Review of Ship Collision Avoidance Guidance Algorithms Using Remote Sensing and Game Control

Abstract

This work provides a mathematical description of a process game for the safe driving of a ship that encounters other ships. State and control constraint variables, as well as a set of acceptable ship tactics, are taken into consideration. Multi-criteria optimization operations are developed as positional and matrix games, based on cooperative, non-cooperative, and classical (non-game) optimal steering control. Adequate algorithms for ship collision avoidance, relating to the above operations, are developed and verified through digital simulation of a real navigational situation using MATLAB/Simulink.

1. Introduction

A key problem with optimal control in maritime transport and logistics processes is the cost of the process: the control function is ambiguous and may be based on the control method and/or some stochastic situation with a fixed stochastic model, or it must be calculated using a specific control method and other undefined parameters [1].

Such a solution to this problem is the analysis of the process with the use of artificial intelligence methods [2] or the methods of game theory [3]. However, a more accurate representation of the actual process of safe ship motion control in collision situations is treating it as the controlling process that gamers use. The main game control methods in sea transport processes concern the control of the locations of moving objects, including the functioning of feedback, which can be represented by positional and matrix games. Such methods can be used to determine the safe control of a ship, avoiding collisions with other ships at sea.

In general, the entire process is performed first by the collision avoidance guidance algorithm and then by the ship motion control algorithm. This article deals with the synthesis methods of collision avoidance guidance algorithms, taking into account the game properties of the process.

However, for the synthesis of ship motion control algorithms, optimal, adaptive, and fuzzy control methods are used [4,5,6,7,8,9]. Then, it becomes important to take into account the influence of wind and sea waves on the ship’s movement [10,11].

Thus, previous publications on the application of game theory elements concerned both cooperative operations—autonomous surface vehicles (ASVs), and autonomous underwater vehicles (AUVs) [12,13,14,15,16,17]—and non-cooperative operations [18,19,20]. Other applications concerned fishery management [21,22], target defense [23], and power systems [20].

The use of game theory methods for the synthesis of collision avoidance guidance algorithms has so far been neglected in the literature; therefore, it has become the aim of this review paper.

This review presents the formulation of various types of algorithms for safe ship control in collision situations, developing the author’s earlier works [24,25,26].

Radar remote sensing data are used to describe and computationally implement various types of game control methods to avoid collisions with multiple objects.

The algorithms developed by the author include an innovative formulation of the control process of a group of ships as a multi-person, multistep matrix game and multistage positional game, then the optimization of this control task using dual linear programming.

For this purpose, Section 2 presents a mathematical game control process model containing: conditions of the control process, state and control limitations, and sets of allowable ship tactics.

Then, in Section 3, on the basis of the formulated model, two basic control algorithms are synthesized—positional and risk game control, in three possible versions—as a multi-stage cooperative game, as a non-cooperative game, and as optimal control outside the game.

Section 4 presents the results of the simulation tests of six algorithms in the navigational situation of passing a dozen or so ships.

The research results are discussed in Section 5, and detailed conclusions are included in Chapter 6, indicating the scope of work to be performed in the future.

2. The Game Control Process Model

A significant feature of the positional game is that the motion of each object depends on the position p(t_k) of the other objects it passes at time k, as shown in Figure 1.

In the process model, possible variations in the speeds and directions of the encountered objects are considered [27,28,29,30,31,32].

2.1. The Condition of the Control Process

The considered ship’s position x₀ (Equation (1)), as well as those of the encountered objects x_j (Equation (2)), determine the present condition of the control process:

$${\mathrm{x}}_0=\left({\mathrm{X}}_0,{\mathrm{Y}}_0\right),$$

(1)

$${\mathrm{x}}_{\mathrm{j}}=\left({\mathrm{X}}_{\mathrm{j}},{\mathrm{Y}}_{\mathrm{j}}\right),\mathrm{j}=1,2,\mathrm{\dots },\mathrm{m}.$$

(2)

