Multi-Ship Collision Avoidance Decision-Making Based on Collision Risk Index

Most maritime accidents are caused by human errors or failures. Providing early warning and decision support to the officer on watch (OOW) is one of the primary issues to reduce such errors and failures. In this paper, a quantitative real-time multi-ship collision risk analysis and collision avoidance decision-making model is proposed. Firstly, a multi-ship real-time collision risk analysis system was established under the overall requirements of the International Code for Collision Avoidance at Sea (COLREGs) and good seamanship, based on five collision risk influencing factors. Then, the fuzzy logic method is used to calculate the collision risk and analyze these elements in real time. Finally, decisions on changing course or changing speed are made to avoid collision. The results of collision avoidance decisions made at different collision risk thresholds are compared in a series of simulations. The results reflect that the multi-ship collision avoidance decision problem can be well-resolved using the proposed multi-ship collision risk evaluation method. In particular, the model can also make correct decisions when the collision risk thresholds of ships in the same scenario are different. The model can provide a good collision risk warning and decision support for the OOW in real-time mode.


Introduction
Shipping has been one of the dominant modes of global trade, which accounts for more than 90% of international cargo trade [1]. Ship collision is a major type of maritime accident that often results in significant casualties, environmental pollution and economic losses. Even though ships have many advanced equipment (e.g., Automatic Identification System, AIS, Automatic Radar Plotting Aids, ARPA and Integrated Navigation Systems, INS), it is stated that, in many marine accident surveys, more than 80% of maritime traffic accidents are related to human factors [2][3][4][5]. Therefore, the impact of human factors on accidents still plays an important role [6][7][8]. It is therefore complex and important to focus on the decision-making process of ship collision avoidance. Particular attention is paid to the fact that the maneuvering difficulty of the officer on watch (OOW) increases exponentially in harbors and sea areas with much higher density ship traffic flows. The International Maritime Organization (IMO) issued International Code for Collision Avoidance at Sea (COLREGs) in 1972 [9]. The COLREGs outline ship-related navigation rules and concepts, which requires all ships engaged in international voyages to comply with the convention. Therefore, Hilgert, H et al. [10] have been

