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Abstract

The collision avoidance and path planning technology is an important issue for submarine research field. It is essential and crucial for submarine to navigate underwater safely. There are lots of drawbacks for ensuring this important mission. For handling with the challenging and imperative issue, a novel and efficient collision risk model is proposed. A serial of motions of the submarine to be processed for fulfilling the collision avoidance based on this approach. The concepts of detection domain and fuzzy logic are adopted for modeling the degree of collision risk. The detection domain is used as the submarine safety scope when navigating underwater while the fuzzy logic is used as the mathematical implement for the analysis and synthesis of relations between obstacles or other submarines that are met in the navigation environment. The collision risk is regarded as one effective evaluation function that reflects which exact action should be done when the threshold of the risk degree is met. In order to verify the performance of proposed method, multisubmarines simulation experiments are concluded in different scenarios.

Keywords

Collision avoidance risk degree fuzzy logic multiobject simulation

Introduction

Submarine, a vessel designed to operate below the surface of the water. Submarines were of major importance in both World Wars and today rank with aircraft carriers as the most important ships in the world’s navies. The development of submarines has revolutionized naval strategy by placing the main emphasis on undersea, rather than surface, warfare. Nonnaval submarines are important tools in ocean research and exploration of the ocean floor. In general, most submarines are mounted with an autonomous navigation system which is used for route planning and path planning to find a safe and reasonable path to the given goal. Route planning aims at deriving waypoints from a start position to the goal position based on environment information, and path planning aims at deriving a new path between waypoints when submarines meet with unknown obstacle or unexpected mission change happens. In the processing of navigation, the submarine has to check the degree of risk for estimating the situation the real-time environment information which is not known. For modeling the degree of risk accurately, we discuss this challenging task in two-dimensions, namely, vertical and horizontal dimension.

Collision risk index (CRI) is one of conventional parameters to describe the degree of risk for marine ship or vessel which sail on the sea surface. Based on our best knowledge, this traditional criterion is also adequate for submarine even though the environment information as well as the dimension of action is different.

CRI is the quantized measurement of the collision probability between submarines and other static or dynamic obstacles. This criterion is crucial for making collision avoidance decision. The reasonable value range of CRI is from 0 to 1. CRI = 0 shows that there is no any risk of collision while CRI = 1 indicates that one inevitable collision will happen regardless of any avoidance actions are made. If the value of CRI can be obtained efficiently and accurately, the submarine manager will be alerted earlier and make more significant decision to void any impendent collisions.

The concept of domain for ship is proposed first by Fujii-Ara when he studied the waterway traffic capacity in 1963. Fujii-Ara established the ship domain model through surveying many marine traffic collisions and analyzing the statistical data as well as described the detailed definition of domain. The domain model is described as an ellipse whose long axis is seven times longer than short axis; the center of the ellipse located on the ship and stern line of the ship is set as the ellipse axis. The size of the domain is varied based on the ship size and navigation situation. Goodwin, Davis, B.A. Colley, and T. G. Coldwell made some efforts to improve the domain model in aspect of size and shape for solving how to reflect the actual dangerous situation precisely. The improved domain model not only considers the encounter situation between ships but also takes into account the psychological factors manipulation of staff (manipulation experience, personal ability, etc.), physical factors (the size of the boat, type, etc.) of the boat, and the current environment (weather conditions, water type, traffic density, etc.).

The collision avoidance and path planning technology is an important branch of Unmanned Autonomous Vehicle (UAV) or submarine research field. The so-called path planning of submarine is that the submarine searches an optimum or approximate optimum noncollision path from start state to goal state according to a certain performance objective. A lot of research work had been done in the robot path planning.^1–6 There are two different types of path planning based on whether the environment information is known. The global path planning that the environment information is known completely and the local path planning that the environment information is not known completely or partially; the local path planning is called the dynamic collision avoidance planning and it uses the global path planning as guidance and detects the operating environment of robot online by sensors, as to obtain the information that of the position, shape, size of obstacle, etc.^7–12 So the local path planning is a process which avoids the arisen unknown obstacle at as short time as possible using the local environmental information of obtained online. The dynamic collision avoidance planning is a kind of mapping that from the perception space to the action space.^12–16 The mapping relation can be realized using different methods, but it is difficult to express using an exact arithmetic formula. The artificial neural network is an information processing algorithm that emulates biologic nervous system;^17–20 it has the ability of strong nonlinear function approach and the ability of strong data fusion. So the neural network has a bigger application potential for the mobile robot’s local path planning in dynamic environment. But the neural network system is affected by the learning sample significantly, it is very difficult to select the sample set which has stronger representativeness, and it is not actual to let the sample set cover the whole sample space, so the selection and design of the sample is a difficult problem. On the other hand, the neural network has some problems such as converge to local minimum; the over learning and the structure of ANN are always decided by experience because it doesn’t have a good guiding theory. Especially, when the number of the training sample is not enough, the predicting accuracy will be influenced.