The control system begins to operate at moment t_k, depending on the data the radar anti-collision system send to it regarding the present positions of the objects being traced:

$$\mathrm{p}({\mathrm{t}}_{\mathrm{k}})=\left[\begin{array}{l}{\mathrm{x}}_0({\mathrm{t}}_{\mathrm{k}})\\ {\mathrm{x}}_{\mathrm{j}}({\mathrm{t}}_{\mathrm{k}})\end{array}\right] \mathrm{j}=1,2,\mathrm{\dots },\mathrm{m} \mathrm{k}=1,2,\mathrm{\dots },\mathrm{K}.$$

(3)

Based on the overall notion of multi-stage games, it is expected that the considered object identifies the positions of other passing objects at time t_k [33,34,35,36,37,38,39].

2.2. State and Control Limitations

Managers of vessel traffic must take into account the actual navigation constraints C, including the outline of the coastline or fairway, permanent obstacles, and waters closed to navigation, which constitute the limitations of the process state (Figure 2):

$$\left\{{\mathrm{x}}_0\left(\mathrm{t}\right),{\mathrm{x}}_{\mathrm{j}}\left(\mathrm{t}\right)\right\}\in \mathrm{C}.$$

(4)

The control constraints consider the kinematics of ship movements, in terms of the rules recommended by COLREGs, in order to maintain a safe distance d_s between ships [40]:

$${\mathrm{u}}_0\in {\mathrm{U}}_0,{\mathrm{u}}_{\mathrm{j}}\in {\mathrm{U}}_{\mathrm{j}},\mathrm{j}=1,2,....,\mathrm{m}.$$

(5)

2.3. Sets of Allowable Ship Tactics

The sets of acceptable tactics for the participants in the game depend on those of each other, that is, the steering method chosen by ship j alters the admissible tactics set U_0,j for the considered ship, consisting of possible changes in its course to port S_01,j or starboard S_02,j:

$${\mathrm{U}}_{0,\mathrm{j}}\left[\mathrm{p}\left(\mathrm{t}\right)\right]={\mathrm{S}}_{01,\mathrm{j}}\cup {\mathrm{S}}_{02,\mathrm{j}}.$$

(6)

Figure 3 depicts the method used to determine the sets S_01,j and S_02,j of acceptable tactics for the considered ship during the passing of the encountered ship j at speed V_j and with course ψ_j.

Inequalities related to the tangents of a circle with radius d_s define the set U_0,j, as follows:

$${\mathrm{f}}_{0,\mathrm{j}} {\mathrm{u}}_{0,{\mathrm{x}}^{\prime }}+{\mathrm{g}}_{0,\mathrm{j}} {\mathrm{u}}_{0,{\mathrm{y}}^{\prime }}\le {\mathrm{h}}_{0,\mathrm{j}},$$

(7)

as well as that confining the arc of a circle with radius V:

$${\mathrm{u}}_{0,{\mathrm{x}}^{\prime }}^2+{\mathrm{u}}_{0,{\mathrm{y}}^{\prime }}^2\le {\mathrm{V}}^2.$$

(8)

The equations below describe the inequality components:

$$\overrightarrow{\mathrm{V}}={\overrightarrow{\mathrm{u}}}_0\left({\mathrm{u}}_{0,{\mathrm{x}}^{\prime }},{\mathrm{u}}_{0,{\mathrm{y}}^{\prime }}\right),$$

(9)

$${\mathrm{f}}_{0,\mathrm{j}}=-{\mathrm{w}}_{0,\mathrm{j}} \mathrm{c}\mathrm{o}\mathrm{s}({\mathrm{q}}_{0,\mathrm{j}}+{\mathrm{w}}_{0,\mathrm{j}} {\mathrm{\delta }}_{0,\mathrm{j}}),$$

(10)

$${\mathrm{g}}_{0,\mathrm{j}}={\mathrm{w}}_{0,\mathrm{j}} \mathrm{s}\mathrm{i}\mathrm{n}({\mathrm{q}}_{0,\mathrm{j}}+{\mathrm{w}}_{0,\mathrm{j}} {\mathrm{\delta }}_{0,\mathrm{j}}),$$

