2.1 Construction of collaborative collision avoidance network

Due to the different levels of ship autonomy, each ship under the mixed scenario has its own understanding of the current situation. To formulate a collaborative collision avoidance strategy between multi-modal ships at sea, it is first necessary to determine a relatively objective calculation method to identify the objects that need to be collaborative and the timing of collaboration in mixed scenarios, build a reasonable collaborative network, and improve the quality of collaboration. In other words, it is necessary to find ships that are currently related to each other in the space centred on the OS, and they may be directly related or indirectly related (through other ships). In this section, we propose a method for constructing a collaborative collision avoidance network under mixed scenarios, which is calculated by a double-matrix model, the collision risk matrix, and the encounter situation matrix. Ship collision risk creates connections between different ships, and the encounters define these connections. At the same time, according to the requirements of the relevant clauses of COLREGS (1972), collision risk is the prerequisite for the formation of the encounter situation, so the collision risk matrix is directly related to the encounter situation matrix (Wang et al., Reference Wang, Zhang and Zheng2021).

First, define $\xi$ as the scene composed of all ships within a radius of 12 nautical miles centred on the OS, as shown in Equation (1), the scene $\xi$ contains $n + 1$ ships in total:

(1)\begin{equation}\xi = \textrm{OS}\mathop \bigcup \limits_{k = 1,\ldots n} \textrm{T}{\textrm{S}_k}\end{equation}

To make $n + 1$ ships map to each other, the collision risk matrix $M_{\textrm{CR}}^\xi$ and the encounter situation matrix $M_{\textrm{ES}}^\xi$ are both constructed, each of which are $({n + 1} )\times ({n + 1} )$ symmetric matrices and the main diagonal elements are zero, as shown in Equations (2) and (3). The matrix elements $\textrm{CR}_a^b$ and $\textrm{ES}_a^b$ respectively represent the collision risk of ship b relative to ship a and the encounter situation between these two ships, where $M_{\textrm{ES}}^\xi$ is determined by $M_{\textrm{CR}}^\xi$:

(2)\begin{align}M_{\textrm{CR}}^\xi & = \left[ {\begin{array}{*{20}{c}} 0& {\textrm{CR}_{\textrm{OS}}^{\textrm{T}{\textrm{S}_1}}}& \cdots & {\textrm{CR}_{\textrm{OS}}^{\textrm{T}{\textrm{S}_k}}}\\ {\textrm{CR}_{T{S_1}}^{OS}}& 0& \cdots & {CR_{\textrm{T}{\textrm{S}_1}}^{\textrm{T}{\textrm{S}_k}}}\\ \vdots & \vdots & \ddots & \vdots \\ {\textrm{CR}_{\textrm{T}{\textrm{S}_k}}^{\textrm{OS}}}& {\textrm{CR}_{\textrm{T}{\textrm{S}_k}}^{\textrm{T}{\textrm{S}_1}}}& \cdots & 0 \end{array}} \right]\end{align}

(3)\begin{align}M_{\textrm{ES}}^\xi & = \left[ {\begin{array}{*{20}{c}} 0& {\textrm{ES}_{\textrm{OS}}^{\textrm{T}{\textrm{S}_1}}}& \cdots & {\textrm{ES}_{\textrm{OS}}^{\textrm{T}{\textrm{S}_k}}}\\ {\textrm{ES}_{T{S_1}}^{OS}}& 0& \cdots & {\textrm{ES}_{\textrm{T}{\textrm{S}_1}}^{\textrm{T}{\textrm{S}_k}}}\\ \vdots & \vdots & \ddots & \vdots \\ {\textrm{ES}_{\textrm{T}{\textrm{S}_k}}^{\textrm{OS}}}& {\textrm{ES}_{\textrm{T}{\textrm{S}_k}}^{\textrm{T}{\textrm{S}_1}}}& \cdots & 0 \end{array}} \right]\end{align}

For the element $\textrm{CR}_a^b$, there are many ways to calculate the collision risk between ships (Szlapczynski and Szlapczynska, Reference Szlapczynski and Szlapczynska2017; Chen et al., Reference Chen, Huang, Mou and Van Gelder2019). The most widely used methods are still based on the distance at closest point of approach (DCPA; unit, nautical miles) and the time to closest point of approach (TCPA; unit, minutes). This set of spatiotemporal parameters can objectively represent the risk and urgency of collision between two ships, so this paper calculates $\textrm{CR}_a^b$ based on these two index values.

