3.1. Framework and Terminology

Distributed ship collision avoidance is made up of two procedures: the control and search procedures. Its framework is given in Figure 1.

Figure 1. Framework of Distributed Ship Collision Avoidance.

By the control procedure, a ship decides whether to proceed to the next position. If a ship does not have any target ship in the area within a certain distance from her current position and also has not yet arrived at her destination, she moves to the next position.

By the search procedure, a ship tries to avoid collision by running a distributed algorithm when she confirms that there is a collision risk. If every ship finds a solution, or if the computational time exceeds a certain time limit, they update the next positions and move to them. We set a time limit on the computational time within which ships can exchange messages with each other to figure out safe courses. When the time has elapsed, they update the next positions (typically, with the courses that may not be safe but have the minimum cost found so far) and move to them. All of the ships alternate the search and control procedures until they arrive at their destinations.

Figure 2 illustrates the basic terms. The own ship located at the centre has a detection range where she can detect target ships. The own ship can exchange messages with a target ship in the detection range, but not with a ship located outside the detection range. The own ship must keep a safety domain between herself and a target ship. The safety domain is a circle with a certain radius depending on ships. If that safety domain is penetrated, we consider they collide with each other. Let us suppose, for example, that a ship sails the ocean at 12 knots. Due to the restriction on ship movement, a sailing ship cannot change her course abruptly. Once she selects a course, she must follow it for a certain period. We set 3 minutes for that period throughout the paper. Namely, a ship is required to consider changing her course every 3 minutes (every 0·6 nautical miles). We consider this 3 minutes a unit of time to be called a time step.

Figure 2. Basic terms.

We set a time window to 15 minutes (five time steps). Namely, a ship makes a 15-minute plan on her future positions based on current positions, headings, and speeds of herself and target ships. Note that this is done every 3 minutes through communication with target ships. When a time window gets larger, a ship has a longer view of future events since she will make a longer plan on her future positions to compute the cost for each candidate course described later.

It is worth noting that distributed algorithms for ship collision avoidance operate on any number of ships with any communication structure. Suppose that three ships encounter each other as shown in Figure 3(a). Ships O and A can exchange messages directly because they are inside each other's detection range. However, ships O and B cannot because they are mutually outside of their detection ranges. A communication structure of these three ships is denoted by the graph in Figure 3(a), whose nodes are ships and edges are the possibilities of direct message exchange. A communication structure of nine ships is also shown in Figure 3(b).

Figure 3. Communication structure among three ships(a, left) and nine ships (b, right).

3.2. Cost and Improvement

Given current positions, headings, and speeds of target ships, a ship computes the cost for each candidate course. A candidate course is chosen from a discrete set of angles for alternating courses. In consideration of typical ship manoeuvring, it ranges from 45^° on the port side (−45^°) to 45^° on the starboard side (+45^°) in steps of 5^°. If the heading angle for a destination exists in these bounds, it is also included as a candidate course.

We propose a cost function that is comprised of the collision risk against a target ship and the relative angle between a candidate course and a destination.

Equation (1) shows the Collision Risk (CR), where crs and j mean a candidate course and a target ship, respectively, and self means the own ship. If self will collide with ship j in its time window of 15 minutes when choosing a course crs, CR _self , for crs and j is computed as TimeWindow divided by TCPA (Time to Closest Point of Approach). Otherwise, it becomes zero.

(1) $$CR_{self}{\lpar crs\comma \; j\rpar }\equiv \left\{\matrix{\displaystyle{TimeWindow \over TCPA_{self}\lpar crs\comma \; j\rpar}\comma \; \hfill & if \quad self\, \, \hbox{will collide with ship} j \hfill \cr 0\comma \; \hfill & \hbox{otherwise} \hfill}\right. $$

Equation (2) computes the cost for a course crs, which is made up of two parts: first, the sum of CR _self over the target ships at risk for crs, and second, the relative angle between crs and a destination. Parameter α is a weight factor that controls the relationship between safety and efficiency. If α gets lower, a ship places more emphasis on efficiency than safety. We set the value of α to one throughout the paper.

(2) $$COST_{self}\lpar crs\rpar \equiv\alpha\sum_{j\in TargetShips}CR_{self}\lpar crs\comma \; j\rpar +\displaystyle{{\vert\theta_{dest}-\theta_{self}\lpar crs\rpar \vert}\over{180^\circ}} $$

When performing the search procedure in Figure 1, a ship tentatively selects one course as her intention that imposes the cost computed by Equation (2). However, she may be able to reduce the cost by changing her intention to another course. Equation (3) computes the largest cost reduction as improvement _self . A ship always tries to select the course that gives the largest cost reduction. Note that a tie in cost reduction is broken by following the COLREGS, which means a ship will select a starboard side when there are multiple courses that give the same largest cost reduction.