For facilitating the sailor manipulation, some intelligent collision avoidance systems are devolved such as the collision avoidance expert system which is carried out by the University of Liverpool.^21–24 Another outstanding collision avoidance system is established by Rensselaer Polytechnic Institute that support more collision case and can simulate various collision avoidance strategies. Some more complicated mathematical models related to degree of risk, collision avoidance actions, path planning, and reasoning decision making are proposed;^25,26 some of these models have been applied in physical system. In the matter of decision making, the multiagent principles are adopted to refine the reasoning performance.

The remainder of the paper is organized as follows: the kinematics and kinetic model of submarine is presented in the next section, the 2D collision risk model is elaborated in “Collision risk model” section, some simulation experiments are carried out in “Submarine collision avoidance simulation” section, and in the final section we conclude this paper and indicates some further work.

Kinematics and kinetic model of submarine

The kinematics and kinetic model of submarine is crucial for submarine to make appropriate decision to achieve the collision avoidance. For simplification analysis, we adopt the conventional submarine standard kinematics and kinetic model which is released by American Taylor Naval Research Center in 1967. Only the important parameters related to hydrodynamic coefficients are considered while other parameters which play a minor role are ignored.

The common equation of submarine

Set the structure of submarine is rigid, the submarine common equation is as follows:

Axial equation

m [\overset{\cdot}{u} - vr + wq - x_{G} (q 2 + r 2) + y_{G} (pq - \overset{\cdot}{r}) + z_{G} (p \overset{\cdot}{r} + \overset{\cdot}{q})] = \frac{1}{2} ρ L 4 [X'_{qq} q 2 + X'_{rr} r 2 + X'_{rp} rp] + \frac{1}{2} ρ L 3 [X'_{\overset{\cdot}{u}} q 2 + X'_{vr} vr + X'_{wq} wq] + \frac{1}{2} ρ L 2 [X'_{uu} u 2 + X'_{vv} v 2 + X'_{ww} w 2] + \frac{1}{2} ρ L 2 u 2 [X'_{δ_{r} δ_{r}} δ_{r}^{2} + X'_{δ_{s} δ_{s}} δ_{s}^{2} + X'_{δ_{b} δ_{b}} δ_{b}^{2}] + \frac{1}{2} ρ L 2 [a_{r} u 2 + b_{r} {uu}_{c} + c_{r} u_{c}^{2}] - (Δ P - Δ B) \sin θ

(1)

Transverse equation

m [\overset{\cdot}{v} - wp + ur - y_{G} (r 2 + p 2) + z_{G} (qr - \overset{\cdot}{p}) + x_{G} (pq + \overset{\cdot}{r})] = \frac{1}{2} ρ L 4 [Y'_{\overset{\cdot}{r}} \overset{\cdot}{r} + Y'_{\overset{\cdot}{p}} \overset{\cdot}{p} + Y'_{p | p |} p | p | + Y'_{pq} pq + Y'_{qr} qr + Y'_{r | r |} r | r |] \times \frac{1}{2} ρ L 3 [Y'_{\overset{\cdot}{v}} \overset{\cdot}{v} + Y'_{r} ur + Y'_{p} up + Y'_{vq} vq + Y'_{wq} wq + Y'_{wr} wr + Y'_{v | r |} \frac{v}{| v |} | (v 2 + w 2) \frac{1}{2} | | r |] + \frac{1}{2} ρ L 2 [Y'_{u 2} u 2 + Y_{v} uv + Y'_{v | v |} v | (v 2 + w 2) \frac{1}{2} | + Y'_{wv} wv] + \frac{1}{2} ρ L 3 [Y'_{| r | δ_{r}} u | r | δ_{r}] + \frac{1}{2} ρ L 2 [Y'_{δ_{r}} u 2 δ_{r}] + (Δ P - Δ B) \cos \sin ϕ