For DCPA, what needs to be explained here is that the value range of the original calculation result $\textrm{DCP}{\textrm{A}^\ast }$ is equal to $({ - \infty , + \infty } )$. The following $\textrm{DCPA}$ is the result of taking the absolute value of $\textrm{DCP}{\textrm{A}^\ast }$. For $\textrm{DCP}{\textrm{A}^\ast }$, its positive and negative values have different meanings: if the TS is on the left side of the OS, when the TS passes the bow of the OS, $\textrm{DCP}{\textrm{A}^\ast } < 0$ and when passing the stern of the OS, $\textrm{DCP}{\textrm{A}^\ast } > 0$; if the TS is on the right side of the OS, when the TS passes the bow of the OS, $\textrm{DCP}{\textrm{A}^\ast } < 0$ and when passing the stern of the OS, $\textrm{DCP}{\textrm{A}^\ast } > 0$.

This paper defines that there is no risk when $\textrm{DCPA} > 0.9$, $\textrm{TCPA} > 20$ or $\textrm{TCPA} \le 0$. In addition to this condition, the time risk function $f({\textrm{TCPA}} )$ and the space risk function $f({\textrm{DCPA}} )$ are constructed, as respectively shown in Equations (4) and (5). The calculation results of collision risk between two ships are obtained by weighting, as shown in Equation (6), where $\textrm{CR}_a^b \in [{0,1} ]$.

(4)\begin{gather}f({\textrm{TCPA}} )= \left\{ {\begin{array}{*{20}{l}} 1& {\textrm{TCPA} \le 12}\\ {{{\left( {\dfrac{{20 - \textrm{TCPA}}}{{20 - 12}}} \right)}^2}}& {12 < \textrm{TCPA} \le 20}\\ 0& {\textrm{TCPA} > 20} \end{array}} \right.\end{gather}

(5)\begin{gather}f({\textrm{DCPA}} )= \left\{ {\begin{array}{*{20}{l}} 1& {\textrm{DCPA} \le 0.5}\\ {\dfrac{1}{2} - \dfrac{1}{2}\sin \left[ {\dfrac{\pi }{{0.9 - 0.5}}\left( {\textrm{DCPA} - \dfrac{{0.9 + 0.5}}{2}} \right)} \right]}& {0.5 < \textrm{DCPA} \le 0.9}\\ 0& {\textrm{DCPA} > 0.9} \end{array}} \right.\end{gather}

(6)\begin{gather}\textrm{CR}_a^b = 0.6\ast f({\textrm{DCPA}} )+ 0.4\ast f({\textrm{TCPA}} )\end{gather}

For the element $\textrm{ES}_a^b$, based on the requirements of relevant clauses from COLREGS, this paper further divides three typical encounter scenarios (head-on, overtaking and crossing) into 16 types. It can reflect the information between ships in a more detailed manner and assist in the formulation of collaborative strategies. The following scene divisions also need to satisfy two preconditions: (1) normal power-driven vessels while underway; (2) vessels in sight of each other. Refer to Figure 2 for symbols and schematic diagrams of encounter situations.

Figure 2.

The division results of 16 encounter situations

To describe different encounter situations, define the bearing of ship b relative to ship a as $\theta _a^b$, and the relative course between the two ships is $\varphi$. Set the blue as ship a and the yellow as ship b. The descriptions on these 16 encounter situations are shown in Table 1.

Table 1.

Definition of different encounter situations

As mentioned above, $M_{\textrm{ES}}^\xi$ is decided by $M_{\textrm{CR}}^\xi$, so we define if the element $\textrm{CR}_a^b > 0.5$ in $M_{\textrm{CR}}^\xi$, then $\textrm{ES}_a^b$ can be obtained by Table 1; otherwise, $\textrm{ES}_a^b = 0$. According to these two matrices, $M_{\textrm{CR}}^\xi$ and $M_{\textrm{ES}}^\xi$, we can build a cooperative collision avoidance topological network, and it follow these rules:

1. (1) under scenario $\xi$, ships in danger of collision are reserved as nodes of the cooperative network;

2. (2) when two ships are in danger of collision, use an edge to connect these two nodes;

3. (3) the connecting edge is directional and stores information about associated ship a and b, including $\textrm{CR}_a^b$ and $\textrm{ES}_a^b$.