(3) $$improvement_{self}\equiv\max_{crs}\lcub {COST}_{self}\lpar {intention}_{self}\rpar -{COST}_{self}\lpar {crs}\rpar \rcub $$

A ship is always aware of absolute angles θ _heading and θ _dest for her heading and destination, respectively. As shown in Equation (4), a course for the destination is computed by θ _dest  − θ _heading to be added into a set of candidate courses only if the course is within the bounds on alterable angles.

(4) $$\theta_{{self\_dest}} \equiv \left\{\matrix{\theta_{dest} - \theta_{heading}\comma \; & \hbox{if} \vert \theta_{dest} - \theta_{heading} \vert \lt 45^{\circ} \cr \hbox{empty}\comma \; & \hbox{otherwise}}\right. $$

where $crs \in \lcub -45^{\circ}\comma \; -40^{\circ}\comma \; \ldots\comma \; -5^{\circ}\comma \; 0^{\circ}\comma \; +5^{\circ}\ldots\comma \; + 40^{\circ}\comma \; + 45^{\circ} \rcub \cup \lcub \theta_{{self}\_{dest}}\rcub \ \theta_{{self}}\lpar {crs}\rpar $ returns θ _heading +crs.

We demonstrate how the own ship computes COST _self and improvement _self using Figure 4. The own ship will collide with Target ₁ after 12 minutes (four time steps) later with her current course. The cost for 000^° (current intention) is computed by $COST \lpar 000^{\circ}\rpar = 15/12 + 0^{\circ}/180^{\circ} = 1{\cdot}25$ , while the cost for 045^° is ${COST}\lpar 045^{\circ}\rpar = 0 + 45^{\circ}/180^{\circ} = 0{\cdot}25$ , and the cost for 005^° is ${COST}\lpar 005^{\circ}\rpar = 0 + 5^{\circ}/180^{\circ}\approx 0{\cdot}028$ . The improvement _self for the own ship is thus computed by $improvement_{self} = \max_{crs} \lcub {COST}\lpar 000^{\circ}\rpar - {COST}\lpar {crs}\rpar \rcub \approx 1{\cdot}222$ , since the cost for 005^° is clearly minimum among the candidate courses. Target ₂ is ruled out for computing the cost because Target ₂ has nothing to do with any collision. Similarly, Target ₁ will also compute her COST _self and improvement _self independently.

Figure 4. Numerical example to compute costs and improvements.

3.3. Distributed Local Search Algorithm

DLSA is a simple distributed algorithm to minimise the total sum of costs over the ships by letting them exchange their current intentions and improvements with each other (Kim et al., Reference Kim, Hirayama and Park2014). It is an incomplete optimisation algorithm, which is not guaranteed to find an optimal solution. Namely, it may end with a sub-optimal solution even if it spends a lot of time searching for an optimum. However, it is often the case that such a sub-optimal solution is quickly found. The basic idea of DLSA is from the core process of the Distributed Breakout Algorithm (DBA) (Yokoo and Hirayama, Reference Yokoo and Hirayama1996; Hirayama and Yokoo, Reference Hirayama and Yokoo2005).

Figure 5 illustrates ship collision avoidance by DLSA. Suppose that three ships encounter each other. Each ship first exchanges their current intentions by ok? messages with target ships to compute COST _self for each candidate course as shown in Figures 5(a) and 5(b). The number of messages exchanged to do so is six in this example. Then, based on COST _self for each candidate course, she computes improvement _self to be sent by improve messages as shown in Figure 5(c). The number of messages exchanged to do so is also six. Finally, as shown in Figure 5(d), the ship that has the largest improvement _self , ship 3, changes her intention, while the other ships maintain their current intentions. We should note that, to reach this state, DLSA consumed 12 messages and two cycles of message exchange.

Figure 5. Procedure for DLSA (a, top-left), (b, top-right), (c, bottom-left), (d, bottom-right).

To prevent an endless loop, only one ship can change her intention at a time among herself and target ships. This process is repeated until the overall cost becomes zero or the computational time exceeds a time limit.Footnote ¹ After all the ships have set their optimal courses as their intentions, they proceed to the next position. At the new position, each ship then starts identifying new target ships within the detection range to avoid collisions by DLSA again.

