A prior information‐based coverage path planner for underwater search and rescue using autonomous underwater vehicle (AUV) with side‐scan sonar

IET Radar, Sonar & NavigationVolume 16, Issue 7 p. 1225-1239 ORIGINAL RESEARCHOpen Access A prior information-based coverage path planner for underwater search and rescue using autonomous underwater vehicle (AUV) with side-scan sonar Chang Cai, Chang Cai orcid.org/0000-0002-8164-2798 School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, ChinaSearch for more papers by this authorJianfeng Chen, Corresponding Author Jianfeng Chen [email protected] School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, China Correspondence Jianfeng Chen, School of Marine Science and Technology, Northwestern Polytechnical University, 127 West Youyi Road, Xi'an, China. Email: [email protected]Search for more papers by this authorQingli Yan, Qingli Yan School of Computer Science and Technology, Xi'an University of Posts and Telecommunications, Xi'an, ChinaSearch for more papers by this authorFen Liu, Fen Liu School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, ChinaSearch for more papers by this authorRongyan Zhou, Rongyan Zhou School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, ChinaSearch for more papers by this author Chang Cai, Chang Cai orcid.org/0000-0002-8164-2798 School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, ChinaSearch for more papers by this authorJianfeng Chen, Corresponding Author Jianfeng Chen [email protected] School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, China Correspondence Jianfeng Chen, School of Marine Science and Technology, Northwestern Polytechnical University, 127 West Youyi Road, Xi'an, China. Email: [email protected]Search for more papers by this authorQingli Yan, Qingli Yan School of Computer Science and Technology, Xi'an University of Posts and Telecommunications, Xi'an, ChinaSearch for more papers by this authorFen Liu, Fen Liu School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, ChinaSearch for more papers by this authorRongyan Zhou, Rongyan Zhou School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, ChinaSearch for more papers by this author First published: 22 April 2022 https://doi.org/10.1049/rsn2.12256Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract The coverage path planning (CPP) technique attracts growing interest in studies on underwater search and rescue (SAR) conducted with an autonomous underwater vehicle (AUV) equipped with a side-scan sonar (SSS). In SAR missions, prior information is crucial. Aiming at the underwater SAR mission with prior information, a new coverage path planner (SAR-A*) is proposed. The ultimate goal is to generate a feasible path for completely covering the task area and preferentially visiting more valuable cells with fewer turns. First, the whole task area is decomposed into hexagon cells as waypoints to be visited for complete coverage. Second, the probability of discovering the target is obtained according to the target presence probability and the SSS detection ability. Under the assumption of prior target information, the target presence probability is modelled as a two-dimension Gaussian distribution based on predicted target locations or trajectories. Then, an optimal next-waypoint selection process is formulated as a multi-objective decision-making problem and solved by the weighted metric method. Finally, simulation and experimental results demonstrate that the generated path can improve the cumulative probability of discovering the target with fewer turns. 1 INTRODUCTION Maritime Search and Rescue (SAR) is a challenging task in dangerous and unknown environments [1]. The purpose of maritime SAR is to locate and rescue the target on the surface [2, 3] or underwater [4]. For the sake of high efficiency and safety in underwater SAR missions, people have utilised a Side-Scan Sonar (SSS) mounted on an Autonomous Unmanned Vehicle (AUV) for wide-area searching for many years [5, 6]. To make the whole process smoothly, the AUV path planning is often an essential stage at the beginning of each SAR mission. Coverage Path Planning (CPP) is the process of generating a feasible path for completely covering an area interested [7]. It is widely used in seabed terrain matchings [8], mine