According to the above methods, this paper gives an example based on one multi-ship encounter scenario ${\xi ^a}$, as shown in Figures 3(a) and 3(b). It should be noted that although there is no direct connection between two ships in the network $\textrm{CR}_a^b = 0$, since they have a common connection node, it is considered that there is an indirect connection between them, as shown in Figure 3, TS-2 and TS-3, they use the OS as the relay node.

Figure 3.

A construction case for collaborative topological network. (a) Multi-ship encounter scenario ${\mathrm{\xi }^\textrm{a}}$. (b) Collaborative network under scenario ${\mathrm{\xi }^\textrm{a}}$

2.2 Multi-ship cooperative game model

2.2.3 Revenue function

Since a ship taking collision avoidance actions will deviate from the original route and disturb the normal navigation state, under the premise of ensuring the safety of navigation, OOWs prefer to use a small range of instructions at a suitable time. If an agreement is reached after negotiation with other ships, the OS's collision avoidance responsibility can be allocated, so that it does not need to afford 100% of the collision avoidance obligation, which is the main way for ships to participate in the coalition to obtain benefits. In addition, reaching an agreement to form a stable collaborative coalition can also ensure that each ship will take reasonable and effective evasive actions to achieve safe passing under mixed scenarios. In this section, we will give the description of the revenue function for sub-coalitions of different dimensions.

Define the benefit of the $k$-th ship in the sub-coalition as $\lambda ({{S_k}} )$, then for the $n$-dimensional sub-coalition ${P^n}$, its revenue function is shown in Equation (7). That is, the benefit of coalition is equal to the sum of the benefit of each ship in this coalition:

(7)\begin{equation}\mu ({{P^n}} )= \mathop \sum \limits_{k = 1}^n \lambda ({{S_k}} )\end{equation}

For one-dimensional coalition ${P^1}$, since only one ship is included, it cannot gain any benefit from coalition, and the revenue function of it is $\mu ({{P^1}} )= 0$.

For two-dimensional coalition ${P^2}$, suppose a two-dimensional coalition $\{{s,t} \}$ includes ship s and ship t, then there is a collision risk $\textrm{CR}_s^t$ and encounter situations $\textrm{ES}_s^t$ and $\textrm{ES}_t^s$ obtained from Table 1. For a typical two-ship coalition, this paper defines that the benefit $\lambda$ obtained by one ship comes from the collision risk $\textrm{CR}_s^t$ that the other ship can share. This is because if a ship does not participate in coalition, since it does not know the intentions and actions of the other ship, in theory, it will have to bear all the obligation to clear the current collision risk. According to COLREGS, we can divide a two-dimensional coalition $\{{s,t} \}$ into the give-way and stand-on ships. For a head on situation, since both ships have the obligation to take actions, each bears 50% of $\textrm{CR}_s^t$. For other situations, the give-way ship needs to undertake the collision avoidance obligation whether it participates or not, so the benefit is equal to 0. The stand-on ship can sail directly with full confidence due to the coordination of the give-way ship, so the benefit is the entire $\textrm{CR}_s^t$. To sum up, the revenue function of two-dimensional coalition is the superposition of $\lambda (s )$ and $\lambda (t )$, as shown in Equation (8):

(8)\begin{equation}\mu ({{P^2}} )= \textrm{CR}_s^t\; ,\; \; {P^2} = \{{s,t} \}\end{equation}