Figure 6 shows the procedure for DLSA. At the initial step, all ships select the current courses as their intentions. After exchanging their current intentions by ok? messages, they compute COST _self and improvement _self . If there exists a ship having non-zero COST _self for her intention, the ship that has a larger improvement _self than those of target ships changes her intention after exchanging improvement _self . This process is repeated until all ships are content with their current intentions. If there is a tie in choosing a ship with the largest improvement _self , it is broken in favour for a smaller ID number of ship.

Figure 6. Procedure for DLSA.

3.4. Distributed Tabu Search Algorithm

We have observed that DLSA is sometimes trapped in a Quasi-Local Minimum (QLM), where a ship is unable to change her course even though a collision risk still exists. To solve this problem, we applied the tabu search technique (Glover, Reference Glover1989), where the ship in QLM puts her current course in a tabu list to prohibit herself from selecting that course for a certain period. The resulting algorithm was called the Distributed Tabu Search Algorithm (DTSA) (Kim et al., Reference Kim, Hirayama and Okimoto2015).

Figure 7 shows the procedure for DTSA. The whole framework is essentially the same as DLSA; only the QLM procedure (dotted red box) is added. This process is repeated until all ships are content with their current intentions. If QLM occurs, a ship calls the QLM procedure, in which she randomly chooses an alternate course excepting any course in the tabu list. This process will be recurred until QLM is resolved.

Figure 7. Procedure for DTSA.

4.1. Motivation

The common drawback of DLSA and DTSA is that they must send a relatively large number of messages in order for the ships to coordinate their actions. In the context of distributed constraint optimisation, the Distributed Stochastic Algorithm (DSA) has been proposed to reduce the number of messages by allowing neighbouring agents to perform simultaneous changes in a stochastic manner (Zhang et al., Reference Zhang, Wang and Wittenburg2002; Reference Zhang, Wang, Xing and Wittenburg2005). They reveal that these simultaneous changes often lead to faster convergence to a sub-optimal solution; furthermore, its stochastic nature excludes the need for a specific method to escape from QLM. The basic idea of DSA is so general that it can be applied to many optimisation problems (like local search and tabu search). In their original papers, DSA was applied to the graph colouring problem (Zhang et al., Reference Zhang, Wang and Wittenburg2002) and the scheduling problem on sensor networks (Zhang et al., Reference Zhang, Wang, Xing and Wittenburg2005). Although several studies have been made on DSA since its development, we are not aware that there is any study that tackles the collision avoidance problem using this idea.

4.2. Detail

Figure 8 shows the procedure for the Distributed Stochastic Search Algorithm for ship collision avoidance (DSSA). First, a ship selects the current course as her intention. After exchanging intentions with target ships, she computes COST _self and improvement _self . If any of the ships is not content with her current intention, she changes it by Rule A described below. This process is repeated in parallel over the ships until all the ships are content with their current intentions.

Figure 8. Procedure for DSSA(p).

A new intention is chosen stochastically by Rule A, where only the ship with positive improvement _self chooses the course with probability improvement _self as her new intention, but keeps her current intention with probability 1 − p. This probability is a parameter that controls the stochastic behaviour of ships. We denote DSSA with a certain value p for this probability as DSSA(p) if necessary.

Table 2 summarises the main features of our distributed algorithms for ship collision avoidance. We should point out that both DLSA and DTSA require two cycles of message exchange for some of the ships to change intentions. Namely, they require one cycle for ok? messages and another cycle for improve messages. However, in DSSA, one cycle suffices for some ships to change intentions since there is no need to exchange improvement _self .

Table 2. Comparison of DLSA, DTSA, and DSSA.

5.1. Four-ship Encounter

Table 3 indicates the values for parameters of four ships. Every ship has the same and constant speed of 12 knots. The radii of detection range and safety domain are set to 12 and 0·5 nautical miles, respectively. All four ships, each sailing in a diagonal direction, are arranged so that they intersect with each other at the centre. Each ship repeats the control and search procedures in Figure 1 every 3 minutes as suggested in Section 3.1. The time limit for the search procedure is controlled over 3–10 seconds in steps of one second in order to observe how the overall performance changes when more time is allocated for the search procedure.

Table 3. Values for parameters of four ships.