(2)

Heave equation

m [\overset{\cdot}{w} - up + vp - z_{G} (p 2 + q 2) + x_{G} (rp - \overset{\cdot}{q}) + y_{G} (rq + \overset{\cdot}{p})] = \frac{1}{2} ρ L 4 [Z'_{\overset{\cdot}{q}} \overset{\cdot}{q} + Z'_{pp} p 2 + Z'_{rr} r 2 + Z'_{rp} rp + Z'_{q | q |} q | q |] + \frac{1}{2} ρ L 3 [Z'_{\overset{\cdot}{w}} \overset{\cdot}{w} + Z'_{q} uq + Z'_{vr} vr + Z'_{vp} vp + Z'_{w | q |} \frac{w}{| w |} | (v 2 + w 2) \frac{1}{2} | | q |] + \frac{1}{2} ρ L 2 [Z'_{0} u 2 + Z'_{w} uw + Z'_{| w |} u | w | + Z'_{w | w |} w | (v 2 + w 2) \frac{1}{2} |] + \frac{1}{2} ρ L 2 [Z'_{ww} w | (v 2 + w 2) \frac{1}{2} | + Z'_{vv} v 2] + \frac{1}{2} ρ L 2 [Z'_{| q | δ_{s}} u | q | δ_{s}] + \frac{1}{2} ρ L 2 [Z'_{δ_{s}} u 2 δ_{s} + Z'_{δ_{b}} u 2 δ_{b}] + (Δ P - Δ B) \cos θ \cos ϕ

(3)

Heeling equation

I_{x} \overset{\cdot}{p} + (I_{z} - I_{y}) qr + m [y_{G} (\overset{\cdot}{w} - uq + vp) - z_{G} (\overset{\cdot}{v} - wp + ur)] = \frac{1}{2} ρ L 5 [K'_{\overset{\cdot}{p}} \overset{\cdot}{p} + K'_{\overset{\cdot}{r}} \overset{\cdot}{r} + K'_{p | p |} p | p | + K'_{qr} qr + K'_{pq} pq + K'_{r | r |} r | r |] + \frac{1}{2} ρ L 4 [K'_{\overset{\cdot}{v}} \overset{\cdot}{v} + K'_{p} up + K'_{r} ur + K'_{vq} vq + K'_{wp} wp + K'_{wr} wr] + \frac{1}{2} ρ L 3 [K'_{0} u 2 + K'_{v} uv + K'_{v | v |} v | (v 2 + w 2) \frac{1}{2} | + K'_{vw} vw] + \frac{1}{2} ρ L 3 [K'_{δ_{r}} u 2 δ_{r}] + ({\bar{y}}_{G} Δ P - {\bar{y}}_{C} Δ B) \cos θ \cos ϕ - ({\bar{z}}_{G} Δ P - {\bar{z}}_{C} Δ B) \cos θ \sin ϕ

(4)

Pitching equation

I_{y} \overset{\cdot}{q} + (I_{x} - I_{z}) rp + m [z_{G} (\overset{\cdot}{u} - vr + wq) - x_{G} (\overset{\cdot}{w} - uq + vp)] = \frac{1}{2} ρ L 5 [M'_{\overset{\cdot}{q}} \overset{\cdot}{q} + M'_{pp} p 2 + M'_{q | q |} q | q | + M'_{rr} r 2 + M'_{rp} rp] + \frac{1}{2} ρ L 4 [M'_{\overset{\cdot}{w}} \overset{\cdot}{w} + M'_{q} uq + M'_{vr} vr + M'_{vp} vp + M'_{w | q |} | (v 2 + w 2) \frac{1}{2} | q] + \frac{1}{2} ρ L 3 [M'_{0} u 2 + M'_{w} uw + M'_{| w |} u | w | + M'_{w | w |} | (v 2 + w 2) \frac{1}{2} | + M'_{ww} | (v 2 + w 2) \frac{1}{2} | + M'_{vv} v 2] + \frac{1}{2} ρ L 4 [M'_{| q | δ_{s}} u | q | δ_{s}] + \frac{1}{2} ρ L 3 [M'_{δ_{s}} u 2 δ_{s} + M'_{δ_{b}} u 2 δ_{b}] - ({\bar{x}}_{G} Δ P - {\bar{x}}_{C} Δ B) \times \cos θ \cos ϕ - ({\bar{z}}_{G} Δ P - {\bar{z}}_{C} Δ B) \sin θ