For multidimensional coalition ${P^w}({w > 2} )$, due to the increase in the number of ships, evasive actions taken by ships in different encounter situations will interact with each other, so this paper follows the strong domination property of COLREGS in the cooperative game model. Set multidimensional coalition ${P^w}$ as $\{{{k_1},{k_2}, \ldots ,{k_w}} \}$, then the revenue function $\lambda (k )$ of each ship k in the multi-dimensional coalition is not only related to the collision risk $\textrm{CR}$ shared by another ship, but also related to the expectation of the other ships’ actions. The benefit of the latter mainly comes from the give-way ship defined in this mixed scenario. If the action of the stand-on ship is an assistant action (in this paper, assistant actions are related to the encounter situation between two ships: when an encounter situation belongs to head on or crossing, the same turning direction is an assistant action; when an encounter situation belongs to overtaking, the opposite turning direction is an assistant), it can be regarded as sharing collision avoidance responsibility of the give-way ship, which can reduce the range of evasive action taken by the give-way ship, so that the give-way ship can benefit from it.

Taking the three-ship coalition in Figure 4 as an example, it can be seen from Table 1 that $\textrm{ES}_{{k_1}}^{{k_2}} = \textrm{C}2$ and $\textrm{ES}_{{k_2}}^{{k_3}} = \textrm{C}2$; therefore, ${k_2}$ is a stand-on ship relative to ${k_1}$ but also a give-way ship relative to ${k_3}$. Considering $\textrm{ES}_{{k_1}}^{{k_3}} = \textrm{H}2$, according to COLREGS, both ${k_2}$ and ${k_3}$ need to alter course to the right side. It is an assistant action relative to ${k_1}$ and ${k_2}$, so ${k_1}$ and ${k_2}$ can turn the original dashed trajectories into solid lines, and the range of evasive actions is reduced, so ${k_1}$ and ${k_2}$ gain additional benefits.

Figure 4.

Benefit to be obtained from expected evasive actions

Therefore, let $G = \{{{g_1},{g_2}, \ldots ,{g_v}} \}$ be the set of give-way ships under different encounter situations in coalition ${P^w}$, then for any one of the ships ${g_i}$, its corresponding stand-on ship is set as ${h_i}$. Take this stand-on ship as the central node to obtain all connected edge information, including encounter situation set $\textrm{E}{\textrm{S}_{{h_i}}}$ and collision risk set $\textrm{C}{\textrm{R}_{{h_i}}}$.

We can index $\textrm{E}{\textrm{S}_{{h_i}}}$ according to Table 2. Since the rules do not clearly stipulate the turning direction in an overtaking situation, the left steering is subject to the right steering. To determine whether ${h_i}$ takes an assistant action can be achieved by comparing with $\textrm{ES}_{{h_i}}^{{g_i}}$. If not, the benefit of ${g_i}$ is equal to 0; if it does, the benefit of ${g_i}$ in the coalition can be calculated from Equation (9):

(9)\begin{equation}\mu ({{g_i}} )= \delta \ast \textrm{CR}_{{h_i}}^{\textrm{max}}\ast \textrm{CR}_{{h_i}}^{{g_i}}\end{equation}

Table 2.

Turning directions in different situations defined by COLREGS

Among them, δ is the tuning factor, which is determined by $\textrm{ES}_{{h_i}}^{{g_i}}$, as shown in Table 3. A larger turning range that ${g_i}$ may take suggests a higher expected benefit of ${h_i}$. In addition, $\textrm{CR}_{{h_i}}^{\textrm{max}}$ represents the maximum collision risk value among ships associated with ${g_i}$. A greater collision risk will result in a greater range of assistant actions that will be taken by ${g_i}$. Therefore, the total revenue of coalition ${P^w}$ from taking actions by other ships is the sum of all benefits from give-way ships. In addition, let the number of connected edges (in danger of collision with each other) of coalition $\{{{k_1},{k_2}, \ldots ,{k_w}} \}$ equal m. Then the benefit of coalition ${P^w}$ generated by the collision risk that can be shared by another ship under the current encounter situation is the sum of collision risks of those connection edges, so the final multi-dimensional coalition revenue function is shown in Equation (10):

(10)\begin{equation}\mu ({{P^w}} )= \mathop \sum \limits_{j = 1}^v \mu ({{g_j}} )+ \mathop \sum \limits_{l = 1}^m \textrm{C}{\textrm{R}_l}\end{equation}

Table 3.