Collision Risk Model
In multi-ship encounters, the real-time CRI with respect to two ships can be evaluated by the relative motion parameters between the ships [21,49,50], i.e., distance to the closest point of approach (DCPA), time to the closest point of approach (TCPA), the relative distance (D), relative bearing (α OT ) and the relative speed (V OT ), as shown in Figure 1.
Based on the distribution and for ease of expression, each ship treats itself as the "Own Ship (OS)" from a first-person perspective. The other ships are considered to be "Target Ship (TS)" if it causes an obstacle to the normal navigation of the OS. In a typical encounter, the collision risk index can usually be calculated using relative motion parameters for two ships [33], as shown in Figure 1.
where SO is the position of OS, ST is the position of TS, θOT is the relative course of TS, D is the relative distance to TS, αT is the true relative position direction to TS, αOT is the angle converted from in the body-fixed coordinate system and VOT is the relative speed of TS. The intermediate parameters in (1)(2)(3)(4) can be calculated by: where Xo = [xo, yo, Vo, θo] T is the motion state of the OS, and XT = [xT, yT, VT, θT] T is the motion state of TS. V x and V y are the relative speed components of the TS on the X and Y axes, respectively, and xOT and yOT are the relative distance components of the TS on the X and Y axes, respectively. The relative speed of the TS is calculated as follows: (8) where θO is the heading of OS (deg), VO is the speed of OS (kn), θT is the heading of TS (deg) and VT is the speed of TS (kn). The relative course and the true course of the TS are calculated as follows [45]: The relative motion parameters are calculated as follows [45]: where S O is the position of OS, S T is the position of TS, θ OT is the relative course of TS, D is the relative distance to TS, α T is the true relative position direction to TS, α OT is the angle converted from in the body-fixed coordinate system and V OT is the relative speed of TS. The intermediate parameters in (1)-(4) can be calculated by: where X o = [x o , y o , V o , θ o ] T is the motion state of the OS, and X T = [x T , y T , V T , θ T ] T is the motion state of TS. V x and V y are the relative speed components of the TS on the X and Y axes, respectively, and x OT and y OT are the relative distance components of the TS on the X and Y axes, respectively. The relative speed of the TS is calculated as follows: where θ O is the heading of OS (deg), V O is the speed of OS (kn), θ T is the heading of TS (deg) and V T is the speed of TS (kn). The relative course and the true course of the TS are calculated as follows [45]: Fuzzy logic is good at expressing the knowledge and experience with unclear boundaries, which is similar with the case of the ship collision risk evaluation. Therefore, fuzzy synthesis judgment is introduced [51,52]. For instance, numerous studies have been done by many researchers using fuzzy logic methods to model the level of collision risk through the membership functions [32,33,53]. In view of the fuzziness and uncertainty of the CRI, many researchers have applied the fuzzy comprehensive evaluation method to the evaluation of the CRI. One of the most important aspects of the fuzzy method is to establish the membership functions of the fuzzy set. In practice, in addition to the TCPA and DCPA, other parameters such as the relative distance, relative bearing and speed ratio also have impact on the OOW to evaluate the level of collision risk. In this study, based on the geometry of ship collision, the above parameters are selected as the influencing factors of the CRI calculation, of which the memberships are defined as follows: (1) The membership function of DCPA [45]: In which the process of d 1 and d 2 are shown as follows [45]: where d 1 represents the minimum distance of the two ships, and d 2 represents the safe encounter range. K is the coefficient determined by the factors as the instability of the ship state and the two ships' incoordination and is generally 1.5-2.0 [32]; K is set to 2.0 in this paper. The OOWs can make adjustments on the value according to the navigation environment in practice.
(2) The membership function of TCPA [45]: where t 1 is the time of ship collision, which represents the time taken by the giving-way ship from the latest moment of avoidance action to the closest point of approach. t 2 is the time of taking attention to the TS, which represents the time when the ship starts to pay attention to the TS. It is generally accepted that 6-8 nautical miles is the distance at which ships can navigate freely [32]. Therefore, for safety reasons, 8 nautical miles is used in this paper as the distance between ships. Then, we set the time corresponding to t 2 when the closest point of approach from the OS to the TS is no larger than 8 nautical miles. V OT is the relative speed between the two ships. D 1 and D 2 represent the latest operation distance and the safe distance for the give-way ship to take collision avoidance measures. t 1 and t 2 are calculated as follows [45]: (3) The membership of D [32]: where the relative distance (D) indicates the risk of collision between ships from a spatial perspective. The smaller the relative distance, the worse the operation effectiveness. When performing steering operations, the closer between the two ships, the smaller the change in DCPA, as well as the higher the probability of collision. D 1 represents the minimum distance to be avoided by the give-way ship taking emergency avoidance action. The value of D 1 generally depends on the size of the give-way ship and its maneuverability. It is generally set to be 12 to 14 times the ship length. The empirical formula for D 2 [54], which is a safe distance to give way to a ship for collision avoidance, is as follows: (4) The membership of a OT : where α OT represents the relative bearing of the TS [32]. The influence of relative bearing on the collision risk index is mainly related to the ease of taking collision avoidance actions. If there is no apparent change in the compass bearing of the incoming ship, it should be assumed that there is a risk of collision. Moreover, the psychological impact of the different bearing on the OOW should be taken into account. The TS is more dangerous when it is on the starboard side of the ship than on the port side and more dangerous when it is in forward of the beam than aft of the beam. Generally speaking, the risk is the largest when the relative bearing of the TS is coming from 19 • of the OS with other conditions unchanged [54].
(5) The membership of K [45]: where θ T and θ 0 are the course of the TS and the OS, respectively. Besides, C ∈ [0, 180] is a constant coefficient [32]. The membership functions of DCPA and D reveal the spatial complexity of the collision risk, and the membership function of TCPA reflects the time complexity of the collision risk. α OT and K are associated with the difficulty of the ship in avoidance of collision and, also, affect the psychology of the OOW.
Based on the above definition, the collision risk between two ships at time t can be obtained based on the fuzzy method. The collision risk model is defined as follows [45]: where f CRI is the collision risk function, and U is the matrix of the membership function of the target factor, which outputs the collision risk model. W is the weight matrix, which belongs to (0,1), and the sum of them is 1. w DCPA , w TCPA , w D , wα OT and w K are the set weights of the membership functions, which are usually set as 0.400, 0.367, 0.133, 0.067 and 0.033, respectively [45]. In addition, a negative TCPA indicates that the two ships have passed each other.

COLREGs
COLREGs are maritime traffic regulations developed by the International Maritime Organization (IMO) to prevent and avoid collisions between ships at sea [9]. According to the COLREGs, the encounter situations of the ship are divided into three categories based on the relative bearing of the TS, e.g., head-on, overtaking and crossing. As shown in Figure 2, with the OS as the center, when the relative bearing of the TS belongs to a OT ∈ [0 where θT and θ0 are the course of the TS and the OS, respectively. Besides, The membership functions of DCPA and D reveal the spatial complexity of the collision risk, and the membership function of TCPA reflects the time complexity of the collision risk. αOT and K are associated with the difficulty of the ship in avoidance of collision and, also, affect the psychology of the OOW.
Based on the above definition, the collision risk between two ships at time t can be obtained based on the fuzzy method. The collision risk model is defined as follows [45]: where fCRI is the collision risk function, and U is the matrix of the membership function of the target factor, which outputs the collision risk model. W is the weight matrix, which belongs to (0,1), and the sum of them is 1. wDCPA, wTCPA, wD, wαOT and wK are the set weights of the membership functions, which are usually set as 0.400, 0.367, 0.133, 0.067 and 0.033, respectively [45]. In addition, a negative TCPA indicates that the two ships have passed each other.