(11)

$${\mathrm{h}}_{0,\mathrm{j}}=-{\mathrm{w}}_{0,\mathrm{j}}\left[{\mathrm{V}}_{\mathrm{j}} \mathrm{s}\mathrm{i}\mathrm{n}({\mathrm{q}}_{\mathrm{j},0}+{\mathrm{w}}_{0,\mathrm{j}} {\mathrm{\delta }}_{0,\mathrm{j}})+\mathrm{V} \mathrm{c}\mathrm{o}\mathrm{s}({\mathrm{q}}_{0,\mathrm{j}}+{\mathrm{w}}_{0,\mathrm{j}} {\mathrm{\delta }}_{0,\mathrm{j}})\right],$$

(12)

$${\mathrm{w}}_{0,\mathrm{j}}=\left\{\begin{array}{l}-1 \mathrm{d}\mathrm{l}\mathrm{a} {\mathrm{S}}_{01,\mathrm{j} }(\mathrm{P}\mathrm{S})\\ 1 \mathrm{d}\mathrm{l}\mathrm{a} {\mathrm{S}}_{02,\mathrm{j}} (\mathrm{S}\mathrm{S})\end{array}\right..$$

(13)

Individual COLREGs can be assigned a certain logical function W_j and, then, its specific value w_0,j. The value of the variable w_{0, j} is assigned to the COLREGs rules regarding the obligation to give way to ships approaching from starboard, or waiting for a ship approaching from port side to give way:

$${\mathrm{W}}_{\mathrm{j}}=\left\{\begin{array}{l}1 \mathrm{t}\mathrm{o} {\mathrm{w}}_{0,\mathrm{j}}= 1\\ 0 \mathrm{t}\mathrm{o} {\mathrm{w}}_{0,\mathrm{j}}=-1\end{array}\right..$$

(14)

Similarly, the steering method chosen for the considered ship alters the set of permissible strategies U_j,0 of the encountered ship j, consisting of the possible changes in its course to port S_j1,0 or starboard S_j2,0:

$${\mathrm{U}}_{\mathrm{j},0}\left[\mathrm{p}\left(\mathrm{t}\right)\right]={\mathrm{S}}_{\mathrm{j}1,0}\cup {\mathrm{S}}_{\mathrm{j}2,0}.$$

(15)

Figure 4 depicts the method for determining the sets S_j1,0 and S_j2,0 of acceptable strategies for the encountered ship j during the passing of the considered ship with speed V and course Ψ_.

The set U_j,0 is defined by the inequalities. Tangents of a circle with radius d_s define the set U_0,j, as follows:

$${\mathrm{f}}_{\mathrm{j},0} {\mathrm{u}}_{\mathrm{j},{\mathrm{x}}^{\prime }}+{\mathrm{g}}_{\mathrm{j},0} {\mathrm{u}}_{\mathrm{j},{\mathrm{y}}^{\prime }}\le {\mathrm{h}}_{\mathrm{j},0},$$

(16)

as well as that confining the arc of a circle with radius V_j:

$${\mathrm{u}}_{\mathrm{j},{\mathrm{x}}^{\prime }}^2+{\mathrm{u}}_{\mathrm{j},{\mathrm{y}}^{\prime }}^2\le {\mathrm{V}}_{\mathrm{j}}^2.$$

(17)

The components of the inequalities are described by the following equations:

$${\overrightarrow{\mathrm{V}}}_{\mathrm{j}}={\overrightarrow{\mathrm{u}}}_{\mathrm{j}}\left({\mathrm{u}}_{\mathrm{j},{\mathrm{x}}^{\prime }},{\mathrm{u}}_{\mathrm{j},{\mathrm{y}}^{\prime }}\right),$$