Tuning factors under different encounter situation (give way)

2.2.4 Assignment of collision avoidance responsibility

In the previous section, the benefit calculation methods from one-dimensional to multi-dimensional coalition were introduced. In a multi-ship cooperative coalition $N = \{{{S_1},{S_2}, \ldots ,{S_n}} \}$, different participants obtain different benefits in this coalition, so it is unreasonable to evenly distribute the collision avoidance responsibility of each ship in this collaborative network when formulating a multi-ship collaborative collision avoidance strategy. The collision avoidance responsibility of each ship should be allocated proportionally according to the contribution of each ship to the total benefit $\mu$ of coalition N, and the contribution of each ship to the coalition is proportional to the distribution of benefits obtained by each ship in this coalition. Therefore, in the multi-ship cooperative collision avoidance strategy, the allocation of collision avoidance responsibility of each ship is directly related to the distribution of the coalition benefit. The collision avoidance responsibility allocation scheme that conforms to the above logic is more likely to be adopted and implemented by ships with different decision-making logic under mixed scenarios, and then promotes cooperation among multiple ships.

To calculate the solution $R = \{{{r_1},{r_2}, \ldots ,{r_n}} \}$ of a multi-ship cooperative game $G({N,\; \mu } )$, this paper uses the Shapley value method (Shapley and Shubik, Reference Shapley and Shubik1954). The most important point of this method is that it reflects the contribution of all parties in the same coalition to the overall goal of cooperation and avoids egalitarianism in the process of benefits distribution. The benefit $\mu (i )$ distributed by the $i$-th ship in the coalition N can be obtained by Equation (11):

(11)\begin{equation}\mu (i )= \mathop \sum \limits_{\begin{array}{*{20}{c}} {i \in P}\\ {P \subseteq N} \end{array}} \frac{{({p - 1} )!({n - p} )!}}{{n!}}[{\mu (P )- \mu ({P - \{i \}} )} ]\end{equation}

According to the description in Section 2.2.1, P is a sub-coalition of N and p is the number of participants in the sub-coalition. By Equation (11), the benefit distribution of all participants in coalition N is obtained, then for the solution $R = \{{{r_1},{r_2}, \ldots ,{r_n}} \}$, the collision avoidance responsibility ${r_i}$ of the $i$-th ship is shown in Equation (12):

(12)\begin{equation}{r_i} = \frac{{\mu (i )}}{{\mu (N )}}\end{equation}

2.3 Calculation of collaborative collision avoidance strategy

When ships do not participate in the collaborative network under a mixed scenario, it is unclear whether other ships will take evasive action. Therefore, to ensure the safety of navigation, most ships take proactive actions, and the responsibility of each ship to bear is still 100%. Due to the construction of collaborative network, ships participating in the same coalition can share collision avoidance responsibility with each other. As mentioned in Section 2.2, the more benefits obtained by participating in the coalition will mean the greater the collision avoidance responsibility is shared. In this section, according to the final result $R = \{{{r_1},{r_2}, \ldots ,{r_n}} \}$ of the cooperative game model, we use the GRVO method to calculate the final strategy for multi-ship collaborative collision avoidance under mixed scenarios, and the strategy form is the steering angle of each ship.

For any ship ${S_i}$ in the coalition $N = \{{{S_1},{S_2}, \ldots ,{S_n}} \}$, with this ship as the central node, we can obtain the set of ships in different encounter situations connected to it as $D = \{{{d_1},{d_2}, \ldots ,{d_m}} \}$, and also the set of encounter situations $\textrm{E}{\textrm{S}_{{S_i}}}$. For any ship ${d_j}$ in set D, the velocity obstacle (VO) (Van den Berg et al., Reference Van den Berg, Lin and Manocha2008) for ${S_i}$ induced by ${d_j}$ is the set of velocities that, if chosen from, will eventually lead to a dangerous situation between two ships. Let C represent the ship position, then $\textrm{RC} = {C_{{d_j}}} - {C_{{S_i}}}$ represents the relative position between these two ships, and let V represent the ship speed, then $\textrm{RV} = {V_{{S_i}}} - {V_{{d_j}}}$ represents the relative speed between these two ships. Now define a ray $\gamma$ that starts from the RC point and launches in the direction of $\textrm{RV}$, then the position of this ray after time t is shown in Equation (13):