COLREGs
COLREGs are maritime traffic regulations developed by the International Maritime Organization (IMO) to prevent and avoid collisions between ships at sea [9]. According to the COLREGs, the encounter situations of the ship are divided into three categories based on the relative bearing of the TS, e.g., head-on, overtaking and crossing. As shown in Figure 2, with the OS as the center, when the relative bearing of the TS belongs to   The ship's responsibility to avoid collisions is divided into two categories under rules 15 and 16 of the COLREGs. When the OS is on the starboard side of the TS, the OS is a stand-on ship and should keep its course and speed. Otherwise, the OS is a give-way ship, which is responsible for avoidance and usually manoeuvers to steer to avoid collision. Moreover, the OS should not cross from ahead of the TS. The rules are not specific to the various encounter situations. Therefore, Zhang et al. [41] proposed a specific decision method for the encounter situations of large and small angle crossings, which is a good way to simplify ship avoidance decisions in multi-ship encounters, i.e., the change of course for large crossing angles and the change of speed for small crossing angles.

DCPA Calculation with Course Alteration
At all stages of the ship voyage, the ship can obtain ship dynamics data such as speed, course, position and relative motion parameters through AIS, APRA and other sensors in real-time mode. It is vital to always maintain a safe distance from the TS in ship trajectory planning. DCPA is one of the key indicators to assess the overall performance of the planned trajectory. However, it should be noted that the ships may change their courses during collision avoidance operations, which would influence the calculations of the DCPA values. This issue should be considered appropriately.
During ship collision avoidance, the ship that needs to take action, whether changing course or speed, needs to stay sufficiently away from the TS to the OS. This means that the value of the DCPA should always be larger than the safe distance after the collision avoidance risk is activated. For instance, after the collision risk initiation, the whole process of the collision avoidance operation between ship 1 and ship 2 is shown in Figure 3. There are three phases, as discussed below. The ship's responsibility to avoid collisions is divided into two categories under rules 15 and 16 of the COLREGs. When the OS is on the starboard side of the TS, the OS is a stand-on ship and should keep its course and speed. Otherwise, the OS is a give-way ship, which is responsible for avoidance and usually manoeuvers to steer to avoid collision. Moreover, the OS should not cross from ahead of the TS. The rules are not specific to the various encounter situations. Therefore, Zhang et al. [41] proposed a specific decision method for the encounter situations of large and small angle crossings, which is a good way to simplify ship avoidance decisions in multi-ship encounters, i.e., the change of course for large crossing angles and the change of speed for small crossing angles.

DCPA Calculation with Course Alteration
At all stages of the ship voyage, the ship can obtain ship dynamics data such as speed, course, position and relative motion parameters through AIS, APRA and other sensors in real-time mode. It is vital to always maintain a safe distance from the TS in ship trajectory planning. DCPA is one of the key indicators to assess the overall performance of the planned trajectory. However, it should be noted that the ships may change their courses during collision avoidance operations, which would influence the calculations of the DCPA values. This issue should be considered appropriately.
During ship collision avoidance, the ship that needs to take action, whether changing course or speed, needs to stay sufficiently away from the TS to the OS. This means that the value of the DCPA should always be larger than the safe distance after the collision avoidance risk is activated. For instance, after the collision risk initiation, the whole process of the collision avoidance operation between ship 1 and ship 2 is shown in Figure 3. There are three phases, as discussed below.

Ship 1
Ship 2 1. d1 (from Ts to T2): This period is the whole process from the starting of Ship 2's changing course to the end. In addition, both ships are changing course. 2. d2 (from T2 to T3): From the end of ship 2's changing course to ship 1's changing course, while ship 2 has returned to its original course. 3. d3 (from T3 to T4): Ship 1 has changed course sufficiently. Both ships have returned to their original course at this stage, while the collision avoidance operation is done. Assuming the OS's initial position is (x0, y0) and the initial position of the TS is (xT, yT), their initial speed and course are V0, θ0 and VT, θT, respectively.
The position of the OS after time t is as follows [41]: The position of the TS after time t is as follows [41]: Assuming the OS's initial position is (x 0 , y 0 ) and the initial position of the TS is (x T , y T ), their initial speed and course are V 0 , θ 0 and V T , θ T , respectively.
The position of the OS after time t is as follows [41]: The position of the TS after time t is as follows [41]: Therefore, the relative distance between the two ships after time t is as follows [41]: As above, the minimum value of D is the bottom of the quadratic function, which corresponds to the time t when taken to the minimum value, as follows: In practice, multi-ship encounter situations are common. The DCPA is used as the basis for collision-avoidance decisions, and the real-time DCPA system calculates the relative distance between ships. In the case of changing the speed of the voyage for collision avoidance, one only needs to change the speed value in the equation.
As shown in Figure 2, according to the COLREGs, when there is a TS on the starboard side of the OS in an encounter situation, it should give way to the TS. Otherwise, the OS is a stand-on ship and should keep the initial speed and course. In this section, the procedure is also designed to ensure that giving-way ships take a more effective approach to collision avoidance in the same amount of time. In accordance with the COLREGs and good seamanship, as well system design requirements, the procedure is mainly based on changing the course and changing the speed. When necessary, the procedure is activated, and the program is shown in Figure 4.

