2.1. Avoidance Regulations and Inland-Ship Domain
In this paper, the collision-avoidance regulations of inland ships are designed according to COLREGs. The encounter situations are also divided into three types: head-on, overtaking, and crossing. The judgment method of the three encounter situations is as follows:
where
represents relative bearing angle.
and
represent the heading angle of the target ship and own ship, respectively. Equations (2)–(4) are the judgment basis of head-on, overtaking, and crossing. The corresponding ship avoidance behaviors in the three encounter situations are shown in
Figure 2.
In
Figure 2, the two ships should pass on the port side in a head-on situation. In the overtaking situation, the overtaking ship is a give-way ship, and the overtaking ship should meet the overtaken ship on the starboard side. In the crossing situation, if there is a target ship coming from the starboard angle of 5° to 112.5°, the own ship is a give-way ship. Then, the own ship shall meet on the port side and pass the stern of the ship that has given way. If there is a ship from the port angle of 5° to 112.5°, the target ship shall be considered a give-way ship. It shall also meet the own ship on the port side and pass the stern of the own ship.
The ship domain is the smallest water area where ships can navigate safely. If a target ship enters the ship domain, it will form a close-quarters situation. At this time, if necessary, both ships should actively avoid or even abandon the inland-river collision-avoidance regulations. Traditional ship domains include the Goodwin, Coldwell, Fujii [
33], and Quaternion models [
34]. Combining the above models with the inland-river environment, this paper designs the inland-ship domain shown in
Figure 3.
In
Figure 3, L represents ship length and
represents drift angle.
Figure 3a shows the ship domain dealing with dynamic obstacles. The inland waterways are narrow and long. Based on ship-type comparisons and weighted averages, the semi-major axis of the ship domain is positioned at 2.8 L and the semi-minor axis at 1.4 L. According to the Goodwin, Coldwell, and Quaternion ship domains, the ship’s position is taken on the left side. Then, according to the long oval ship domain of the Fujii model, the ship’s position is moved downward, and the elliptical ship domain is established at the gray virtual ship. This meets the inland narrow channel, puts the ship in the safest position in the ship domain, and meets the requirements of avoidance regulations. In
Figure 3b, the ship is still in the center since static obstacles do not need to consider avoidance regulations. The circular ship domain is designed to consider inland-river banks. The radius of the circle is 3.5 L. With such improvements, the inland ship domain will change in response to environmental changes. The dynamic design makes collision avoidance more flexible.
2.2. Ship Motion Model
In addition to the rudder force and propeller force, inland ships need to consider the forces generated by external wind, currents, bank effects, and shallow-water effects. Therefore, to truly reflect the navigation trajectory of a ship, accurate ship models are needed for prediction. Inland-ship models with three degrees of freedom can be selected, including surging, swaying, and yawing. The hydrodynamic forces or moments corresponding to the three degrees of freedom are represented by
,
, and
, respectively. The ship-maneuvering model is shown in
Figure 4.
In
Figure 4, the coordinate system consists of two right-hand coordinate systems, in which
is the inertial coordinate system fixed on the earth’s surface, which is used to mark the geographical position of the ship. The
-axis points north, the
-axis points east, and the
-axis points to the earth’s center.
is a motion coordinate system with its origin fixed at the ship’s center of gravity, which is used to record the motion state of the ship. The
-axis points to the bow, the
-axis points to the starboard, and the
-axis is perpendicular to the waterplane. The horizontal view of the coordinate system is shown in
Figure 5, and physical parameters are shown in
Table 1.
The MMG model was proposed by the Japanese Towing Tank Conference Commission (JTTC) in 1972. The ship model is established by considering the separate hydrodynamic forces or moments on the bare hull, propeller, and rudder. In this paper, we extended it to inland rivers. The improved modeling idea is as follows:
where subscripts
,
, and
represent the force or moment from the bare hull, propeller, and rudder, respectively. Subscript
represents the environmental impact on the inland ship, such as wind, currents, and bank effects. The correction of each part of the force in shallow water is as follows:
- (1)
Correction of bare-hull force
The hydrodynamic coefficients in deep water can be estimated by the Inoue and Kijima models, while they need to be corrected in shallow water [
35]. The water depth correction function is as follows:
where
represents the estimation formula of deep water,
represents the estimation formula of shallow water,
represents the water-depth-correction function,
represents the average ship draft,
represents the water depth, and
represents the water-depth draft ratio.
Different hydrodynamic coefficients have different depth-correction functions and constant values, including , , , , and :
where
is a constant coefficient. For other hydrodynamic coefficients, the depth-correction function can be expressed as:
where
,
, and
represent constant coefficients.
- (2)
Correction of propeller force
In shallow water, the calculation formula of
remains unchanged. However, the thrust deduction factor
and wake factor
will be affected to some extent.
will slightly decrease with the decrease in water depth. In contrast,
will significantly decrease with the reduction in water depth. Hence, the correction formula is as follows:
where
is a block coefficient.
- (3)
Correction of rudder force
In shallow water, the calculation formula of
remains unchanged. However, the flow-straightening factor
, flow-increasing factor
, and the distance from the fluid force point to the ship’s center of gravity
will be affected.
will increase with the decrease in water depth and decrease after the turning point. As the water becomes shallower,
increases and
decreases. Therefore, the correction formula is as follows:
- (4)
Analysis of external environmental forces
External impacts on ships often include wind, currents, and bank effects. In general, the effects of wind forces and moments on the hull are as follows:
where
represents air density,
represents relative wind speed,
represents the positive projection area above the waterline, and
represents the side projection area.
and
represent the wind-pressure coefficient, and
represent the wind-moment coefficient, which can be determined by the Isherwood formula.