(18)

$${\mathrm{f}}_{\mathrm{j},0}=-{\mathrm{w}}_{\mathrm{j},0} \mathrm{c}\mathrm{o}\mathrm{s}({\mathrm{q}}_{\mathrm{j},0}+{\mathrm{w}}_{\mathrm{j},0} {\mathrm{\delta }}_{\mathrm{j},0}),$$

(19)

$${\mathrm{g}}_{\mathrm{j},0}={\mathrm{w}}_{\mathrm{j},0} \mathrm{s}\mathrm{i}\mathrm{n}({\mathrm{q}}_{\mathrm{j},0}+{\mathrm{w}}_{\mathrm{j},0} {\mathrm{\delta }}_{\mathrm{j},0}),$$

(20)

$${\mathrm{h}}_{\mathrm{j},0}=-{\mathrm{w}}_{\mathrm{j},0} \mathrm{V} \mathrm{s}\mathrm{i}\mathrm{n}({\mathrm{q}}_{0,\mathrm{j}}+{\mathrm{w}}_{\mathrm{j},0} {\mathrm{\delta }}_{\mathrm{j},0}),$$

(21)

$${\mathrm{w}}_{\mathrm{j},0}=\left\{\begin{array}{l}-1 \mathrm{d}\mathrm{l}\mathrm{a} {\mathrm{S}}_{\mathrm{j}1,0} (\mathrm{P}\mathrm{S})\\ 1 \mathrm{d}\mathrm{l}\mathrm{a} {\mathrm{S}}_{\mathrm{j}2,0} (\mathrm{S}\mathrm{S})\end{array}\right..$$

(22)

3. Multi-Criteria Game Control Algorithms

Safe and optimal ship control, in relation to all encountered ships, can be determined from the total set of admissible controls:

$${\mathrm{U}}_0={\displaystyle \underset{\mathrm{j}=1}{\overset{\mathrm{m}}{\cap }}{\mathrm{U}}_{0,\mathrm{j}}\hspace{1em}\mathrm{j}=1,2,\mathrm{\dots },\mathrm{m}}.$$

(23)

Optimal steering of a ship’s movement u₀^*(t) at time t is related to its current position p(t) and constitutes optimal position control u₀^*(p).

Two sets of admissible tactics can be formulated: firstly, the admissible tactics U_j,0[p(t_k)] of the j ships encountered, relative to the considered ship; and second, the set of admissible tactics U_0,j[p(t_k)] for the considered ship, with respect to the j met ships.

Based on each j-th encountered ship, the steering u_0,j and u_j,0 are calculated, as well as the optimal positional tactics u₀^* for the considered ship, using the condition Q^*, denoting control optimality [41,42,43,44,45,46,47].

3.1. Algorithm ANCPGC of Non-Cooperative Positional Game Control

The optimization criterion is used by the multi-stage positional game control:

$$\begin{array}{l}{\mathrm{Q}}_{\mathrm{A}\mathrm{N}\mathrm{C}\mathrm{P}\mathrm{G}\mathrm{C}}^{\ast }=\underset{{\mathrm{u}}_0^{\ast }\in {\mathrm{U}}_0={\displaystyle \underset{\mathrm{j}=1}{\overset{\mathrm{m}}{\cap }}{\mathrm{U}}_{0,\mathrm{j}}}}{{\displaystyle \min }}\left\{\underset{{\mathrm{u}}_{\mathrm{j},0}\in {\mathrm{U}}_{\mathrm{j},0}}{{\displaystyle \mathrm{m}\mathrm{a}\mathrm{x}}}\hspace{1em}\underset{{\mathrm{u}}_{0,\mathrm{j}}\in {\mathrm{U}}_{0,\mathrm{j}}({\mathrm{u}}_{\mathrm{j}})}{{\displaystyle \mathrm{m}\mathrm{i}\mathrm{n}}}\mathrm{d}\left[{\mathrm{x}}_0({\mathrm{t}}_{\mathrm{k}}),{\mathrm{P}}_{\mathrm{k}}\right]\right\}={\mathrm{d}}_{0,\mathrm{n}\mathrm{c}}^{\ast }\\ \mathrm{j}=1,2,\mathrm{\dots },\mathrm{m}\end{array}$$

(24)

where d₀ is the distance from the considered vessel to the nearest turning point P_k along the pre-determined voyage route of the vessel.