Procedures for Changing Course Decisions
Generally, a giving-way ship to avoid collision by changing course is a major part of the procedure and is also suggested by the rules. In this part of the procedure, the focus is on the course change range and the time range of the course change. The ships are capable of safely avoiding collision under the prevailing circumstances and avoiding the end of the collision. However, it should be noted that, when the ship changes course too much or sails on the new course for too long, the ship's trajectory will deviate too much from its original trajectory, which is also undesirable. Therefore, the procedure sets the maximum and minimum values of the relevant parameters in this paper, e.g., the range of change of the course is set for 25 • to 45 • ; the navigation time on the new course is set to 3 to 10 min. Meanwhile, we need to make sure that the TS can see the actions taken by our ship. Therefore, the change of the course amplitude and steering time should not be too small, either. Therefore, the proposed course change procedure is able to select the optimal offset and the number of decisions as far as possible in accordance with good seamanship while ensuring safety.

Procedures for Changing Speed Decisions
When a small steering angle is required to avoid collision, reducing the speed may be a more effective way to avoid the collision [41]. Therefore, when there is a small crossing angle encounter situation between the two ships, it is designed to avoid collision by reducing the speed in this paper. When it is judged necessary to activate a change of the speed program, the speed will be reduced by 5% each time to update the calculation of the closest point of approach (CPA) until it is possible to provide an effective enough way to avoid the collision between the two ships. Allow more space and time for avoidance actions. When the owned ship as a giving-way ship is encountering another ship with a small crossing angle, it is better to take the decision of changing speed rather than changing course. Therefore, the deceleration procedure is another part of the decision-making procedure.
the OS in an encounter situation, it should give way to the TS. Otherwise, the OS is a stand-on ship and should keep the initial speed and course. In this section, the procedure is also designed to ensure that giving-way ships take a more effective approach to collision avoidance in the same amount of time. In accordance with the COLREGs and good seamanship, as well system design requirements, the procedure is mainly based on changing the course and changing the speed. When necessary, the procedure is activated, and the program is shown in Figure 4.

Real-Time Decision Support Procedure
In order to perform a distributive collision avoidance decision-making procedure, the procedure is used individually for each ship. Ships can use navigational aids to obtain static and dynamic information on the nearby ships. Based on the ship dynamics information, the real-time collision risk index is calculated. The decision is activated when the collision risk index of the two ships exceeds the threshold, and the procedure is used separately for each ship. At the same time, the real-time collision avoidance decision procedure automatically selects a decision to change course or change speed based on the angle of the encounter. On the contrary, when the real-time CRI does not exceed the collision threshold, the ships would keep their courses and speeds while monitoring the environment. In practice, collision avoidance manoeuvers between ships are usually one-off. The risky situation can best be defused by a single change of course or speed [41]. Therefore, it is essential to avoid taking a series of small manoeuvers to avoid collision, which is not recommended by the rules. It should be noted that the designed procedure assumes the ship returns to its original course when it finishes collision-avoidance operations. This strategy is widely accepted in open waters. This can also explain that the deviation of the ship's course caused by steering to avoid the collision is negligible compared to the entire distance travelled by the ship from a macro perspective. After the collision avoidance is completed, the ship will continue on the new track to determine the collision risk index of nearby ships in real-time until the end of the voyage.

Case Studies
In this section, the effectiveness of the proposed approach is evaluated through a series of simulation experiments. The CRI value is tested through comparison simulations under the same encounter situations. Table 1 represents the initial multi-ship encounter scenario among four ships. The ships in such encounter scenario would collide with each other without taking any collision avoidance action. The encounter scenario consists of four ships, and the initial conditions include the position, speed, course and CRI of the ships. Ships are compliant with COLREGs and good seamanship in collision avoidance decision-making. In the simulations, real-time decisions are made using the proposed CRI approach. The decision action procedure switches between course alteration and speed change due to different crossing angles. It achieves a better collision avoidance effect in a shorter period of time and improves the efficiency of collision avoidance.
In the next subsection, the performance of the proposed collision-avoidance decision procedure based on the CRI is evaluated by simulations and correlation analyses of scenarios under different CRI thresholds.