countermeasures [9], search and rescues [4], pipeline inspections [10], hydrographic surveys [11], and so on. Generally, CPP algorithms for surface SAR can be used underwater [2]. However, different from surface SAR tasks, underwater SAR missions have special environment and operational constraints. Hence, it is necessary to develop a specialised method to meet specific requirements. The coverage path planners can be typically divided into two categories, the cellular decomposition-based methods [12-15] and the data/information-based methods [16-20]. Cellular decomposition-based methods decompose the task area into several parts and cover subareas through a basic coverage algorithm. In Ref. [14], a CPP algorithm for covering multi regions is transformed into the Travelling Salesman Problem (TSP) and the Lawn-Mowing (LM) pattern [21] was adopted to cover each region. An artificially weighted spanning tree coverage algorithm was proposed in Ref. [15], which can provide a quasi-optimal solution, whereas the impact of existing information on the CPP problem is not fully taken into account. Additionally, data/information-based methods guide the robot by real-time sensor measurements. An efficient online CPP algorithm put forward in Ref. [20] was based on sensor information to dynamically adapt to unknown environments. Considering the dynamic marine environment, Ai et al. [4] applied reinforcement learning to solve the CPP problem. However, besides real-time sensor information, the impact of prior information in real applications on the CPP problem is important but rarely discussed. In SAR missions, most of these coverage path planners mentioned above can be applied; however, it is noticed that they commonly pay no attention to the prior target information, which is available in most SAR cases. Prior target information usually plays a crucial role in designing target search strategies [10, 22-24]. Take Ref. [10] as an example, the search guidance by the prior target information is beneficial to the whole task. In real SAR missions, predicted prior information from rescue specialists is often available and the prior information helps to put priorities to valuable areas, which is especially important in large-scale SAR missions. In this work, the above issue is addressed and a novel CPP method named SAR-A* is developed. According to the predicted target location by the rescue specialists, a probabilistic map of target presence is constructed. Combined with the imperfect detection capability and operational rules of side-scan sonar, a waypoint selection strategy is formulated as a multi-objective problem and solved by the weighted metric method. The criteria for selecting the next-waypoint are smaller turning angles, shorter distances, and a higher probability of discovering the target. The core idea of traditional A* algorithm is introduced to guide the AUV to cells close to the most possible target position. Under the A* framework, the coverage path is generated by choosing optimal next waypoints recurrently according to the waypoint selection strategy. Simulation results show that the path generated by the proposed method can rapidly improve the probability of discovering the target and decrease total turns. The main contributions of this research study are summed up as follows: (1) The proposed SAR-A* method takes the advantage of the target prior information, which is modelled as the target presence probability map. It also takes the capability of the equipped side-scan sonar into account, which is normally ignored in previous CPP methods. The exploration of these two resources makes the method more useful and feasible in real SAR cases. (2) The core of the proposed method, that is, the next-waypoint selection process, is formulated as a multi-objective optimisation problem and solved by the weighted metric method. Therefore, the resulted path exhibits outstanding performance—higher probability of discovering the target with fewer turns path. (3) There are few restrictions on prior target information. Besides the prior target location, the target trajectory can be well used for building the target presence probability maps easily. The rest of this article is organised as follows. In Section 2, we define the problem and models. Section 3 presents the proposed method including the objective function and the algorithm framework. Following that, the simulations are conducted and the results are given in Section 4. We discuss the general conclusions and future work in Section 5. 