Water wave optimization algorithm for autonomous underwater vehicle path planning problem

1 Introduction

Due to the continuous advancement and development of science and technology, the discovery and exploitation of marine resources has become increasingly important. As effective pieces of marine equipment, AUVs have been widely used to solve certain problems, such as marine hydrographic survey and mine detection, military investigation and rescue, undersea inspection and data collection, ocean exploration, anti-submarine, drilling support, subsea construction, and underwater equipment laying and maintenance. AUV path planning is the process of determining an optimal path from the starting point to the target point under certain specific constraints [1–4]. AUV path planning is not only a manifestation of autonomy and intelligence but also a guarantee of safety and reliability. AUV can effectively avoid all threat areas to obtain an optimal or near-optimal path according to autonomous navigation and obstacle avoidance capabilities. The core concept of AUV path planning is to find the shortest path that satisfies a series of constrains under the premise of comprehensively considering factors, such as fuel consumption, threat level, threat area and navigation area. In recent years, some meta-heuristic optimization algorithms have been used to solve the path planning problems, such as artificial bee colony (ABC) [5], flower pollination algorithm (FPA) [6], bat algorithm (BA) [7], particle swarm optimization (PSO) [8], and moth-flame optimization algorithm (MFO) [9].

Zhuang et al. designed a two-stage cooperative path planner for multiple autonomous underwater vehicles operating in a dynamic environment. The proposed method is capable of reacting quickly to a dynamic ocean environment and can successfully avoid collisions [10]. Lim et al. proposed the method of using selectively hybridized particle swarm optimization algorithms to solve constrained path planning of autonomous underwater vehicles. The method had a low computational requirement and an excellent solution quality, so that the proposed is effective and feasible to obtain the optimal path from the starting point to the target point [11]. Wu tried to solve the coordinated path planning for an unmanned aerial-aquatic vehicle and an autonomous underwater vehicle in an underwater target strike mission. The coordinated path derived from the proposed method is close to the optimal solution in theory [12]. Barua et al. described the algorithms for guidance and control of an autonomous underwater vehicle for a specific identification mission. Their approach obtained better optimization results and the optimal or near-optimal path [13]. Taheri et al. applied a closed-loop rapidly exploring random tree algorithm to solve the path planning problem of an autonomous underwater vehicle, which provided a feasible planned path [14]. Pi et al. provided a search-based motion planning algorithm applied to the coordinated manipulation problem of a dual-arm intervention AUV [15]. Li et al. introduced an autonomous underwater vehicle optimal path planning method for seabed terrain matching navigation to avoid problem areas, and the proposed method had higher matching accuracy and lower time consumption [16]. Ding et al. demonstrated the optimal anti-submarine search path of an AUV by maximizing the cumulative detection probability. The optimal path was effective and suggestive for anti-submarine operation [17]. Lin et al. conducted a study on multiobjective particle swarm optimization for the AUV route planning problem, and the proposed method required less sailing time and energy consumption [18]. Zeng et al. designed a quantum-behaved particle swarm optimization algorithm to solve the optimal path planning problem of an AUV, and their proposed method obtained the best path [19]. Khan et al. presented a protocol that utilized AUVs to save energy wasted by sensor nodes during clustering [20]. MahmoudZadeh et al. applied evolutionary algorithms to solve the path planning problem. The proposed algorithms generated an optimal and collision-free path to avoid obstacles [21]. Gan et al. adopted the quantum-behaved particle swarm optimization algorithm to solve the dynamic trajectory tracking problem for AUVs, where the proposed method balanced exploration and exploitation to obtain a global optimal solution [22]. Zhu et al. showed a biologically inspired self-organizing map algorithm to achieve task assignment and path planning of an AUV system, and the effectiveness and feasibility of the algorithm were verified [23]. Wang et al. utilized the improved gray wolf optimizer to solve the monitoring trajectory optimization problem for USVs, and found that the proposed algorithm consumed minimal energy and exhibited obstacle avoidance [24]. Zhou et al. designed an improved flower pollination algorithm to solve the UUV path planning problem, the proposed method obtain the optimal or near-optimal path [25].