Simulation Scenario 1
In order to better present the specifics of the ship's collision avoidance behaviors, some details of the ships' decision are shown in Table 2, which presents the specific operations for a ship with a CRI threshold of 0.6, e.g., the time of starting to changing course, speed and the new heading; the amount of course change and the nearest relative distance of ship 1 from the other ships, as well as the corresponding moment. Figure 5 shows the typical ship trajectories and positions from the simulations at several typical time points.    Figure 5 show the positions and paths of the simulation ships involved in the experiment. When t = 2 s, S3 makes the first decision and operation when it encounters S2 with a small crossing angle. S3 reduces speed by 30% and keeps for 1498 s. Besides, S3 also encounters S4 with a large crossing angle. S3 is on the port side of S4. Therefore, she turns to starboard for 30 ° and keeps for 597 s on the new course. Similarly, since S1 and S2 are encountered with a large crossing angle, S1 makes the decision of turning to the starboard by 33 ° and keeps on the new course for 405 s. As shown in Figure 6a,d, the DCPA of S1 with S2, S3 and S4 increases, and the real-time CRI of S1 with S2 and S4 decreases. S1 reduces the speed to 60% of the initial speed to pass the aft of S3 at t = 689 s, which is also stipulated by COLREGs and good seamanship. According to Figure 6a, the CRI between S1 and S2 is significantly lower at 282 s and 689 s, indicating that the action is beneficial for collision avoidance. At last, S4 encounters S1 with a large crossing angle at 625 s. S4 turns 25° to starboard and keeps the new course for 158 s. As shown in Figure 6b, the TCPA values between S1 and the other ships were changed to negative at 687 s, 1134 s and 1103 s, respectively, which indicates that the collision avoidance actions are effective. In the simulation, S2 did not take any actions. According to the distributed multi-ship collision avoidance and COLREGs, although S2 is on the port side of S4, they did not incur a collision risk in this scenario. In this simulation, each ship's collision avoidance actions are helpful and do not increase the collision risk.   Figure 5 show the positions and paths of the simulation ships involved in the experiment. When t = 2 s, S 3 makes the first decision and operation when it encounters S 2 with a small crossing angle. S 3 reduces speed by 30% and keeps for 1498 s. Besides, S 3 also encounters S 4 with a large crossing angle. S 3 is on the port side of S 4 . Therefore, she turns to starboard for 30 • and keeps for 597 s on the new course. Similarly, since S 1 and S 2 are encountered with a large crossing angle, S 1 makes the decision of turning to the starboard by 33 • and keeps on the new course for 405 s. As shown in Figure 6a,d, the DCPA of S 1 with S 2 , S 3 and S 4 increases, and the real-time CRI of S 1 with S 2 and S 4 decreases. S 1 reduces the speed to 60% of the initial speed to pass the aft of S 3 at t = 689 s, which is also stipulated by COLREGs and good seamanship. According to Figure 6a, the CRI between S 1 and S 2 is significantly lower at 282 s and 689 s, indicating that the action is beneficial for collision avoidance. At last, S 4 encounters S 1 with a large crossing angle at 625 s. S 4 turns 25 • to starboard and keeps the new course for 158 s. As shown in Figure 6b, the TCPA values between S 1 and the other ships were changed to negative at 687 s, 1134 s and 1103 s, respectively, which indicates that the collision avoidance actions are effective. In the simulation, S 2 did not take any actions. According to the distributed multi-ship collision avoidance and COLREGs, although S 2 is on the port side of S 4 , they did not incur a collision risk in this scenario. In this simulation, each ship's collision avoidance actions are helpful and do not increase the collision risk.

Simulation Scenario 2
In this scenario, the threshold of the CRI is set to 0.7 in the simulations. Figure 7 shows the trajectories and positions of the ships at several typical time points.
In this simulation, it can be seen that the ships' collision avoidance actions shown in Figure 7 and Table 3 are delayed compared with the first simulation. When t = 429 s, S1 encounters S2 with a large crossing angle, and the CRI is equal to 0.7 in Figure 8. S1 turns 40° to starboard and keeps on the new course for 599 s. As shown in Figure 8a,d. The real-time CRI of S1 and S2 and S4 are reduced at 429 s. Meanwhile, S1 makes the decision of turning to the starboard and the stern of S3, which results in a slight increase in the collision risk index with S3. The DCPA value of S1 with the other ships is increased in Figure 8d. S3 encounters S2 with a small crossing angle at 77 s. S3 reduces the speed for 35% and keeps on for 1424 s. S3 and S4 form a large crossing angle encounter situation; thus, S3 is turning to the starboard by 32° and remains on the new heading for 581 s. In Figure 8b, the TCPA values between S1 and the other ships change to negative at 687 s, 1134 s and 1103 s, respectively.