2 PROBLEM DEFINITION AND MODELS The purpose of this study is to design a coverage path planner for an AUV equipped with side-scan sonar while performing a large-scale SAR mission. To explain the addressed problem, details about the coverage path planning problem including definitions and assumptions are described in this section. The task area A to be surveyed measuring LA × WA is defined as an open flat area without obstacles. The whole task area is decomposed into Nc small grid cells, denoted by Call = {c1, c2, …, cNc} A ⊆ C a l l = ⋃ i = 1 N c c i (1)where ci denotes ith grid cell. The task area is covered after traversing all decomposed cells. The AUV moves in an x-y plane at a fixed speed and altitude. While the AUV moves, side-scan sonar generates seabed sonar images along the trajectory. The operational rules of SSS suggest that straight movement helps to reduce the distortion of generated sonar images. The AUV with side-scan sonar and corresponding coverage swath are shown in Figure 1. The blind spot of the side-scan sonar, also called the nadir, is assumed to be successfully compensated. The SSS detection capability is considered in this study, which is analysed in Section 3.2.2. FIGURE 1Open in figure viewerPowerPoint 3D model of an autonomous underwater vehicle (AUV) equipped with a side-scan sonar (SSS) system Aiming at the SAR mission, we assume that target location and search scope at SAR time are provided by rescue specialists. This assumption is rational and can provide helpful information for navigation. The given target information considers all possible time-varying environment factors. When performing tasks, the AUV periodically surfaces to update location and prior information. Therefore, the path planner doesn't need to deduce target information further. The target presence probability stated in Section 3.2.1 incorporates the prior task information. The path planner continuously determines the optimal next-waypoint according to the objective function h elaborated in Section 3.3. According to the objective function h, the sequential waypoints obtained by the proposed path planner are denoted as Pa = {pa1, pa2, …, paNc}, where pai presents the ith waypoint. Certainly, each waypoint can appear only once in the generated path. Finally, the problem is solved until all cells are contained in Pa: min C a l l h p a i s.t. p a i ⊆ C a l l p a i ≠ p a j ∀ i , j ∈ 1 , 2 , … , N c . $\begin{array}{rl}\hfill \underset{{C}_{all}}{\mathrm{min}}& h\left(p{a}_{i}\right)\hfill \\ \hfill \,\text{s.t.}\,& p{a}_{i}\subseteq {C}_{all}\hfill \\ \hfill & p{a}_{i}\ne p{a}_{j}\hfill \\ \hfill & \forall \ i,j\in \left\{1,2,{\mathellipsis},{N}_{c}\right\}.\hfill \end{array}$ (2) 3 THE SAR-A* COVERAGE PATH PLANNER In this section, we present a novel coverage path planner named SAR-A* for the underwater SAR mission. The variables based on prior information are given to combining the prior information with the proposed path planner. The optimal next-waypoint selection strategy chooses the appropriate next-waypoint iteratively until all waypoints are visited. Task area decomposition As one of the most popular ways in complete coverage, task area decomposition divides the task area into a large number of cells. The area coverage is achieved by traversing all cells. Hexagon decomposition is applied in the SAR-A* method. Figure 2 describes the hexagon decomposition, where r = WS/2 denotes the radius of hexagons, and Ws is the swath width of SSS. FIGURE 2Open in figure viewerPowerPoint Hexagon decomposition. Point A can reach six neighbours at the same distance. The overlapping area is presented by red shadow There are plenty of advantages to using hexagon decomposition. First of all, each cell has six choices in different directions and the same distance. Second, the overlap area (see the shaded area in Figure 2) can significantly improve the final detection probability. According to the study in Ref. [25], the confidence of identifying targets decreases with the lateral range from the AUV trajectory. While the lateral range is larger than 70% × WS, it's poorly detectable. Hence, an overlapping edge area is necessary for the task's success. Finally, the next waypoints in each optional heading are equidistant, which leads to less computational burden to compute distance. Variables based on prior information The prior information considered includes two categories: prior target information and prior task information. Based on the prior target information, the target presence probability is moulded. The prior task information such as environment, targets, and used tools is considered and formulated as the SSS detection capability. 