The WWO algorithm based on shallow wave theory simulates propagation, refraction and breaking to perform global optimization [26]. The WWO algorithm accelerates convergence speed and improves calculation accuracy. The WWO algorithm has strong stability and robustness. The WWO algorithm is applied to solve the AUV path planning problem, which not only expands the search space to avoid falling into the local optimum but also balances the global search ability and the local search ability to obtain the optimal solution. The experiments show that the convergence speed and calculation accuracy of the WWO algorithm are better than those of other algorithms, and the WWO algorithm is an effective and feasible method for solving the AUV path planning problem.

This article is divided into following sections. Section 2 introduces the problem description and formulation. Section 3 reviews the WWO algorithm. The experimental results and analysis are provided in Section 4. Finally, conclusions are drawn and future research is proposed in Section 5.

2 Problem description and formulation

An AUV system principally contains two main propellers, four auxiliary thrusters, a horizontal rudder and a vertical rudder. The two main propellers provide navigation power (surge) and are installed at the stern of the vehicle. Two of the auxiliary thrusters provide sway power and yaw momentums and are installed on a transverse layout, and the other two auxiliary thrusters provide power for heave and moments for pitching and are installed on a vertical layout. The horizontal rudder and vertical rudder are used to change the heading angle of the horizontal direction and vertical direction, respectively. Therefore, surge, sway, heave, pitch and yaw are controllable while roll is not.

The North-East-Down (NED) coordinate and body-fixed coordinate is given in Fig. 1. The NED coordinate is used to describe the position information of the AUV. The coordinate origin O _E is a random point at sea lavel; the ξ axis is always horizontal and pointing to the north, the η axis is kept perpendicular to the ξ axis and points to the east, and the ζ axis is kept perpendicular to the ζη plane and points to the center of the Earth. The coordinate origin O _B is used as the original coordinate point of the AUV, the x axis points to the bow along the longitudinal section, the y axis is perpendicular to the longitudinal section and points to the right chord, and the z axis points to the bottom of the boat along the longitudinal section. To determine the mutual correspondence between the NED coordinate and the body-fixed coordinate, the kinematic model of AUV is as follows:

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Fig. 1

Earth-inertial frames and body-fixed reference frames.

[ x · y · z · ] = [ cos ψ cos θ cos ψ sin θ sin ϕ - sin ψ cos ϕ cos ψ sin θ cos ϕ + sin ψ sin θ sin ψ cos θ sin ψ sin θ sin ϕ + cos ψ cos ϕ sin ψ sin θ cos ϕ - cos ψ sin ϕ - sin θ cos θ sin ϕ cos θ cos ϕ ] · [ u v w ](1) where ϕ, θ and ψ are important control parameters, and the value ranges are - π 2 < ϕ < π 2 , - π 2 < θ < π 2 and -π < ψ < π. [ ϕ · θ · ψ · ] = [ 1 sin ϕ tan θ cos ϕ tan θ 0 cos ϕ - sin ϕ 0 sin ϕ / cos θ cos ϕ / cos θ ] · [ p q r ](2)

Assuming that the roll movement is uncontrollable and the structure of AUV is bilaterally symmetrical, take w = 0, v = 0, and we rearrange to obtain [ x · y · z · ] = [ cos ψ cos θ sin ψ cos θ - sin θ ] · u(3) { θ · = q ψ · = r / cos θ(4)

As an active sonar, the forward looking sonar (FLS) is necessary for AUV safe navigation. FLS not only detects and perceives information about the environment surrounding an AUV but also identifies threat areas and sends out danger signals, which is beneficial for an AUV to successfully avoid threat areas and reach the target point safely. FLS based on the characteristics of sound waves propagating in water is used to complete underwater target detection and underwater communication through signal conversion and processing. FLS uses discrete data points to output and save data for detecting underwater obstacles. During the navigation of the AUV, the FLS is used to perceive the underwater road conditions. The data of the detected obstacles uses the sonar position as a reference point to effectively process more stored echo information.