First, the ship’s steering is calculated, in order to ensure that it has shortest transient course; that is, there should be minimal path loss (criterion: min) for non-cooperative met ships. This contributes greatly to extending the trajectory (criterion: max) of the ship.

Finally, from the controls set for the considered ship with regard to the j distinct encountered ships, the ship control sets are calculated for all met ships, in order to ensure there are the least path losses (criterion: min).

In the control algorithm, a triple linear programming method is utilized, based on the three optimization conditions described above (criterion: min, max, min).

The smallest losses in the considered ship’s path while safely evading the encountered ships are obtained through the maximum projection of the velocity vector onto the given ship’s course.

The simplex method is used to calculate the optimal control multiple times at each discrete stage of motion.

3.2. Algorithm ACPGC of Cooperative Positional Game Control

An optimization criterion is used for the multi-stage positional cooperative game control, as follows:

$$\begin{array}{l}{\mathrm{Q}}_{\mathrm{A}\mathrm{C}\mathrm{P}\mathrm{G}\mathrm{C}}^{\ast }=\underset{{\mathrm{u}}_0^{\ast }\in {\mathrm{U}}_0={\displaystyle \underset{\mathrm{j}=1}{\overset{\mathrm{m}}{\cap }}{\mathrm{U}}_{0,\mathrm{j}}}}{{\displaystyle \min }}\left\{\underset{{\mathrm{u}}_{\mathrm{j},0}\in {\mathrm{U}}_{\mathrm{j},0}}{{\displaystyle \mathrm{m}\mathrm{i}\mathrm{n}}}\hspace{1em}\underset{{\mathrm{u}}_{0,\mathrm{j}}\in {\mathrm{U}}_{0,\mathrm{j}}({\mathrm{u}}_{\mathrm{j}})}{{\displaystyle \mathrm{m}\mathrm{i}\mathrm{n}}}\mathrm{d}\left[{\mathrm{x}}_0({\mathrm{t}}_{\mathrm{k}}),{\mathrm{P}}_{\mathrm{k}}\right]\right\}={\mathrm{d}}_{0,\mathrm{c}}^{\ast }\\ \mathrm{j}=1,2,\mathrm{\dots },\mathrm{m}\end{array}$$

(25)

The modified ACPGC algorithm takes into account interactions between ships, in order to prevent the collision of any ships, and to replace the second condition (i.e., max to min).

3.3. Algorithm ANGPC of Non-Game Positional Control

The following optimization criterion is used for the multi-stage non-game control:

$$\begin{array}{l}{\mathrm{Q}}_{\mathrm{A}\mathrm{N}\mathrm{G}\mathrm{P}\mathrm{C}}^{\ast }=\underset{{\mathrm{u}}_0^{\ast }\in {\mathrm{U}}_0={\displaystyle \underset{\mathrm{j}=1}{\overset{\mathrm{m}}{\cap }}{\mathrm{U}}_{0,\mathrm{j}}}}{{\displaystyle \min }}\left\{\mathrm{d}\left[{\mathrm{x}}_0({\mathrm{t}}_{\mathrm{k}}),{\mathrm{P}}_{\mathrm{k}}\right]\right\}={\mathrm{d}}_{0,\mathrm{o}\mathrm{c}}^{\ast }\\ \mathrm{j}=1,2,\mathrm{\dots },\mathrm{m}\end{array}$$

(26)

The criteria given in Equations (24)–(26) make it possible to determine the optimal changes in the course and speed values that form the considered ship’s optimal trajectory in a collision situation. This ensures the smallest loss of the path for the safe passage of the met ships at a distance no less than reference value d_s. The dynamic properties of the ship, while implementing course and speed changes, are taken into account through consideration of the maneuver advance time t_m [48,49,50].

