig. 22. The heading angle of own ship. 493
X. Wang et al.
Ocean Engineering 146 (2017) 486–497
Fig. 23. The velocity of own ship.
Fig. 26. The ratio of actual DCPA to expected DCPA.
If own ship alter her course to starboard in a certain degree, e.g. the own ship alter her course to 020 , the alteration angle of course is 20 . The simulation results are shown in Figs. 27–31. The trajectories of two ships and the target ship relative to own ship at time t ¼ 100s, 300s, 500s, 700s and 900s are shown in Figs. 27 and 28, respectively. Fig. 29 shows the curves of the expected DCPA, the dynamic DCPA and the actual DCPA between the own ship and target ship, respectively. It can be observed from Fig. 29 that the expected DCPA is 0.77 n miles by using traditional geometrograph, while actual DCPA is 0.74 n miles by using collision avoidance dynamic support system. Fig. 30 gives the curves of the expected TCPA and the actual TCPA, respectively. From Fig. 30, it can be seen that expected TCPA is 513s, while actual TCPA is 535s. The curve of the relative distance of two ships is shown in Fig. 31. If own ship alter her course to starboard in a proper range, e.g. the range of the own ship's alteration angle of course is from 1 to 60 . The simulation results are shown in Figs. 32–34. The curves of the expected DCPA and the actual DCPA with increasing alteration angle of course are shown in Fig. 32. It can be seen from Fig. 32 that the expected DCPA and actual DCPA are constant if the alteration angle of course is more than a certain degree, such as 34 in this scenario, the main reason is the initial status of encounter is the Closest Point of Approach (CPA) of two ships, i.e. the initial relative distance is the DCPA. Fig. 33 gives the curves of the expected TCPA and the actual TCPA, respectively. And the curve of the ratio of actual DCPA to expected DCPA is given in Fig. 34.
Fig. 24. The varied trend of expected and actual DCPA.
4.4. Discussions and analysis The ratio of actual DCPA to expected DCPA of ships meeting in closequarters situation for head-on situation, crossing situation and overtaking situation are shown in Figs. 16, 26 and 34, respectively. According to Figs. 16, 26 and 34, it is obvious that the curves of the ratio of actual DCPA to expected DCPA vary with alteration angle of course for each scenario are vary different. For example, for head-on situation, it can be observed from Fig. 16 that the ratio of actual DCPA to expected DCPA is decrease with increasing alteration angle of course apparently, i.e. the ratio is decrease from 98% to 77% with alteration angle of course of own ship increase from 1 to 60 . For crossing situation, it can be observed from Fig. 26 that the ratio is decrease smoothly with increasing alteration angle of course overall, i.e. the ratio is decrease from 95% to 88% with alteration angle of course of own ship increasing from 1 to 60 . And it can be observed from Fig. 34 that the ratio of actual DCPA to expected DCPA is very high and fluctuating between 95% and 98% with increasing alteration angle of course for overtaking situation. Furthermore, no matter in term of COLREGs or in practice, a
Fig. 25. The varied trend of expected and actual TCPA. 494
X. Wang et al.
Ocean Engineering 146 (2017) 486–497
Fig. 30. The TCPA between two ships. Fig. 27. Trajectories of two ships.
Fig. 31. Relative distance between two ships. Fig. 28. Relative trajectories of target ship.
Fig. 29. The DCPA between two ships.
Fig. 32. The varied trend of expected and actual DCPA. 495
X. Wang et al.
Ocean Engineering 146 (2017) 486–497
Meanwhile, based on the collision avoidance dynamic support system, the quantitative results of ship maneuverability effect on the action which is taken to avoid collision in close-quarters situation are revealed in this study. 5. Conclusions In this article, a ship maneuverability based collision avoidance dynamic support system in close-quarters situation is presented. The dynamic calculation model of collision avoidance parameter is employed to calculate the dynamic DCPA and TCPA in real-time when ship is maneuvering. Then the collision avoidance dynamic support system is developed by combining the mathematical model of ship maneuvering motion, the control mechanism of ship maneuvering motion as well as the dynamic calculation model of collision avoidance parameter. Following this approach, the proposed system is able to eliminate the insufficiency of neglect of ship maneuverability in the process of avoiding collision. Moreover, by incorporating the close-quarters situation into the proposed collision avoidance dynamic support system, simulation examples consisting three encounter scenarios of two ships in closequarters situation are applied to demonstrate the significance and necessity of ship maneuverability in the process of collision avoidance and illustrate the merits and effectiveness of the proposed system. The simulation results show that the proposed dynamic support system is a reasonable, effective and practicable system for collision avoidance, particularly in close-quarters situation. Furthermore, since the different velocity ratio of two meeting ships is another factor in the process of collision avoidance, our future research will be concentrated on the influence of different velocity ratio of two meeting ships on the process of collision avoidance.
Fig. 33. The varied trend of expected and actual TCPA.
Acknowledgements This work is supported in part by National Natural Science Foundation of China (Grant 51309041, 51179019 and 61374114), and the Fundamental Research Program for Key Laboratory of the Education Department of Liaoning Province (Grant. LZ2015006). References Fig. 34. The ratio of actual DCPA to expected DCPA.