For distributed threat areas, the goal of AUV path planning is to obtain an optimal or near-optimal path under certain constraints. The transformation of the coordinate system is given in Fig. 2.

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Fig. 2

Transformation of the coordinates system.

The coordinate system O _xy is transformed to the coordinate system O_x′y′. The positive direction of the horizontal axis is the line from the starting point to the target point. The transformation relationships can be defined as: γ = arcsin y 2 - y 1 | AB → |(5)

[ x y ] = [ cos γ sin γ - sin γ cos γ ] · [ x ′ y ′ ] + [ x 1 y 1 ](6) where point (x, y) denotes the coordinate in the O _xy , point (x′, y′) denotes the new coordinate in the O_x′y′, and γ denotes the rotation angle of the coordinate system. On the horizontal axis x′, the line segment from the starting point to the target point is divided into equal D segments. These lines are parallel to the axis y′, the intersections with axis x′ are represented by small rectangles, D abscissas are obtained, then a point is elected on each line to obtain D ordinates. Finally, nodes with horizontal coordinates and vertical coordinates are obtained, thus a path is designed by connecting points.

Threat cost J _t and fuel cost J _f are main evaluation criteria for AUV path planning, and the calculation formulas can be defined as: J t = ∫ 0 L w t dl(7) J f = ∫ 0 L w f dl(8) where w _t and w _f denote the threat cost and fuel cost of the neutron segment, respectively. L denotes the total length of the path. An approximate method is applied to calculate the optimal path. Each subsegment is divided into five smaller subsegments, and the threat cost on each subsegment is concentrated at the end point of the subsegment. The model of the AUV threat cost is given in Fig. 3.

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Fig. 3

Model of the AUV threat cost.

If the distance between the threat point and the end of the subsegment is less than the radius of the threat area, the threat cost of a subsegment can be defined as: w t , L i = L i 5 · ∑ k = 1 N t t k · ( 1 d 0.1 , i , k 4 + 1 d 0.3 , i , k 4 + 1 d 0.5 , i , k 4 + 1 d 0.7 , i , k 4 + 1 d 0.9 , i , k 4 )(9)

where N _t denotes the number of the threat regions, L _i denotes the length of ith segment, d_0.1,i,k denotes a distance from 1/10 point on the ithsegment to the kth threat center. Assuming that the speed of AUV is constant, the fuel cost J _f denotes proportional to the path length. The total cost of the path planning can be defined as: J = kJ t + ( 1 - k ) J f(10) where k ∈ (0, 1) denotes the important weight coefficient. As k is close to 0, the planned path is shorter. As k is close to 1, the planned path has a higher safety factor. In this paper, k is set to 0.5.

The WWO algorithm mainly simulates wave motions including propagation, refraction and breaking for effective searching. The process of wave movement is from deep regions to shallow regions. For a wave object, wave height h and wavelength λ are two important attributes, and the fitness value varies with the unevenness of the seabed. In deep regions, a wave obtains lower fitness value, lower wave height, and longer wavelength. In shallow regions, the wave obtains higher fitness value, higher wave height, and shorter wavelength, as illustrated in Fig. 4. The correspondence between problem space and population space is given in Table 1.

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Fig. 4

Different wave shapes in deep and shallow water.

3.3 Breaking

As the energy of water wave continues to increase, the wave crest will continue to steepen. When the velocity of the wave crest exceeds the propagation velocity of the water wave, the wave will break up into a series of solitary waves. In the WWO algorithm, the optimal wave x^* is used to perform breaking operations. The specific way is to randomly select k dimensions from D, and selecting each dimension creates solitary waves. The updated location can be defined as: x ′ ( d ) = x ( d ) + N ( 0 , 1 ) · β L ( d )(15) where β denotes the breaking coefficient. If the fitness value of all solitary waves is worse than that of the optimal wave x^*, x^* is retained. Otherwise, wave x^* is replaced by an optimal solitary wave.