3.2.1 Target presence probability Prior target information from rescue specialists plays a crucial role in a SAR mission. Based on professional SAR simulation results [26, 27], the target fallen in the task area is most likely to appear in the predicted location. Figure 3 shows an example of a SAR scope estimation system [28]. FIGURE 3Open in figure viewerPowerPoint The target trajectory and search area evaluated by rescue specialists The target presence probability gives the probabilities pe that the target presents at any cell ci. It decreases with distance from the estimated target location, which can be depicted by the Gaussian distribution. Two-dimensional Gaussian distribution is applied to describe the probability that the target exists in each cell. Both the target location and the possible target trace can be applied to build the target presence probability map. The target presence probabilities have a two-dimensional Gaussian distribution with location vector L = {l1, l2, …, lNc}, mean vector μ and covariance matrix Σ, written as p e ∼ N ( L , μ , Σ ) ${p}_{e}\sim \mathcal{N}(L,\mu ,{\Sigma})$ . Let the parameters l i = x i , y i T ${l}_{i}={\left({x}_{i},{y}_{i}\right)}^{T}$ denote the location of cell ci, μ = x t , y t T $\mu ={\left({x}_{t},{y}_{t}\right)}^{T}$ denote the prior target location, xt and yt are coordinates in x-axis and y-axis, Σ = σ 1 2 σ 1 σ 2 σ 2 σ 1 σ 2 2 ${\Sigma}=\left(\begin{array}{cc}\hfill {\sigma }_{1}^{2}\hfill & \hfill {\sigma }_{1}{\sigma }_{2}\hfill \\ \hfill {\sigma }_{2}{\sigma }_{1}\hfill & \hfill {\sigma }_{2}^{2}\hfill \end{array}\right)$ , and σ1 and σ2 are constants. The target presence probability at cell ci is approximated by p e l i ; μ , Σ = 1 2 π | Σ | 1 / 2 exp − 1 2 l i − μ T Σ − 1 l i − μ . ${p}_{e}\left({l}_{i};\mu ,{\Sigma}\right)=\frac{1}{2\pi \vert {\Sigma}{\vert }^{1/2}}\mathrm{exp}\left(‐\frac{1}{2}{\left({l}_{i}‐\mu \right)}^{T}{{\Sigma}}^{‐1}\left({l}_{i}‐\mu \right)\right).$ (3) In this model, the parameter selection of N ( L , μ , Σ ) $\mathcal{N}\ (L,\mu ,{\Sigma})$ combines with the predicted target location and the estimated trajectory, which are reflected by the shape of the isocontours. Generally, the shape of the isocontours of a bivariate Gaussian distribution is a rotated ellipse which is presented by x i − μ 1 σ 1 2 − 2 ρ x i − μ 1 σ 1 y i − μ 2 σ 2 + y i − μ 2 σ 2 2 = k 2 ${\left(\frac{{x}_{i}‐{\mu }_{1}}{{\sigma }_{1}}\right)}^{2}‐2\rho \left(\frac{{x}_{i}‐{\mu }_{1}}{{\sigma }_{1}}\right)\left(\frac{{y}_{i}‐{\mu }_{2}}{{\sigma }_{2}}\right)+{\left(\frac{{y}_{i}‐{\mu }_{2}}{{\sigma }_{2}}\right)}^{2}={k}^{2}$ (4)where k is a constant. The vector μ is the centre of the ellipse, which is set to the predicted target position. Σ, that is, σ1 and σ2, is determined by the falling trajectory. When only the predicted target location is provided, Σ is a scalar matrix. Take Figure 4 for example, Figure 4a describes the target presence probability distribution according to the prior target position, and Figure 4b depicts the distribution when a horizontal estimated trajectory is assumed. FIGURE 4Open in figure viewerPowerPoint Two contours of different bivariate Gaussian distributions describing different prior information. (a) The target presence probability distribution according to the prior target position. (b) The target presence probability distribution according to the horizontal estimated trajectory 3.2.2 SSS detection capability The SSS is essential for the underwater SAR mission, which directly determines the mission result. In practice, the sonar data quality is rarely flawless, which results in the uncertain quality of generated sonar images [29]. The SSS detection ability