To develop the ACPGC, ANGPC, and ANCPGC algorithms, the lp linear programming function from the Optimization Toolbox of the MATLAB/Simulink software was used (Algorithm A1) (Figure 5).

Appendix A shows the MATLAB/Simulink software used to calculate the optimal maneuver.

3.4. Algorithm ANCRGC of Non-Cooperative Risk Game Control

The ship collision risk concept (Equation (27)) can be defined using the present condition of the approach, explained by the parameters d^j_min and t^j_min, in order to evaluate whether the situation can be assumed to be safe. The situation is considered safe if the ships pass at a distance no smaller than the pre-determined safe value d_s and in a time no shorter than t_s:

$${\mathrm{r}}_{\mathrm{j}}={\left[{\mathrm{k}}_1{\left(\frac{{\mathrm{d}}_{\mathrm{m}\mathrm{i}\mathrm{n}}^{\mathrm{j}}}{{\mathrm{d}}_{\mathrm{s}}}\right)}^2+{\mathrm{k}}_2{\left(\frac{{\mathrm{t}}_{\mathrm{m}\mathrm{i}\mathrm{n}}^{\mathrm{j}}}{{\mathrm{t}}_{\mathrm{s}}}\right)}^2+{\left(\frac{{\mathrm{d}}_{\mathrm{j}}}{{\mathrm{d}}_{\mathrm{s}}}\right)}^2\right]}^{-\frac{1}{2}}.$$

(27)

The values of the parameters k₁ and k₂ are influenced by the state of visibility at sea and the dimensions of the ship (i.e., its length l_d and breadth b_d), as follows:

$$0\le [{\mathrm{k}}_1({\mathrm{l}}_{\mathrm{d}},{\mathrm{b}}_{\mathrm{d}}),{\mathrm{k}}_2({\mathrm{l}}_{\mathrm{d}},{\mathrm{b}}_{\mathrm{d}})]\le 1,$$

(28)

$${\mathrm{l}}_{\mathrm{d}}=1.1(1+0.345 {\mathrm{V}}^{1.6}),$$

(29)

$${\mathrm{b}}_{\mathrm{d}}=1.1(\mathrm{b}+0.767 \mathrm{L}{\mathrm{V}}^{0.4}).$$

(30)

The formula below is used for the control goal:

$${\mathrm{Q}}_{\mathrm{A}\mathrm{N}\mathrm{C}\mathrm{R}\mathrm{G}\mathrm{C}}^{*}=\underset{{\mathrm{u}}_0}{\mathrm{m}\mathrm{i}\mathrm{n}} \underset{{\mathrm{u}}_{\mathrm{j}}}{\mathrm{m}\mathrm{a}\mathrm{x}} {\mathrm{r}}_{\mathrm{j}}.$$

(31)

Then, some form of the probability matrix P = [p_j(u₀, u_j)] can be formulated for the use of pure tactics [51].

The highest probability tactic can be used to solve the control problem:

$${\mathrm{u}}_0^{*}={\mathrm{u}}_0^{}\left\{{[{\mathrm{p}}_{\mathrm{j}}({\mathrm{u}}_0,{\mathrm{u}}_{\mathrm{j}})]}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right\}.$$

(32)

3.5. Algorithm ACRGC of Cooperative Risk Game Control

The following optimization criterion is used in the cooperative risk game control:

$${\mathrm{Q}}_{\mathrm{A}\mathrm{C}\mathrm{R}\mathrm{G}\mathrm{C}}^{*}=\underset{{\mathrm{u}}_0}{\mathrm{m}\mathrm{i}\mathrm{n}} \underset{{\mathrm{u}}_{\mathrm{j}}}{\mathrm{m}\mathrm{i}\mathrm{n}} {\mathrm{r}}_{\mathrm{j}}.$$

(33)

3.6. Algorithm ANGRC of Non-Game Risk Control

The above criterion is usually reduced for non-game control, as follows:

$${\mathrm{Q}}_{\mathrm{A}\mathrm{N}\mathrm{G}\mathrm{R}\mathrm{C}}^{*}=\underset{{\mathrm{u}}_0}{\mathrm{m}\mathrm{i}\mathrm{n}} {\mathrm{r}}_{\mathrm{j}}.$$

(34)

To develop the ANCRGC, ACRGC, and ANGRC algorithms, the lp linear programming function from the Optimization Toolbox of the MATLAB/Simulink software was used (Algorithm A2) (Figure 6).