Since the influence of waves on inland ships is limited, only the impact of currents is considered. Therefore, the cross-flow formula is as follows:
where
represents water density,
represents the relative speed of the ship to water,
represents the lateral velocity of the ship relative to water,
and
are unknown coefficients, which can be determined according to the low-aspect-ratio wing theory.
In addition, when a ship is sailing in a narrow channel, due to the fluid change caused by the hull, the flow velocity near the bank is faster, and the water pressure is lower. As a result, the bank thrust at the bow and the bank suction at the stern form a turning moment, and the ship receives bank suction. The formula for the effect of the bank force on the ship is as follows:
where
represents the width of the ship, and
represents the distance from the ship’s center to the bank.
In this paper, the turning and Z-type tests verify the inland-ship model. The ship data are from the merchant ship “HUAIJI River.” The scale parameters of the real ship and ship model are shown in
Table 2:
To test the trajectory of an inland-ship model in shallow water, the turning test simulation was performed for four draft ratios. The ship was operated with the right full rudder at a 35° rudder angle. The simulation results obtained by MATLAB are shown in
Figure 6 and
Table 3.
In
Figure 6, the ship starts from the coordinate origin and turns to the right. The advance is 3.8–5.5 L, and the transfer is 1.7–2 L. The simulation results show that, when the water is infinitely deep, the tactical diameter is about 3 L. The shallow-water effect becomes more apparent with the continuous reduction of the water depth. The resistance gradually increases, increasing the difficulty of ship maneuverability. As a result, the tactical diameter gradually increases. When the draft ratio approaches 1.0, the tactical diameter of the ship is about 4 L.
Furthermore, the Z-type test is performed on the ship model. The Z-type test is an important method for detecting ship maneuverability and predicting ship trajectory. In this paper, the rudder angle is set to ±10°, and the simulation result of the right 10°/10° Z-type maneuver test is shown in
Figure 7.
In
Figure 7, the solid line indicates the change in the rudder angle. The heading angle of the ship model is the chain line, and the dash line shows the real ship data. An angle greater than zero refers to the starboard side, and less than zero refers to the port side. Through comparison, it can be seen that within the first 100 s, the heading angle difference between the simulation data and the real ship data is about 1°, and the time difference is within 2 s. It is evident that the heading angles of the two groups of data are almost consistent with the change of rudder angle. This proves that the ship model can roughly capture the maneuvering motions in a reasonable time and is useful for the maneuvering predictions of inland ships.
2.3. Collision Risk Model
To prioritize collision avoidance for different obstacles, this paper adopts the fuzzy logic theory and improves it to quantify the degree of ship collision risk. The design of the fuzzy set is as follows:
where
refers to the shortest encounter distance of the ship,
refers to the shortest encounter time,
refers to the distance between the two ships,
refers to the ship speed ratio, and
refers to the distance from the ship’s center of gravity to the bank. As mentioned above, inland ships are affected by bank effects. The suction generated by the bank will affect the ship’s maneuverability. The narrower the channel, the higher the collision risk. Therefore, the membership function of the channel width is as follows:
where
is the nearest avoidance distance from the bank, and
is the safe avoidance distance from the bank. Finally, the computing method of inland-ship collision risk is as follows:
where
is the weight, which is set to 0.32, 0.30, 0.16, 0.10, and 0.12 according to the Analytic Hierarchy Process, respectively, for the above membership degrees. Furthermore, the degree of risk of dynamic obstacles is transformed from quantitative to qualitative. The collision risk level is shown in
Table 4:
The ships with Classes IV and V may not take collision-avoidance measures, namely the free navigation stage, but those with Class IV risk levels are considered potential hazards. Medium risk indicates a collision risk between the own ship and the target ship. Hence, it is necessary to make a collision-avoidance decision for the own ship and see whether the navigation route needs to be changed according to the avoidance regulations. When the risk level is maximum or high risk, the ship has entered a close-quarters situation. Therefore, ships should take collision-avoidance decisions immediately and turn with full rudder. If necessary, they can even deviate from the avoidance regulations.
Figure 8 shows the simulation and verification of the collision-risk model.
In
Figure 8, the unit length of the coordinate axis is 2 km. The own ship is at its origin, and there are four target ships around it, forming different encounter situations with the own ship. The risk level of the target ship is analyzed using the risk model, and the result is shown in
Table 5:
Ship S3 has the highest collision risk around S1, which is close to Class II and needs to be avoided first. Ship S2, with the second-highest risk, is close to S1 in its current position, but its risk is slightly lower than S3 due to its higher speed and smaller relative velocity. S4 is a head-on ship. Although there is a distance from S1, the relative velocity is fairly high, and the collision risk reaches Class III. Finally, S5, the farthest ship, has the lowest risk. In general, the ship does not need to avoid a collision when the risk level is Class IV. However, the ship in Class IV has a potential hazard. Therefore, as discussed in
Section 3.2, the potential hazard still needs to be considered when making collision-avoidance decisions.