(5)

Yaw equation

I_{z} \overset{\cdot}{r} + (I_{y} - I_{x}) pq + m [x_{G} (\overset{\cdot}{v} - up + ur) - y_{G} (\overset{\cdot}{u} - vr + wp)] = \frac{1}{2} ρ L 5 [N'_{\overset{\cdot}{r}} \overset{\cdot}{r} + N'_{\overset{\cdot}{p}} \overset{\cdot}{p} + N'_{p | p |} p | p | + N'_{r | r |} r | r | + N'_{pq} pq + N'_{qr} qr] + \frac{1}{2} ρ L 4 [N'_{\overset{\cdot}{v}} \overset{\cdot}{v} + N'_{up} up + N'_{r} ur + N'_{wr} wr + N'_{wp} wp + N'_{vq} vq + N'_{| v | r} | (v 2 + w 2) \frac{1}{2} | r] + \frac{1}{2} ρ L 4 [N'_{| r | δ_{r}} u | r | δ_{r}] + \frac{1}{2} ρ L 3 [N'_{δ_{r}} u 2 δ_{r}] + \frac{1}{2} ρ L 3 [N'_{0} u 2 + N'_{v} uv + N'_{vw} vw + N'_{v | v |} v | (v 2 + w 2) \frac{1}{2} |] + ({\bar{x}}_{G} Δ P - {\bar{x}}_{C} Δ B) \cos θ \sin ϕ - ({\bar{y}}_{G} Δ P - {\bar{y}}_{C} Δ B) \sin θ

(6)

Motion equation in six axles

\overset{\cdot}{ξ} = u \cos ψ \cos θ + v (\cos ψ \sin θ \sin ϕ - \sin ψ \cos ϕ) + w (\cos ψ \sin θ \cos ϕ + \sin ψ \sin ϕ)

(7)

\overset{\cdot}{η} = u \sin ψ \cos θ + v (\sin ψ \sin θ \sin ϕ + \cos ψ \cos ϕ) + w (\sin ψ \sin θ \cos ϕ - \cos ψ \sin ϕ)

(8)

\overset{\cdot}{ζ} = - u \sin ψ + v \cos θ \sin ϕ + w \cos θ \cos ϕ)

(9)

\overset{\cdot}{ϕ} = p + q \tan θ \sin ϕ + r \tan θ \cos ϕ

(10)

\overset{\cdot}{θ} = q \cos ϕ + r \sin ϕ

(11)

\overset{\cdot}{ψ} = \frac{q \sin ϕ}{\cos θ} + \frac{r \cos ϕ}{\cos θ}

(12)

where

X' (), Y' (), Z' ()

are nondimensionalized hydrodynamic coefficient.

K' (), M' (), N' ()

are nondimensionalized hydrodynamic torque coefficient.

u, v, w

are the axle, transverse, and vertical velocity, respectively, unit:

m / s

p, q, r

are the rolling, pitching, and heading rotational velocity, unit:

rad / s

. h is metacenter of the submarine, unit:

m

δ_{s}

δ_{b}

δ_{r}

are stem, tail, and rudder of submarine helm angle, unit:

rad

. φ, ψ, θ are heeling angle, course angle, trim angle, respectively, unit:

rad

. ζ, ξ, η are axial, transverse, heave displacement, unit:

m

. L is the length of submarine, unit:

m

. m is the weight of submarine, unit:

kg

. I_x, I_y, I_z are the

X, Y, Z

axis moment of inertia, unit

N m 2

Δ P, Δ B

are the variation of gravity and buoyancy, unit

N

The common formulation of submarine

The equation of the axial direction is as follows

U \approx u \approx U_{0} (1 - e - 0.52 R / L) = U_{0} (1 - e - 0.52 u / | \overset{\cdot}{ψ} | L)

(13)

The equation of the transverse direction is as follows

m [\overset{\cdot}{v} + ur] = \frac{1}{2} ρ L 4 [Y'_{\overset{\cdot}{r}} \overset{\cdot}{r} + Y'_{\overset{\cdot}{p}} \overset{\cdot}{p}] + \frac{1}{2} ρ L 3 [Y'_{\overset{\cdot}{v}} \overset{\cdot}{v} + Y'_{r} ur] + \frac{1}{2} ρ L 2 [Y_{v} uv + Y'_{v | v |} v | (v 2 + w 2) \frac{1}{2} |] + \frac{1}{2} ρ L 2 [Y'_{δ_{r}} u 2 δ_{r}]