Ahn, J.H., Rhee, K.P., You, Y.J., 2012. A study on the collision avoidance of a ship using neural networks and fuzzy logic. Appl. Ocean Res. 37, 162–173. Cahill, R.A., 1982. The avoidance of close quarters in clear weather. J. Navig. 35 (1), 151–159. Chai, T., Weng, J.X., Xiong, D.Q., 2017. Development of a quantitative risk assessment model for ship collisions in fairways. Saf. Sci. 91, 71–83. Cockcroft, A.N., Lameijer, J.N.F., 2011. A Guide to the Collision Avoidance Rules: International Regulations for Preventing Collisions at Sea, seventh ed. Butterworth–Heinemann. Eleftheria, E., Apostolos, P., Markos, V., 2016. Statistical analysis of ship accidents and review of safety level. Saf. Sci. 85, 282–292. Goerlandt, F., Kujala, P., 2014. On the reliability and validity of ship–ship collision risk analysis in light of different perspectives on risk. Saf. Sci. 62, 348–365. Goerlandt, F., Montewka, J., 2015. A framework for risk analysis of maritime transportation systems: a case study for oil spill from tankers in a ship–ship collision. Saf. Sci. 76, 42–66. Goerlandt, F., Montewka, J., Kuzmin, V., Kujala, P., 2015. A risk-informed ship collision alert system: framework and application. Saf. Sci. 77, 182–204. Hilgert, H., 1983. Defining the close-quarters situation at sea. J. Navig. 36 (3), 454–461. Hong, B.G., Yang, L.J., 2012. Ship Handing. Dalian Maritime University Press, Dalian. Hwang, C.N., 2002. The integrated design of fuzzy collision-avoidance and H∞Autopilots on ships. J. Navig. 55 (1), 117–136. International Maritime Organization (IMO), 1981. Definition of the Term “Close-quarters Situation”. Note by the Government of the German Democratic Republic, NAV 26/4/ 2. International Maritime Organization (IMO), 2001. International Regulations for Preventing Collisions at Sea (COLREGs) 1972. Le, M.D., Nguyen, S.H., Nguyen, L.A., 2004. Study on a new and effective fuzzy PID ship autopilot. Artif. Life Robot. 8, 197–201. Li, B., Pang, F.W., 2013. An approach of vessel collision risk assessment based on the D-S evidence theory. Ocean. Eng. 74, 16–21. Perera, L.P., Carvalho, J.P., Soares, C.G., 2011. Fuzzy logic based decision making system for collision avoidance of ocean navigation under critical collision conditions. J. Mar. Sci. Technol. 16, 84–99.
substantial alteration of course is the most common and effective action when a ship involved in a close-quarters situation. Significant differences are also observed for a certain substantial alteration angle of course in each scenario (as shown in Figs. 16, 26 and 34). For instance, when the alteration angle of course is 60 , the ratio of actual DCPA to expected DCPA in head-on situation is 77% which is nearly three quarters, it means that the difference between actual DCPA and expected DCPA is very large and visible. At the same time, the ratio of actual DCPA to expected DCPA is 88% in crossing situation, it means that the difference between actual DCPA and expected DCPA is also large. Meanwhile, the ratio of actual DCPA to expected DCPA is 96% in overtaking situation, it means that the difference between actual DCPA and expected DCPA is small. On the basis of above analyses, it is obvious that the difference between the actual DCPA and expected DCPA of meeting ships are universal exist in the close-quarters situation due to ship maneuverability. Moreover, it can be seen that the maximum difference between actual DCPA and expected DCPA and the lowest ratio of actual DCPA to expected DCPA are indicated in the head-on situation on the whole, followed by the crossing situation, and the last is the overtaking situation. As a result, it is easily to find out that the main scenario which is influenced by ship maneuverability during collision avoidance is head-on situation. The main reason is that the absolute value of course difference and relative velocity of two meeting ships are maximum in head-on situation by comparing with the crossing situation and overtaking situation. 496
X. Wang et al.
Ocean Engineering 146 (2017) 486–497 Yang, Y.S., 1995. Study on mathematical model for ship collision avoidance dynamic system. J. Dalian Marit. Univ. 21 (1), 30–36. Yang, Y.S., Yu, X.L., Jia, X.L., 1999. Robust PID autopilot for ships. J. Dalian Marit. Univ. 25 (4), 11–15. Yang, Y.S., Ren, J.S., 2003. Adaptive fuzzy robust tracking controller design via small gain approach and its application. IEEE Trans. Fuzzy Syst. 11 (6), 783–795. Yoshimura, Y., Nomoto, K., 1978. Modeling of manoeuvring behaviour of ships with a propeller idling, boosting and reversing. J. Soc. Nav. Archit. Jpn. 144, 57–69 (in Japanese). Zhang, Y.K., Zhao, L.D., 1991. The concept and model of close-quarters situation. J. Dalian Marit. Univ. 17 (1), 7–13. Zhang, J.F., Zhang, D., Yan, X.P., Haugen, S., Soares, C.G., 2015. A distributed anticollision decision support formulation in multi-ship encounter situations under COLREGs. Ocean. Eng. 105, 336–348.
Tam, C.K., Bucknall, R., Greig, A., 2009. Review of collision avoidance and path planning methods for ships in close range encounters. J. Navig. 62, 455–476. Tam, C.K., Bucknall, R., 2010a. Collision risk assessment for ships. J. Mar. Sci. Technol. 15, 257–270. Tam, C.K., Bucknall, R., 2010b. Path-planning algorithm for ships in close-range encounters. J. Mar. Sci. Technol. 15, 395–407. Tam, C.K., Bucknall, R., 2013. Cooperative path planning algorithm for marine surface vessels. Ocean Eng. 57, 25–33. Tsou, M.C., Kao, S.L., Su, C.M., 2010. Decision support from genetic algorithms for ship collision avoidance route planning and alerts. J. Navig. 63 (1), 167–182. Wang, F.C., 1991. A discussion on the explanation of “close-quarters situation”. J. Dalian Mar. Coll. 17 (1), 1–6. Xue, Y.Z., Clelland, D., Lee, B.S., Han, D.F., 2011. Automatic simulation of ship navigation. Ocean. Eng. 38 (17–18), 2290–2305.
497