(13)\begin{equation}\gamma ({\textrm{RC,RV}} )= \{{\textrm{RC} + \textrm{RV}\ast t\; |\; t \ge 0} \}\end{equation}

Define the safe encounter distance $\textrm{SD}$ between two ships to be positively correlated with ship distance $|{\textrm{RC}} |$, as shown in Equation (14), units are in nautical miles. Then use the symbol $\psi$ to represent the circular area with ${d_j}$ as the centre and $\textrm{SD}$ as the radius. Therefore, the velocity obstacle $\textrm{V}{\textrm{O}_{{S_i}|{d_j}}}$ for ${S_i}$ induced by ${d_j}$ can be expressed as Equation (15):

(14)\begin{gather}\textrm{SD} = 0.2\ast |{\textrm{RC}} |,\; \; \; \textrm{S}{\textrm{D}_{\textrm{max}}} = 1,\; \; \textrm{S}{\textrm{D}_{\textrm{min}}} = 0.5\end{gather}

(15)\begin{gather}\textrm{V}{\textrm{O}_{{S_i}|{d_j}}}({{V_{{d_j}}}} )= \{{{V_{{S_i}}}\; |\; \gamma ({\textrm{RC,RV}} )\cap \psi \ne \emptyset } \}\end{gather}

The definition of $\textrm{V}{\textrm{O}_{{S_i}|{d_j}}}$ is based on the fact that ship ${S_i}$ bears 100% of the collision avoidance responsibility relative to ${d_j}$. In other words, ${S_i}$ takes evasive action alone to ensure that these two ships pass at a safe distance $\textrm{SD}$. However, in a multi-ship collaborative network under a mixed scenario, each ship is assigned with different collision avoidance responsibility according to the result $R = \{{{r_1},{r_2}, \ldots ,{r_n}} \}$ of the cooperative game. Let ${r_S}_i$ and ${r_d}_j$ be the collision avoidance responsibility of ${S_i}$ and ${d_j}$ assigned in a collaborative network, respectively, then the relative responsibility of ${S_i}$ relative to ${d_j}$ can be calculated by Equation (16):

(16)\begin{equation}r_{{S_i}}^\ast{=} \frac{{{r_S}_i}}{{{r_S}_i + {r_d}_j}}\end{equation}

Therefore, the original velocity obstacle $\textrm{V}{\textrm{O}_{{S_i}|{d_j}}}$ can be updated to $\textrm{GRV}{\textrm{O}_{{S_i}|{d_j}}}$ based on the definition of GRVO. Let the current velocity of ${S_i}$ be $V_{{S_i}}^\ast$, then $\textrm{GRV}{\textrm{O}_{{S_i}|{d_j}}}$ can be expressed as Equation (17), and the schematic is shown in Figure 5 (with a responsibility of 50%):

(17)\begin{equation}\textrm{GRV}{\textrm{O}_{{S_i}|{d_j}}} = \left\{ {V_{{S_i}}^\ast \; |\; \frac{1}{{r_{{S_i}}^\ast }}V_{{S_i}}^\ast{+} \left( {1 - \frac{1}{{r_{{S_i}}^\ast }}} \right){V_{{S_i}}} \in \textrm{V}{\textrm{O}_{{S_i}|{d_j}}}({{V_{{d_j}}}} )} \right\}\end{equation}

Figure 5.

Generalised reciprocal velocity obstacle

In addition, when situation $\textrm{ES}_{{S_i}}^{{d_j}}$ meets the conditions listed in Table 2, $V_{{S_i}}^\ast$ will also be constrained by the turning direction. Set the turning direction constraint imposed by ${d_j}$ on ${S_i}$ to be ${\sigma ^{{d_j}}}$, then the functions for right turning constraint $\sigma _r^{{d_j}}$ and left turning constraint $\sigma _l^{{d_j}}$ can be shown in Equation (18). Here, ‘${\times}$’ represents the vector cross product and $\varphi$ represents the relative course.