Appendix B shows the MATLAB/Simulink program used to create COLREG rules.

4. Computer Simulation of Control Algorithms

The multi-criteria optimization algorithms were used to determine a ship’s safe trajectories under the conditions of encountering 15 other ships in the Skagerrak Strait, under restricted visibility at sea of d_s = 1.4 nm (nautical miles); see

Table 1. Motion variables of the considered ship and 15 encountered ships.

Ship j	Distance dj (nm)	Bearing νj (o)	Speed Vj (kn)	Course Ψj (o)
0	-	-	19	0
1	9.1	321	14	90
2	1.9	11	16	180
3	8.1	10	15	200
4	11.9	35	17	275
5	7.1	270	14	50
6	8.1	100	8	6
7	10.9	315	10	90
8	13.1	325	7	45
9	6.9	45	19	10
10	14.9	23	6	275
11	15.1	23	7	270
12	4.2	175	4	130
13	12.8	40	0	0
14	7.3	59	16	20
15	8.5	119	12	30

and Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14.

5. Discussion

The courses of individual safe trajectories differed mainly in terms of the final deviation from the initial course of the ship. This deviation is a measure of the extension of the ship’s route, which entails an increase in the operational costs of the ship, through additional fuel consumption by the ship’s propulsion system.

The deviation of the safe trajectory from its initial direction, with restricted visibility at sea, for which the adopted safe distance of passing ships with the value d_s = 1.4 nm ranges from 3.8 nm for the cooperative position game ACPGC to 6.1 nm for the non-cooperative matrix game ANCRGC. On the other hand, trajectories for non-game control are characterized by a smaller final deviation of about 2.2 nm, but they do not take into account the possible risk of collision due to subjective erroneous maneuvering decisions of other ships.

The risk game control algorithm, both cooperative ACRGC and non-cooperative ANCRGC, is more sensitive and careful to ensure safe passing of ships.

In further studies, it is necessary to compare safe trajectories in different states of visibility at sea, good visibility for the safe passing distance of ships from 0.1 nm to 1.0 nm and for restricted visibility at sea in the range of safe passing from 1.1 to 3.0 nm.

6. Conclusions

The aim of the presented review of multi-criteria optimization algorithms used for the ship collision avoidance process was to determine the best solutions that simultaneously satisfy a set of many—often contradictory—optimization criteria. Similar examples include profit maximization and risk minimization.

Multi-stage optimization deals with problems that change over time; that is, practical optimization operations where the objective control function depends on the state of the transport process changing over time.

Due to the high complexity of the general model for the differential game related to the process of controlling moving objects, simplified models were used for the synthesis of practical control algorithms. Optimization of individual process models allowed for appropriate algorithms for determining the safe trajectory of a ship when it is very close to other (stationary and moving) ships to be obtained, which can be used for the computer-aided decision-making of the captain of the ship.

When developing a mathematical description of the control process in the form of a model of the appropriate type of game, to be used for safely controlling a ship that passing many other ships, one must take into account the degree of ambiguity of the situation caused by imperfect priority rules and the subjectivity of the navigator who makes the final decision to avoid a collision.

In light of this, multi-criteria optimization of safe ship control allowed for the synthesis of appropriate algorithms for cooperative, non-cooperative, and non-game control.

In future studies, the use of selected methods of computational intelligence can be analyzed; in particular, fuzzy control, allowing for better adjustment of control to the environmental conditions, and neural network, allowing for more accurate mapping of objects encountered as mobile obstacles.

The potential application of the research presented in this article is especially indicated for the improvement of maritime autonomous surface ship (MASS) control software.