(14)

The equation of the vertical direction is as follows

m [\overset{\cdot}{w} - up + vp] = \frac{1}{2} ρ L 4 [Z'_{\overset{\cdot}{q}} \overset{\cdot}{q}] + \frac{1}{2} ρ L 3 [Z'_{\overset{\cdot}{w}} \overset{\cdot}{w} + Z'_{q} uq + Z'_{vp} vp] + \frac{1}{2} ρ L 2 [Z'_{0} u 2 + Z'_{w} uw + Z'_{w | w |} w | (v 2 + w 2) \frac{1}{2} | + Z'_{vv} v 2] + \frac{1}{2} ρ L 2 [Z'_{δ_{s}} u 2 δ_{s} + Z'_{δ_{b}} u 2 δ_{b}]

(15)

The heeling equation is as follows

\frac{1}{2} ρ L 3 [K'_{v} uv] + \frac{1}{2} ρ L 3 K'_{δ_{r}} u 2 δ_{r} - mgh \cos θ \sin ϕ

(16)

The pitching equation is as follows

I_{x} \overset{\cdot}{p} = \frac{1}{2} ρ L 4 [K'_{\overset{\cdot}{v}} \overset{\cdot}{v} + K'_{r} ur] + I_{y} \overset{\cdot}{q} + (I_{x} - I_{z}) rp = \frac{1}{2} ρ L 5 [M'_{\overset{\cdot}{q}} \overset{\cdot}{q} + M'_{rp} rp] + \frac{1}{2} ρ L 4 [M'_{\overset{\cdot}{w}} \overset{\cdot}{w} + M'_{q} uq + M'_{w | q |} | (v 2 + w 2) \frac{1}{2} | q] + \frac{1}{2} ρ L 3 [M'_{0} u 2 + M'_{w} uw + M'_{vv} v 2] + \frac{1}{2} ρ L 3 [M'_{δ_{s}} u 2 δ_{s} + M'_{δ_{b}} u 2 δ_{b}] - mgh \sin θ

(17)

The yawing equation is as follows

I_{z} \overset{\cdot}{r} = \frac{1}{2} ρ L 5 [N'_{\overset{\cdot}{r}} \overset{\cdot}{r}] + \frac{1}{2} ρ L 4 [N'_{\overset{\cdot}{v}} \overset{\cdot}{v} + N'_{r} ur + N'_{| v | r} | (v 2 + w 2) \frac{1}{2} | r] + \frac{1}{2} ρ L 3 [N'_{v} uv] + \frac{1}{2} ρ L 3 [N'_{δ_{r}} u 2 δ_{r}]

(18)

According to the six equations mentioned above, the basic motion characteristics can be achieved which can restrict some extreme unpractical motions which the submarine cannot implement.

Collision risk model

Membership function

We demand that the mathematical model can fully reflect the impact of various factors on the collision risk. The distance to closest point of approach (DCPA), time to closest point of approach (TCPA), and the RANGE (distance between the submarine and obstacles) are selected as the major parameter to construct the membership function. Based on the definition of the DCPA, TCPA, and RANGE, the smaller of their value, the more dangerous about the submarine. This to say, the larger of the risk degree. So we use these three parameters to construct the membership functions of the collision risk. The fuzzy statistics method which is similar to probability statistics approach is adopted to describe the fuzzy risk in the membership function.

For the TCPA and RANGE, we take $ξ = (ξ_{1}, ξ_{2})$ to $(0, + \infty)$ on the random interval. $ξ_{1}$ and $ξ_{2}$ are independent continuous random variables, then, where $F_{1} (x)$ is the distribution function of $ξ_{1}$ on interval $(0, + \infty)$ , when $ξ_{1} = 0$ , then $F_{1} (x) = 1$ . Where $F_{2} (x)$ is the distribution function of $ξ_{2}$ on interval $(0, + \infty)$ . Let $ξ_{2}$ obey the uniform distribution in the interval $(a_{1}, a_{2})$ , the probability density function is

f (x) = {\frac{1}{a_{2} - a_{1}} a_{1} < x < a_{2} 0 x \leq a_{1}, x \geq a_{2}

(19)