In general, the propagation operation modulates the fitness values of different regions to effectively balance exploration and exploitation; the refraction operation removes the exhausted wave from the population and obtains the optimal wave, which can avoid search stagnation and accelerate convergence speed; and the breaking operation enhances the intensive search near the optimal solution and improves calculation accuracy. These three operations provide a good search mechanism for the WWO algorithm to obtain the global optimal solution. The correspondence between AUV path planning and the WWO algorithm is given in Table 2. For the swarm intelligence optimization algorithm, the search process and optimization process simulate the evolution and foraging process of individuals, the points in the search space simulate individuals in nature, the objective function of the problem is analogous to the individual’s ability to adapt to the environment, and the individual foraging process or the survival of the fittest is analogous to the iterative process of replacing the less feasible solutions with more feasible solutions in the search and optimization process. The solution procedure of the AUV path planning problem is given in Table 3. The solution methodology flow chart of the AUV path planning problem is provided in Fig. 5.

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Table 2

The correspondence between the AUV path planning and the WWO algorithm

The AUV path planning space	The WWO algorithm space
A collection contains all the optimization schemes to solve AUV path planning problem	A wave population with
An optimal or near-optimal path	An optimal wave
The total cost of the AUV path planning problem	The fitness function of the WWO algorithm

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Table 3

The solution procedure of the AUV path planning problem

Step 1. According to the environmental model, initialize the detailed information about the AUV path planning task, starting point coordinates, target point coordinates, threat center, threat radius and threat level. Transform coordinate system according to Equation (5) and (6).

Step 2. Initialization: Randomly initialize a wave population Pof n waves (solutions), the wavelength λ, the wave height hmax, the wavelength reduction coefficient α, the breaking coefficient β, maximum number kmax of breaking directions, and the maximum number itermax.

Step 3. Calculate the total cost of each water wave by Equation (10) based on water wave parameters. The smaller the cost value is, the better performance the AUV path maintains.

Step 4. Obtain the optimal wave

While stop criterion is not satisfied do

Step 5. for each wave x ∈ P do

Step 5.1. Propagate wave x to a new wave x′ by Equation (11).

Step 5.2. if f (x′) < f (x) then

Step 5.2.1. if f (x′) < f (x*), then wave x′ perform breaking operation by Equation (15), wave x* updates to wave x′.

Step 5.2.2. water wave x is replaced by wave x′ in the population.

Step 5.3. else, the wave height of wave x is decreased by one to express the energy loss. Ifx · h = 0, then wave x perform refraction operation by Equation (13) and (14).

Step 5.4. Update the wavelength of each wave by Equation (12).

Step 6. Calculate the total cost of each water wave by Equation (10).

Step 7. Set iter = iter + 1. Ifiter < itermax, go to step 5.

Step 8. Inversely transform the coordinates in final path into the original coordinate and output the optimal solution.

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Fig. 5

Solution methodology flow chart of AUV path planning.

To verify effectiveness and feasibility, the WWO algorithm was applied to solve the randomly generated threat areas and generated fixed threat areas. The WWO algorithm has faster convergence speed and higher calculation accuracy, so that it can effectively obtain an optimal or near-optimal path in solving the path planning problem. The numerical experiment is set up on a computer with an Intel Core i7-8750 H 2.2 GHz CPU, a GTX1060, and 8 GB memory running on Windows 10.

4.1 Simulation experiments of AUV path planning in random threat areas

The AUV sails in the deep sea, and the threat regions encountered are randomly distributed. The generated threat regions are as follows:

Threat point coordinates: x = rand ( n , 1 ) · λ + m(16) y = rand ( n , 1 ) · λ + m(17) where n denotes the number of the threat regions, λ denotes the path length, and m denotes the adjustment parameter to ensure the generation of the starting point.