(18)\begin{equation}\left\{ {\begin{array}{@{}c} {\sigma_r^{{d_j}} = \{{V_{{S_i}}^\ast |\; {{({\textrm{RP} \times \varphi_{{S_i}}^{{d_j}}} )}_z} < 0} \}}\\ {\sigma_l^{{d_j}} = \{{V_{{S_i}}^\ast \; |{{({\textrm{RP} \times \varphi_{{S_i}}^{{d_j}}} )}_z} > 0} \}} \end{array}} \right.\end{equation}

Therefore, for any ship ${d_j}$ in set D, ${S_i}$ can provide the relative velocity obstacle area $\textrm{GRV}{\textrm{O}_{{S_i}|{d_j}}}$ and turning constraint ${\sigma ^{{d_j}}}$. Let the velocity vector corresponding to ${S_i}$ at different turning angles $\varepsilon$ be $V_{{S_i}}^{{\varepsilon ^\ast }}$, then the initial collision avoidance decision ${\varepsilon _{{S_i}}}$ of ship ${S_i}$ in the cooperative collision avoidance coalition N can be calculated by Equation (19):

(19)\begin{equation}{\varepsilon _{{S_i}}} = \textrm{min}\left\{ {V_{{S_i}}^{{\varepsilon^\ast }}|\; V_{{S_i}}^{{\varepsilon^\ast }} \notin \mathop \sum \limits_{j = 1}^m ({\textrm{GRV}{\textrm{O}_{{S_i}|{d_j}}}\; \cup {\sigma^{{d_j}}}} )} \right\}\end{equation}

The initial decision ${\varepsilon _{{S_i}}}$of ${S_i}$ in coalition N can meet the requirements in most scenarios, considering that the definition of Table 2 above is in accordance with the requirement of ‘left turning obeys right turning’. So, for the decision ${\varepsilon _{{S_i}}}$, if ship ${d_j}$ is regarded as the overtaken ship, and the decision of ${S_i}$ relative to this ship under collision avoidance responsibility $r_{{S_i}}^\ast$ is to turn right, then the assistant action that ${d_j}$ is expected to share is to turn left. However, the final decision of ${d_j}$ may be a right turn in the global situation. Therefore, for each ship $d_o^j$ in the subset ${D_o}$ formed by the overtaken ships from set D, it is necessary to review the initial decision ${\varepsilon _{{S_i}}}$. If none of ${\varepsilon _{{S_i}}}$ is located in each ship's velocity obstacle area $\textrm{GRV}{\textrm{O}_{{S_i}|d_o^j}}$ in the set ${D_o}$, then ${\varepsilon _{{S_i}}}$ is feasible; if not, let the new velocity be $V_{{d_j}}^\ast$, which corresponds to the decision of ship $d_o^j$ in the coalition, then calculate the new velocity obstacle area $\textrm{GRV}{\textrm{O}_{{S_i}|{d_j}}}$ and get the updated final decision ${O_{{S_i}}}$.

In summary, for the collaborative collision avoidance coalition $N = \{{{S_1},{S_2}, \ldots ,{S_n}} \}$ under a mixed scenario, the final solution of the cooperative collision avoidance strategy is obtained as $O = \{{{O_{{S_1}}},{O_{{S_2}}}, \ldots ,{O_{{S_n}}}} \}$.

3.1 Scenario I

The initial parameters of each ship under this scenario are shown in Table 4, where the ship position is a relative coordinate with $({0,0} )$ as the centre point. According to the collaborative network construction method proposed in Section 2.1, the collision risk matrix and the encounter situation matrix among these three ships are shown in Tables 5 and 6, respectively. According to the above information, we can get a topological network of cooperative collision avoidance between ships under this scenario, as shown in Figure 6. It can be seen that these three ships in this scenario are all connected to each other, and the decision of any ship will have an impact on the other ships.

Table 4.

Initial state parameters of each ship in Scenario I

Table 5.

Collision risk matrix of Scenario I

Table 6.

Encounter situation matrix of Scenario I

Figure 6.

A collaborative topology network for Scenario I

According to this network, we can get the sub-coalitions with different dimensions and their benefits, as shown in Table 7. At the same time, according to the Shapley value method, the allocation ratio of collision avoidance responsibility of each ship under this scenario and the final collaborative decision of each ship can be obtained, and the results are shown in Table 8.