The distribution function of $ξ_{2}$ is

F_{2} (x) = {0 0 < x \leq a_{1} \frac{x - a_{1}}{a_{2} - a_{1}} a_{1} < x \leq a_{2} 1 a_{2} < x

(20)

So the membership function is

μ (x) = {1 0 < x \leq a_{1} \frac{a_{2} - x}{a_{2} - a_{1}} a_{1} < x \leq a_{2} 0 a_{2} < x

(21)

We get the membership function of DCPA as

μ (x) = {1 0 < x \leq a_{1} \frac{a_{2} - x}{a_{2} - a_{1}} a_{1} < x \leq a_{2} 0 a_{2} < x

(22)

Set the value intervals of DCPA, TCPA, and RANGE as U_D, U_T, andU_R which also are taken as the collision risk fuzzy set that is described: A_D, A_T, and A_R. Then the membership functions are

u_{A_{T}} = {1 | TCPA | < t_{1} (\frac{t_{2} - | TCPA |}{t_{2} - t_{1}}) 2 t_{1} \leq | TCPA | \leq t_{2} 0 t_{2} < | TCPA |

(24)

u_{A_{R}} = {1 | Range | < r_{1} (\frac{r_{2} - | Range |}{r_{2} - r_{1}}) 2 r_{1} \leq | Range | \leq r_{2} 0 r_{2} < | Range |

(25)

Based on equations (23) to (25), the figures of membership functions are shown in Figure 1.

Figure 1.

Membership functions of DCPA, TCPA, and RANGE.

Analysis of collision risk model

For the domain, we define the expression as follows

d_{1} = (K_{1} \cdot DLA + T \cdot V_{R} \cdot F_{1}) \cdot F_{2} \cdot F_{3} = Domain

(26)

where Distance Latest Action (DLA) is the latest taken action to avoid collision distance between the two submarines. K₁ indicates the encounter category of the two or more submarines. If one submarine surpasses another one

K_{1} = 1.0

, otherwise, if the submarine is surpassed by others,

K_{1} = 0.8

. T is the consuming time while the submarine turn 90° around, V_R is the relative velocity of the target submarine, F₁ is determined by the encounter situation

where

DE = {Thecurrentorientation DCPA \leq 0.5 nmile TheorientationofCPA DCPA > 0.5 nmile

(28)

F₂ is the visibility factor

F_{2} = {1.0 favorable 1.25 unfavorable

(29)

F₃ is the situation of water

F_{3} = {1.0, generallywater 0.75, turbidwater

(30)

For limited visibility, expand domain value to increase the range of early alert. In the turbid water environment, due to limited visibility, so the domain value should be reduced lightly

d_{2} = K_{2} \cdot d_{1}

(31)

where K₂ is the coefficient that is determined by situation of the submarine.

Horizontal risk model

Based on equation (31), set the d₁ and d₂ as the safety area radius. Figure 2 shows collision risk area of space horizontal plane. When $DCPA < d_{1}$ , it indicates there is one potential collision risk may occurred. On the other hand, when $DCPA > d_{2}$ , there is no any collision risk may happened. When $d_{1} < DCPA < d_{2}$ , the degree of risk is obtained through the membership functions.

Figure 2.

Collision risk area of space horizontal plane.

Vertical risk model

Taking into account the characteristics of submarine movement, we decompose the submarine movement into horizontal and vertical space. Similar to the horizontal risk area, we separate the vertical space into three levels based on h₁ and h₂. As shown in Figure 3, for the vertical distance HCPA

U_{A_{h}} = {1 | HCPA | < h_{1} (\frac{h_{2} - | HCPA |}{h_{2} - h_{1}}) 2 h_{1} \leq | HCPA | \leq h_{2} 0 h_{2} < | HCPA |

(32)

Figure 3.

Collision risk area of vertical.

where, $h_{1} = 0.01$ n mile, $h_{2} = 2 h_{1}$ .

Submarine collision risk model

According to horizontal and vertical risk model obtained above, we adopt the criteria that when the DCPA and TCPA are not equal to zero, one collision may have occurred as well as one collision avoidance action need to be taken immediately. For instance, if the $DCPA = 0$ in horizontal plane, it means $U_{A} (DCPA, TCPA, R_{T}) = 1$ , this may have two cases as follows:

If $HCPA = 0$ , then $U_{A} (HCPA) = 1$ , it means submarines and targets along the Z-axis direction of motion in the horizontal projection of the intersection of the encounter happened. There are two collisions in the 3D space.