Simulation Scenario 2
In this scenario, the threshold of the CRI is set to 0.7 in the simulations. Figure 7 shows the trajectories and positions of the ships at several typical time points.
In this simulation, it can be seen that the ships' collision avoidance actions shown in Figure 7 and Table 3 are delayed compared with the first simulation. When t = 429 s, S 1 encounters S 2 with a large crossing angle, and the CRI is equal to 0.7 in Figure 8. S 1 turns 40 • to starboard and keeps on the new course for 599 s. As shown in Figure 8a,d. The real-time CRI of S 1 and S 2 and S 4 are reduced at 429 s. Meanwhile, S 1 makes the decision of turning to the starboard and the stern of S 3 , which results in a slight increase in the collision risk index with S 3 . The DCPA value of S 1 with the other ships is increased in Figure 8d. S 3 encounters S 2 with a small crossing angle at 77 s. S 3 reduces the speed for 35% and keeps on for 1424 s. S 3 and S 4 form a large crossing angle encounter situation; thus, S 3 is turning to the starboard by 32 • and remains on the new heading for 581 s. In Figure 8b

Simulation Scenario 3
In this simulation, the threshold of the CRI for all ships is set to 0.9 for the collision avoidance decision-making procedure given the same encounter situations. The trajectories of the four ships and their positions at typical time points are shown in Figure 9.

Simulation Scenario 3
In this simulation, the threshold of the CRI for all ships is set to 0.9 for the collision avoidance decision-making procedure given the same encounter situations. The trajectories of the four ships and their positions at typical time points are shown in Figure 9.
According to the data of Table 4, S 3 encounters S 4 with a large crossing angle at 309 s, which turns 40 • to starboard and keeps on the new angle for 599 s. Meanwhile, S 3 also encounters S 4 with a small crossing angle at 309 s. S 3 reduces the speed to 45% of the initial speed. Similarly, when t = 577 s, S 1 turns 45 • to starboard to avoid S 2 . As shown in Figure 10a,d, the DCPA values of S 1 and S 2 , S 3 and S 4 are increased, and the CRI values of S 1 and S 3 and S 4 decrease at 577 s, but due to the late change of course, the CRI values of S 1 and S 2 continue increasing. The TCPA values between S 1 and S 2 , S 3 and S 4 change to negative at 716 s, 994 s and 970 s in Figure 10b, respectively. The collision avoidance actions undertaken by the ships are effective. However, S 1 takes a large angle to change course in Figure 9. In the collision avoidance decision when the threshold of the CRI is 0.9, the ships are not properly coordinated, and the actions between the ships are relatively passive, which results in this scenario are much more difficult than scenario 1 and 2. Therefore, the highest risky situation occurs at 715 s in this scenario, when the distance between S 1 and S 2 is 574 m in Figure 10c. Although the two ships are almost parallel with the opposite course, it is still an undesirable situation. This is due to the large threshold value results in the late decision to perform a collision-avoidance action. According to the data of Table 4, S3 encounters S4 with a large crossing angle at 309 s, which turns 40° to starboard and keeps on the new angle for 599 s. Meanwhile, S3 also encounters S4 with a small crossing angle at 309 s. S3 reduces the speed to 45% of the initial speed. Similarly, when t = 577 s, S1 turns 45° to starboard to avoid S2. As shown in Figure 10a,d, the DCPA values of S1 and S2, S3 and S4 are increased, and the CRI values of S1 and S3 and S4 decrease at 577 s, but due to the late change of course, the CRI values of S1 and S2 continue increasing. The TCPA values between S1 and S2, S3 and S4 change to negative at 716 s, 994 s and 970 s in Figure 10b, respectively. The collision avoidance actions undertaken by the ships are effective. However, S1 takes a large angle to change course in Figure 9. In the collision avoidance decision when the threshold of the CRI is 0.9, the ships are not properly coordinated, and the actions between the ships are relatively passive, which results in this scenario are much more difficult than scenario 1 and 2. Therefore, the highest risky situation occurs at 715 s in this scenario, when the distance between S1 and S2 is 574 m in Figure 10c. Although the two ships are almost parallel with the opposite course, it is still an undesirable situation. This is due to the large threshold value results in the late decision to perform a collision-avoidance action.

Simulation Scenario 4
It should be noted, in this scenario, that when facing with the same encounter scenario, the CRI selected by different OOWs during collision avoidance decision-makings may vary a lot due to their different navigational experience and their different psychological states, as well as other types of influencing factors. That is to say, the OOWs may make different decisions in the same encounter scenario. Therefore, the thresholds of different CRI values are applied to the four encounter ships during the collision avoidance decision-making to verify the performance of the proposed approach. The trajectories and positions of the simulated ships at typical time points are presented in Figure 11.