Table 7.

Sub-coalitions of Scenario I and their benefits

Table 8.

Collision avoidance responsibility and collaborative decision for each ship under Scenario I

We assume that each ship forms a cooperative network at the position in Table 4, and the global collision avoidance responsibility assigned to each ship under this mixed scenario also reaches an agreement at this moment, that is, all three ships will take evasive actions at these coordinates in Table 4 at the same time. The final results of the simulation verification of this model in the first scenario are shown in Figure 7(a)–7(f). Figure 7 records the whole process of each ship's collision avoidance decision and the trajectory of each ship under Scenario I. The starting point of each ship is set at the position 720 s before the coordinates in Table 4, and the end point of each ship is a point on the extension line of the starting point of each ship along the ship course. To reflect the ship's manoeuvrability characteristics, we set the steering rate of each ship to 2°/10 s during the whole simulation test. In addition, we have also simplified the strategy for resuming voyages after each ship has finished evasive action, that is, if any ship meets the condition of $\textrm{TCPA} < 0$ with other ships that have been in danger of collision, then this ship ends evasion and navigates to the destination (returns to the original route). The DCPA values relative to each other when these ships take evasive action are shown in Table 9.

Figure 7.

Collision avoidance trajectory of each ship under Scenario I. (a) T = 500 s. (b) T = 1000 s. (c) T = 1500 s. (d) T = 2000 s. (e) T = 2500 s. (f) T = 3433 s (Reach Destination)

Table 9.

DCPA relative to other ships under Scenario I

In addition, the relative distances with other ships during the whole process are shown in Figure 8(a)–8(c). It can be seen from the figure that although each ship imposes a steering rate, the relative distance between these ships is within the range of the safe encounter distance, which can improve the collision avoidance efficiency and effectiveness.

Figure 8.

Change of relative distance between ships under Scenario I. (a) Ship1. (b) Ship2. (c) Ship3

3.2 Scenario II

Scenario II is a more complex mixed situation, including a total of 5 ships, and the initial state parameters of each ship are shown in Table 10. The collision risk matrix and the encounter situation matrix calculated by this model, according to the information in Table 10, are shown in Tables 11 and 12, respectively. According to the above information, the topological collaborative network formed by these ships in Scenario II can be obtained, as shown in Figure 8.

Table 10.

Initial state parameters of each ship in Scenario II

Table 11.

Collision risk matrix of Scenario II

Table 12.

Encounter situation matrix of Scenario II

It can be seen from Figure 9, the five-ship coalition can be divided into 28 sub-coalitions according to different dimensions (one-dimensional to five-dimensional), and the benefit of each sub-coalition is shown in Table 13. According to Table 13, the global collision avoidance responsibility of each ship assigned by this model and the final multi-ship cooperative collision avoidance strategy under this mixed scenario can be calculated as shown in Table 14.

Figure 9.

A collaborative topology network for Scenario II

Table 13.

Sub-coalitions of Scenario II and their benefits

Table 14.

Collision avoidance responsibility and collaborative decision for each ship under Scenario II

Consistent with the settings in Scenario I, we assume that each ship forms a cooperative network at the positions in Table 10, and at the same time, these five ships under this mixed scenario reach an agreement on the global collision avoidance responsibility assigned by this model. The final results of model simulation verification are shown in Figure 10(a)–10(f). This figure records the entire process of each ship's collision avoidance decision and the navigation trajectory. The DCPA values relative to each other when these ships take evasive action are shown in Table 15.

Figure 10.

Collision avoidance trajectory of each ship under Scenario II. (a) T = 500s. (b) T = 1000s. (c) T = 1500s. (d) T = 2000s. (e) T = 2500s. (f) T = 3946s (Reach Destination)

Table 15.

DCPA relative to other ships under Scenario II

At the same time, we also recorded the relative distances between ships in this scenario, as shown in Figure 11(a)–11(e), respectively. According to the distance change curve, it can be known that the cooperative collision avoidance strategy generated by this model can make ships safely pass and clear.

Figure 11.

Change of relative distance between ships under Scenario II. (a) Ship1. (b) Ship2. (c) Ship3. (d) Ship4. (e) Ship5