If $HCPA = 0$ , then $U_{A} (HCPA) = 1$ , it means that the submarines and targets along the Z-axis motion does not meet each other even though there exists one intersection point in the horizontal space. No any collision may have occurred definitely.

Based on above, we give submarine collision risk model as follows

Submarine collision avoidance simulation

For evaluating the performance of collision avoidance risk model, we carried out one multiobject collision avoidance simulation system to illustrate the submarine collision avoidance decision making.

We initialize the submarine motion angle as 0°, speed is 10 kn, the initial coordinate is (0,0,−0.1), pitch angle is 0°. In our simulation scenario, we lay out four static obstacles around the submarine. For No. 1 obstacle, the course angle is 0°, the speed is 0 kn, the coordinate is (−1.5,−0.4,−0.1), the pitch angle is 0°; No. 2 obstacle’s course is 0°, the speed is 0 kn, the coordinate is (1,−1,−0.1), the pitch angle is 0°; No. 3 obstacle’s course is 330°, the speed is 10 kn, the coordinate is (0.8,0,−0.3), the pitch angle is 10°; No. 4 obstacle’s course is 120°, the speed is 8 kn, the coordinate is (−2.5,−1,−0.3), the pitch angle is 10°. The visibility of this simulation is set to good and the water situation is set to general.

Based on our proposed 2D risk model the horizontal degree of risk of the four obstacles is 0.55, 0.42, 0.51, and 0.63 while the vertical degree of risk is 1.0, 1.0, 0.78, and 0.83, respectively. The layout of the simulation initial situation is shown in Figure 4.

Figure 4.

Layout of the submarine and obstacles.

In the process of simulation, we set a safety margin parameter as 0.8 unit. This is to say, if the position of the obstacle fluctuate ±0.8 unit randomly, the submarine is not necessary to make any collision avoidance actions.

For submarine collision avoidance decision making, two categories of avoidance strategies are introduced:

Strategy 1:

Avoid the nearest obstacle 1 under turn right 29°, slow down to 5 kn, then turn left 42° in 3 min, speed up to 10 kn; turn right 10° in 6 min, keep speed at 10 kn. The simulation result is shown in Figures 5 and 6.

Figure 5.

Simulation collision avoidance under strategy 1 in 3D view.

Figure 6.

Simulation collision avoidance under strategy 1 in horizontal view.

Strategy 2:

Avoid the obstacle 2 under turn left 20°, slow down to 5 kn, then turn left 20° in 3 min, speed up to 10 kn; turn right 10° in 6 min, keep speed at 10 kn. The simulation result is shown in Figures 7 and 8.

Figure 7.

Simulation collision avoidance under strategy 2 in 3D view.

Figure 8.

Simulation collision avoidance under strategy 2 in horizontal view.

Along with the moving of submarine, the degree of the risk related to the four obstacles is also changing. The value of the risk degree is shown in Figures 9 and 10 under strategy 1 and 2, respectively.

Figure 9.

Risk curves of each obstacle under strategy 1.

Figure 10.

Risk curves of each obstacle under strategy 2.

We also carried out some comparison between strategies 1 and 2 taking speed fluctuation, times of direction change, and time consuming into account. The result of the simulation shows that the strategy 2 is superior to strategy 1 significantly.

Conclusion

The main purpose of this paper is to handle the submarine collision avoidance. We proposed a novel and efficient collision risk model as well as discussed a serial of motions pattern of the submarine to be processed for fulfilling the collision avoidance based on this model. The concepts of detection domain and fuzzy logic are adopted for modeling the degree of collision risk in vertical and horizontal space. The detection domain is used as the submarine safety scope when navigating underwater while the fuzzy logic is used as the mathematical implement for the analysis and synthesis of relations between obstacles or other submarines which encounter each other in the navigation environment. One efficient simulation system is carried out to verify the performance of proposed method in different scenarios.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the National Natural Science Foundation of China (No. 61203255), Natural Science Foundation of Heilongjiang Province of China (No. F201414) and Education Apartment Foundation of Heilongjiang Province (No. 12541725).

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