Simulation Scenario 4
It should be noted, in this scenario, that when facing with the same encounter scenario, the CRI selected by different OOWs during collision avoidance decision-makings may vary a lot due to their different navigational experience and their different psychological states, as well as other types of influencing factors. That is to say, the OOWs may make different decisions in the same encounter scenario. Therefore, the thresholds of different CRI values are applied to the four encounter ships during the collision avoidance decision-making to verify the performance of the proposed approach. The trajectories and positions of the simulated ships at typical time points are presented in Figure 11. Table 5 and Figure 12 illustrate the ships' action details and Parameter changes. Since each ship selects different thresholds, the results are quite different from scenarios 1-3. At t = 233 s, S 1 and S 3 encounter each other with a small crossing angle, while S 1 and S 2 encounter with a large crossing angle. S 1 turns 25 • to starboard and reduces to 85% of its original speed. The collision risk reference threshold for S 1 is 0.6, and the timing of the action is seen to be on time, i.e., the steering and speed range are relatively small. When t = 732 s, S 1 makes the decision to continue to reduce speed in order to safely sail from aft of S 3 and reduces the speed to 45% of the original speed. Similarly, S 3 encounters at a large crossing angle with S 4 and a small crossing angle with S 2 at 309 s. S 1 makes the decision to turn 40 • to starboard, as well as navigates on the new course for 597 s, which reduces the speed to 45% of the original speed to complete the collision avoidance manoeuver. The timing of the decision is related to the ship's initial position and state of motion. The decision of S 4 is the last, and its first decision occurs at 605 s, which includes turning 25 • to starboard. S 4 crosses from the aft region of S 1 , and S 4 crosses from the aft region of S 1 . All the ships' behaviors are consistent with the COLREGs. In this scenario, the higher the threshold of CRI chosen by S 3 , the greater the magnitude of the collision-avoidance manoeuver. threshold for S1 is 0.6, and the timing of the action is seen to be on time, i.e., the steering and speed range are relatively small. When t = 732 s, S1 makes the decision to continue to reduce speed in order to safely sail from aft of S3 and reduces the speed to 45% of the original speed. Similarly, S3 encounters at a large crossing angle with S4 and a small crossing angle with S2 at 309 s. S1 makes the decision to turn 40° to starboard, as well as navigates on the new course for 597 s, which reduces the speed to 45% of the original speed to complete the collision avoidance manoeuver. The timing of the decision is related to the ship's initial position and state of motion. The decision of S4 is the last, and its first decision occurs at 605 s, which includes turning 25° to starboard. S4 crosses from the aft region of S1, and S4 crosses from the aft region of S1. All the ships' behaviors are consistent with the COLREGs. In this scenario, the higher the threshold of CRI chosen by S3, the greater the magnitude of the collisionavoidance manoeuver.

Simulation Scenario 5
In this simulation, it is supposed that the OOW of S 1 may select a larger CRI threshold value, and the OOW of S 4 has less experience and uses a smaller threshold value. Moreover, the reference collision risk reference threshold varies for each ship, and the simulation test is conducted to verify the effectiveness of the procedure. The trajectories of the simulated ships at typical time points is presented in Figure 13.
According to Table 6 and Figure 14, S 3 and S 2 encounter with a small crossing angle at 77 s, and S 3 reduces speed to 65% of the original speed. According to the COLREGs, S 3 should be the give-way ship of S 4 . Therefore, S 3 turns starboard to keep clear and avoid S 4 at 182 s. Similarly, S 4 turns 42 • to starboard to avoid collision with S 1 and S 4 , which keeps on the new angle for 598 s. Finally, S 1 makes the decision of turning starboard for 45 • to avoid collision with S 1 and S 2 . Due to the reasoning that the collision risk threshold of S 1 is large, S 1 takes the last and largest action to avoid collision, which is not recommended by the COLREGs. As shown in Figure 14a,d. The collision risk of S 1 and S 4 decrease, and the DCPA values of S 1 and S 2 , S 3 and S 4 increase at 577 s. In Figure 14b, the TCPA values between S 1 and S 2 , S 3 and S 4 reduce to negative values at 716 s, 975 s and 842 s, respectively. The relative distances of the ships are shown in Figure 14c. When the CRI threshold is selected as a large value, a small relative distance can occur. Even though the collision is avoided, the encounter situation is more complicated. With higher demands on the OOWs' maneuvering levels, the OOWs are more likely to pay close attention to the ship dynamics of other ships in the vicinity. starboard to avoid collision with S1 and S4, which keeps on the new angle for 598 s. Finally, S1 makes the decision of turning starboard for 45° to avoid collision with S1 and S2. Due to the reasoning that the collision risk threshold of S1 is large, S1 takes the last and largest action to avoid collision, which is not recommended by the COLREGs. As shown in Figure 14a,d. The collision risk of S1 and S4 decrease, and the DCPA values of S1 and S2, S3 and S4 increase at 577 s. In Figure 14b, the TCPA values between S1 and S2, S3 and S4 reduce to negative values at 716 s, 975 s and 842 s, respectively. The relative distances of the ships are shown in Figure 14c. When the CRI threshold is selected as a large value, a small relative distance can occur. Even though the collision is avoided, the encounter situation is more complicated. With higher demands on the OOWs' maneuvering levels, the OOWs are more likely to pay close attention to the ship dynamics of other ships in the vicinity.