The threat radius: r = rand ( n , 1 ) / 4 · λ(18) where r denotes the randomly generated radius. In this paper, the radius of the randomly generated threat regions is not less than 10.

Threat coefficient: k = round ( rand · r )(19) where round () denotes the result of rounding the specified number of decimal places.

The starting point coordinates and the target point coordinates are: start = [ min ( x ) - r , 0 ](20) target = [ max ( x ) + r + Δ x , round ( rand · λ ) + Δ y ](21) where the starting point and target point are guaranteed to be outside the threat regions. Δx and Δy adjust the distance of the target point to the threat regions.

In this section, the WWO algorithm is used to solve the randomly generated threat areas, and the number of the threat points is 6. Since comparison experiments cannot be performed, the simulation tests of the WWO algorithm is 5. The population size is 20, the maximum number of iterations is 200, and the dimension of the problem is 14. Table 4 gives some relevant parameters about threat center, threat radius and threat level. The starting point and the target point coordinates are given in Table 5. The experimental results for random threat areas are given in Table 6.

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Table 4

A randomly generated parameter table with six threat points

Parameters	1	2	3	4	5	6
Threat center	(98,35)	(73,68)	(28,72)	(92,99)	(101,69)	(27,55)
Threat radius	23	12	13	17	10	13
Threat level	9	5	5	7	4	5
Threat center	(30,48)	(109,56)	(113,79)	(61,90)	(55,27)	(79,55)
Threat radius	10	21	25	10	17	25
Threat level	5	10	12	5	8	12
Threat center	(106,75)	(56,46)	(79,75)	(51,81)	(85,33)	(71,60)
Threat radius	10	10	10	10	24	10
Threat level	9	9	9	9	21	9
Threat center	(52,33)	(28,29)	(82,49)	(54,85)	(24,76)	(21,64)
Threat radius	20	10	15	16	16	10
Threat level	9	4	6	7	7	4
Threat center	(109,52)	(81,53)	(101,64)	(49,107)	(109,109)	(38,60)
Threat radius	10	12	10	17	10	10
Threat level	4	5	4	7	4	4

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Table 5

The starting point and the target point coordinates

Parameters	1	2	3	4	5
Starting point	(14,0)	(20,0)	(40,0)	(7,0)	(27,0)
Target point	(121,103)	(139,108)	(116,108)	(98,101)	(119,106)

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Table 6

Experimental results for random threat areas

Threat numbers	Population	Iteration	1	2	3	4	5
6	20	200	75.6242	86.4332	71.9365	70.5012	71.1435

For five randomly generated threat areas, we conducted five independent experiments on AUV path planning. Figures 6–10 are the effect diagrams of the path. The WWO algorithm can effectively avoid the threat areas and obtain an optimal path from the starting point to the target point in response to randomly generated threat regions. Figure 11 gives the convergence graph of five independent experiments, which indicates that the WWO algorithm has a faster convergence speed in path planning. Since the randomly generated threat regions are different, the optimal path obtained will be different. The WWO algorithm cannot be compared with other optimization algorithms due to different threat regions. Therefore, we select three sets of fixed threat regions for AUV path planning in the following section.

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Fig. 6

The path planning result using WWO, D = 14.

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Fig. 7

The path planning result using WWO, D = 14.

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Fig. 8

The path planning result using WWO, D = 14.

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Fig. 9

The path planning result using WWO, D = 14.

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Fig. 10

The path planning result using WWO, D = 14.

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Fig. 11

The evolution curves of fitness values, D = 14.