To clarify the importance of exchanging intentions explicitly among ships, we first illustrate in Figure 9 the simulated trajectories of (a) “non-cooperative” and (b) “cooperative” ships for this instance.Footnote ² By a “non-cooperative” ship, we mean the ship with the RADAR/ARPA system that always selects the best course (in terms of our cost function), which is computed through sensing data of target ships. On the other hand, by a “cooperative” ship, we mean the ship with our distributed algorithm that always selects the best course (in terms of our cost function), which is computed through exchanging intentions with target ships. The trajectories of four “non-cooperative” ships are shown in Figure 9(a), which clearly indicates that the ships change their courses suddenly and significantly. These behaviours can generally lead to unstable and longer trajectories. On the other hand, those of four “cooperative” ships with DSSA are also shown in Figure 9(b), which clearly indicates that the ships go smoothly to their destinations without collisions. The “cooperative” ships are likely to have a shorter average distance than the “non-cooperative” ships.

Figure 9. Simulated trajectories of four ships by non-cooperative (a, left) and cooperative (b, right).

Then, we compare the performance of DLSA, DTSA, and DSSA(0·5) for this instance while varying a time limit from 3 to 10 seconds. The results of this simulation are shown in Figure 10. The bar indicates the average distance, which is the average length of trajectories over the ships, and the line indicates the number of messages exchanged among the ships. Compared to DLSA and DTSA, DSSA(0·5) showed better performance in both measures. In terms of average distance, DLSA and DTSA showed almost the same results. The average distance for DSSA was lower than for DLSA and DTSA. In terms of the number of messages, the longer the time limit, the greater the number of messages for every algorithm. In comparison with those for DLSA and DTSA, the number of messages for DSSA was significantly lower regardless of time limit. This implies that DSSA did not exceed any time limit to find optimal courses for all ships, while DLSA and DTSA often did so. DSSA enables multiple ships to alter their intentions simultaneously, leading to fast convergence to the optimum within one second.

Figure 10. Performance of algorithms for four-ship encounter.

In order to prove that collision never occurred in this experiment, we show DCPA (Distance at Closest Point of Approach) for any pair of four ships with DSSA in Table 4. Note that the minimum DCPA was 0·5177 nm between ships 2 and 3, which is larger than the radius of safety domain (0·5 nm). Figure 11 also shows the trajectories of four ships.

Figure 11. Simulated trajectories of four ships by DSSA(0·5).

Table 4. DCPA for any pair of four ships. (unit: nm)

5.2. 12-ship Encounter

Table 5 indicates the values for parameters of 12 ships, which are the same as in the previous experiment except for origin/destination/heading of ships. We arranged 12 ships as in Figure 12, which also shows the simulated trajectories by DSSA. All the ships arrived at their destinations without any collision, since the minimum DCPA was 1·0007 nm between ships 6 and 9 (see Table 6). It also demonstrates how much a ship's decision is affected by target ships. The ships in the middle that are surrounded by many target ships altered their courses significantly while other ships altered their courses only a little.

Figure 12. Simulated trajectories of 12 ships by DSSA(0·5).

Table 5. Values for parameters of 12 ships.

Table 6. DCPA for any pair of 12 ships. (unit: nm).

Figure 13 shows the average distance and the number of messages for this instance. In terms of average distance, all algorithms showed a similar result. In terms of the number of messages, DSSA had many fewer than DLSA and DTSA.

Figure 13. Performance of algorithms for 12-ship encounter.

In addition to the above experiments, we have examined the performance of DLSA, DTSA and DSSA on various artificial settings, such as the one with 100 ships with randomly generated origins, destinations, and headings. We should point out that we have obtained similar results to the above.

5.3. Real data from AIS in the Strait of Dover

To demonstrate the applicability to real scenarios, we applied DSSA to the real data from AIS in the Strait of Dover as shown in Figure 14, which includes eight ships with different speeds, safety domains, origins, destinations, and headings. Details are also shown in Table 7. These data were collected from the website (vesselfinder.com). Note that the time limit for the search procedure was fixed to 3 seconds.

Figure 14. Eight ships in the Strait of Dover. (source: www.vesselfinder.com)

Table 7. Values for parameters of eight ships in the Strait of Dover.

Figure 15 indicates the trajectories of eight ships computed by DSSA. Table 8 shows DCPA for any pair of eight ships, where the minimum DCPA is 0·5544 nm between ships 4 and 8. Although these eight ships have different safety domains, Table 8 clearly indicates that all ships arrived at their destinations without collision.

Figure 15. Simulated trajectories of eight ships in the Strait of Dover by DSSA.

Table 8. DCPA for any pair of eight ships. (unit: nm)