Analysis of trajectory safety
In the above cases, when different CRI thresholds are selected, the relative distances derived from taking the decision are quite different in Figures 6c, 8c and 10c. As shown in Figure 10c, the relative distances between S 1 and S 2 and S 1 and S 3 are much smaller than those in Figures 6c and 8c with the same encounter situations. These indicate that the collision-avoidance actions of the ships are more coordinated when the CRI threshold chosen is relatively low. S 3 reduces the speed to 70% and 65% of the initial speed at t = 2 s and t = 77 s, respectively, in Tables 2 and 3. However, S 3 reduces the speed to 45% of the initial speed at 309 s in Table 4, which brings a new challenge to the ship's speed reduction maneuver. Therefore, when the reference threshold is set to 0.9, the ship is too late to make the decision to avoid collision. Safety is low, and the OOW needs to be highly alert to his surroundings and places higher demands on the ship's maneuverability.

Analysis of Trajectory Efficiency
Tables 2-4 reveal the parameters of the ships' action measures in the collision avoidance process at different reference thresholds, respectively. In Figure 5, S 1 turns 33 • to starboard at 282 s and navigates on the new heading for 405 s, while, in Figure 7, the action is taken later, i.e., S 1 turns 40 • to starboard at 429 s and navigates on the new course for 599 s, due to the large CRI threshold. Table 7 presents the ship maneuvering details for the experimental results of Scenarios 1-3. According to the data in Table 5, when the selected CRI is 0.6, S 1 's trajectory deviates 1.315 nm from the original trajectory. However, with the increase of the CRI, the steering angle of the ship increases as well. When the selected CRI are 0.7 and 0.9, S 1 's trajectory deviates 2.513 nm and 2.990 nm. That is to say, the former is more economical. Therefore, the collision risk threshold that is too high will cause the ship to deviate from the original course to a large degree. As shown in Table 7, the deviation of the track at a collision risk threshold of 0.9 is more than twice as large as the case with a threshold of 0.6. This is also not recommended by good seamanship if action is taken too late. Tables 5 and 6 present the information on ship operations for different ships that have different CRI thresholds. Meanwhile, ships can successfully complete collision avoidance decisions even when the reference thresholds are different in Figures 11 and 13. Besides, in both scenarios, we also find that the larger CRI of the selected threshold leads to a later decision and a larger magnitude of manipulation.

Analysis of Maneuver Difficulty
According to Tables 2-4, when the threshold is 0.6, the steering amplitude taken is relatively small. On the contrary, when the thresholds are 0.7 and 0.9, the ships take larger steering actions. In multi-ship encounter scenario, taking excessive steering actions can easily create a secondary collision risk with other ships, which would result in one more operation. In Table 4 and Figure 9, S 3 takes multiple steering actions to avoid collision. In practice, it is much more difficult for the OOW to maneuver in this scenario. Furthermore, the OOW is prone to making the wrong decisions in such a stressful encounter situation. In scenarios 4 and 5 with different CRI thresholds, S 3 only takes one turn and is more effective, as shown in Figures 11 and 13. In reality, it is also common for different decision-makers to choose different thresholds. If this collision risk and collision avoidance decision model is actually used on-board a ship, it can provide the OOW with the best time to make a collision avoidance decision or raise an alarm if the real-time collision risk exceeds the threshold. Moreover, the OOW can get real-time access to path planning for collision-avoidance decisions.

Conclusions
In this paper, a distributed multi-ship collision avoidance decision-making on the premise of satisfying the COLREGs and good shipman ship is studied, which focuses on comparing the collision-avoidance decisions made by each ship choosing the same and different collision risk thresholds from its own perspective. It compensates for the excessive subjectivity of the OOW in the decision-making process. The model can provide a potential way to harmonize collision avoidance decision-making for the OOW in dealing with multi-ship encounter situations. The results indicate that: (1) The timing of the ship taking collision-avoidance action is closely related to the selected value of the CRI. This paper evaluates the validity of the proposed multi-ship collision-avoidance decision model based on the collision risk index through five groups of comparative experiments and carries out a detailed parameter analysis and finds that, when the collision risk threshold value taken by the ship is small (such as CRI = 0.6), the ship takes a more timely collision avoidance action, and the magnitude is in-line with the recommendation of the COLREGs. Therefore, through discussion and analysis, it is suggested that the threshold of the CRI be within (0.6, 0.7), which will also be affected by other factors such as visibility, sea state, etc. The OOWs need to adjust the CRI to the real-time navigational environment. Meanwhile, the OOW's experience in maneuvering ships should not be ignored as an aid to collision avoidance decision-making. (2) In multi-ship collision avoidance, different collision risk thresholds are set for experimentation, especially when different ships adopt different collision risk thresholds in the same scenario; the model can also provide decision aid and collision-avoidance warning for the OOW. Ship collision risk assessment models are the basis for collision-avoidance decisions. The integration of relevant data analysis indicators into visualization systems will be a way forward in future research. In addition, the consideration of the effects of ship maneuverability and static obstacles on collision-avoidance decisions should also be investigated as a key focus of future research.