4.2 Simulation experiments of AUV path planning in fixed threat areas

To test the performance of the WWO algorithm for solving AUV path planning problems, the WWO algorithm was compared with other intelligent optimization algorithms, including ABC, FPA, BA, PSO and MFO. The population size was 20, the maximum number of iterations was 200, and the number of independent runs was 20. Best, Worst, Mean and Std. represented the optimal value, the worst value, the mean value and the standard deviation. The parameters of all algorithms were empirical values and these parameters were derived from the original algorithm. The initial parameters of all algorithms are given in Table 7.

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Table 7

Initial parameters of all algorithms

Algorithms	Parameters
ABC [5]	Maximum number of searches t = 5D
FPA [6]	Switching probability p = 0.8
BA [7]	Maximum pulse intensity A = 0.5, maximum pulse frequency r0 = 0.5, pulse attenuation coefficient γ = 0.95, pulse frequency increase factor α = 0.05
PSO [8]	Constant inertia ω = 0.7298, first acceleration factor c1 = 1.4962, second acceleration factor c2 = 1.4962
MFO [9]	A constant for defining the shape of logarithmic spiral b = 1, a random number t ∈ [-1, 1], convergence constant r ∈ [-2, - 1]
WWO [26]	Wavelength λ = 0.5, wave height hmax = 6, wavelength reduction coefficientα = 1 . 0026, breaking coefficient β ∈ [0.01, 0.25], maximum number of breaking directions kmax = Min (12, D/2)

Eight threat regions were used to verify the effectiveness of the WWO algorithm, and the starting point coordinates and target point coordinates were (0,0) and (100,100). Tables 8 and 9 give information on fixed threat regions and experimental results, respectively. The smallest standard deviation is shown in bold. The dimension was 10, the standard deviation of the WWO algorithm was worse than that of ABC and FPA, and the optimal value of WWO was better than that of ABC, FPA, BA and MFO. The dimensions were 15, 20, 25, 50 and 100, the standard deviation of the WWO algorithm was the smallest of all the algorithms, which indicates that WWO has better stability for solving the path planning problem. As the dimension increases, the optimal value, worst value, mean value and standard deviation increased accordingly. Figures 12–14 show that for path planning results, the WWO algorithm can travel along the edge of the threat regions to obtain the shortest optimized path. Figures 15–20 show that the convergence speed and the calculation accuracy of the WWO algorithm are significantly better than other optimization algorithms. Figures 21–26 show a comparison with other algorithms and indicate that the WWO algorithm obtains a smaller standard deviation, which indicates that WWO has higher stability and is more robust. Table 10 lists the results of the p-value Wilcoxon rank-sum [27]. Most of the values are less than 0.05, which indicates that the results are not accidental.

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Table 8

Information of fixed threat regions

Parameter	Values
Threat center	(30,20)	(50,15)	(56,55)	(0,24)	(50,80)	(75,90)	(100,70)	(50,36)
Threat radius	10	21	10	17	12	10	20	11
Threat level	5	5	5	5	5	5	5	5

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Table 9

Experimental results in different Dim

Dim	Result	ABC	FPA	BA	PSO	MFO	WWO	Rank
10	Best	77.46911	81.66488	131.1134	73.20228	73.70884	73.43460	3
	Worst	158.1165	122.9144	968.3899	183.2854	196.3991	136.7001
	Mean	88.19875	97.5196	412.7734	124.5051	117.8808	89.73958
	Std	17.6521	11.57983	196.7997	35.97221	39.12623	27.44364
15	Best	77.8581	91.70764	144.4140	72.52665	77.17022	72.48798	1
	Worst	147.9426	176.6483	935.7004	139.4299	202.8406	104.6581
	Mean	98.40757	129.4548	345.8716	92.2564	112.9956	89.85825
	Std	17.65153	24.53874	207.5458	17.21917	33.54773	15.23420
20	Best	83.52200	110.0399	109.1176	72.27397	82.3559	73.09791	1
	Worst	198.0079	244.2918	1231.615	163.9923	230.8629	97.26798
	Mean	112.0809	163.3028	388.1526	86.42976	122.9919	81.99995
	Std	33.3744	30.08485	254.4486	19.20993	43.07774	6.280959
25	Best	95.39054	123.9522	218.915	72.53236	79.23822	71.76682	1
	Worst	249.8128	304.9924	1586.78	133.9232	712.9751	85.89939
	Mean	136.5002	215.5197	653.6681	81.71362	187.4807	78.58791
	Std	34.32822	53.42395	376.2454	12.97656	155.6567	2.527923
50	Best	376.3448	296.3874	448.5348	118.9048	256.0572	82.68874	1
	Worst	2776.805	2501.877	5069.379	526.7492	1874.872	119.8188
	Mean	1016.975	1323.017	2336.23	246.4334	606.5542	90.59792
	Std	669.7158	685.2483	1319.189	122.2871	462.2646	8.415053
100	Best	5043.015	3429.677	3014.341	697.1723	2254.459	183.4793	1
	Worst	30226.61	14821.71	20856.22	5890.888	12627.57	334.0300
	Mean	12462.78	8569.250	9102.742	2383.068	5800.794	253.2770
	Std	5869.216	3444.647	4860.132	1460.63	2978.801	37.97445

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Fig. 12

The path planning result using WWO, D = 10.

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Fig. 13

The path planning result using WWO, D = 15.

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Fig. 14

The path planning result using WWO, D = 20.

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Fig. 15

The evolution curves of algorithms, D = 10.

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Fig. 16

The evolution curves of algorithms, D = 15.

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Fig. 17

The evolution curves of algorithms, D = 20.

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Fig. 18

The evolution curves of algorithms, D = 25.

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Fig. 19

The evolution curves of algorithms, D = 50.

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Fig. 20

The evolution curves of algorithms, D = 100.

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Fig. 21

The ANOVA of algorithms, D = 10.

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Fig. 22

The ANOVA of algorithms, D = 15.

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Fig. 23

The ANOVA of algorithms, D = 20.

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Fig. 24

The ANOVA of algorithms, D = 25.

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Fig. 25

The ANOVA of algorithms, D = 50.

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Fig. 26

The ANOVA of algorithms, D = 100.

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Table 10

Results of the p-value Wilcoxon rank-sum

Algorithm	Dim=10	Dim=15	Dim=20	Dim=25	Dim=50	Dim=100
WWO vs ABC	0.004703	0.04388	3.07E-06	6.8E-08	6.8E-08	6.8E-08
WWO vs FPA	0.007113	5.87E-06	6.8E-08	6.8E-08	6.8E-08	6.8E-08
WWO vs BA	1.43E-07	6.8E-08	6.8E-08	6.8E-08	6.8E-08	6.8E-08
WWO vs PSO	0.081032	0.903116	0.989209	0.818149	7.9E-08	6.8E-08
WWO vs MFO	0.001625	0.006557	1.58E-06	2.56E-07	6.8E-08	6.8E-08

In this paper, a novel WWO algorithm is proposed to solve the optimal path planning problem in a complex ocean environment. The generated optimal path ensures that the AUV safely reaches the target point with minimum fuel cost. For randomly generated threat regions, the WWO algorithm can successfully avoid all threat regions and obtain an optimal path from the starting point to the target point. For generated fixed threat regions, the WWO algorithm has faster convergence speed and higher calculation accuracy compared with other intelligent evolutionary algorithms, which can better solve the optimal path planning problem. For these regions, the WWO algorithm can effectively balance exploration and exploitation to avoid all threat regions for obtaining the optimal or near-optimal path under the constraint condition. The experimental comparison shows the stability and superiority of the WWO algorithm over the ABC, FPA, BA, PSO and MFO algorithms, which provides an effective approach to solve the optimal path planning problem.

For future studies, the WWO will be used to solve the path planning problem that generates more threat areas randomly. The WWO algorithm will be applied to solve AUV path replanning in a spatiotemporal ocean environment, cooperative path planning for multiple AUVs, and the three-dimensional optimal path planning problem.