refers to the synthetic appraisal that the system can accurately detect and identify a target. Prior task information including environment, targets and used tools is considered in SSS detection capability. As shown in Figure 5, factors affecting the performance of the SSS are grouped into four categories: target characteristics (Tg), underwater environment (En), SSS parameters (SSS) and AUV state (AS). Furthermore, it can be seen that factors in underwater environment, SSS parameters and target characteristics are given in prior and do not change with different AUV paths. With all given task details and existing professional methods qualifying the SSS performance [30], the detection capability pd is represented as a numeral result and considered in the algorithm designing, where pd ranges between 0 and 1. The grading level set of the SSS detection capability is {very strong, stronger, medium, weaker, very weak}, and the corresponding fuzzy scores set is ( 0 , 0.2 ) , ( 0.21 , 0.4 ) , ( 0.41 , 0.6 ) , ( 0.61 , 0.8 ) , ( 0.81 , 1 ) $\left\{(0,0.2),(0.21,0.4),(0.41,0.6),(0.61,0.8),(0.81,1)\right\}$ . FIGURE 5Open in figure viewerPowerPoint Factors affecting the side-scan sonar (SSS) detection capability 3.2.3 Probability of discovering the target The probability of discovering the target, denoted as pl, refers to the capability of successfully detecting and identifying the target. The pDT is calculated by the following formula p D T c i = p e c i × p d c i ${p}_{DT}\left({c}_{i}\right)={p}_{e}\left({c}_{i}\right)\times {p}_{d}\left({c}_{i}\right)$ (5)where pe(ci) and pd(ci) are target presence probability and SSS detection capability in cell ci. Optimal next-waypoint selection strategy Based on the discrete task area and prior information, the area coverage is achieved by visiting all waypoints. In this part, the optimal next-waypoint selection strategy is designed to determine appropriate next-waypoint. We establish three detailed criteria: (1) shorter distance from the current position and the cell with highest probability of discovering the target, (2) smaller turning angle to avoid distortion of sonar images, and (3) higher probability of discovering the target to improve efficiency. The corresponding mathematical formulae are presented as follows. Additionally, to reduce the computational burden, the pre-calculated available list Cava is the intersection of neighbours of the current cell and unvisited cells openList. The first objective function, denoted by f1, presents the shorter distance from both the current cell and the cell with the highest probability of discovering the target, which can avoid unnecessary distance and guide the AUV close to the possible target location min f 1 c i = w d 0 d c c c i d u + w d h d c h c i d u , c i ∈ C a v a $\mathrm{min}\ {f}_{1}\left({c}_{i}\right)={w}_{{d}_{0}}\frac{d\left({c}_{c}{c}_{i}\right)}{{d}_{u}}+{w}_{{d}_{h}}\frac{d\left({c}_{h}{c}_{i}\right)}{{d}_{u}},\quad {c}_{i}\in {C}_{ava}$ (6)where cc denotes the current position, ci indicates the candidate cells in Cava, and ch is the cell with the maximum pDT. Additionally, d(⋅) represents Euclidean distances between two cells, du is the unit distance between every two adjacent cells, w d 0 ${w}_{{d}_{0}}$ and w d h ${w}_{{d}_{h}}$ are weights, and w d 0 = w d h = 0.5 ${w}_{{d}_{0}}={w}_{{d}_{h}}=0.5$ . Take the case in Figure 6a as an example, the main direction is B, however, the AUV at point A should not consider point B as the next-waypoint if point C and D have not been visited. FIGURE 6Open in figure viewerPowerPoint Criteria for selecting the next-waypoint. (a) Shorter distance. (b) Fewer turns. (c) Higher target discovering probability The second objective function f2 is intended to avoid turns since turns lead to unstable navigation and distortion of sonar images [25]. Fewer turns are able to ensure sonar data quality at a path planning level. The f2 is described as min f 2 c i = θ a c c c i − θ a c p a r c c 180 , c i ∈ C a v a $\mathrm{min}\ {f}_{2}\left({c}_{i}\right)=\frac{{\theta }_{a}\left({c}_{c}{c}_{i}\right)‐{\theta }_{a}\left({c}_{par}{c}_{c}\right)}{180},\quad {c}_{i}\in {C}_{ava}$ (7)where θa(⋅) stands for the included angle between the candidate direction and last heading, cpar is the parent cell of cc. In Figure 6b, while the last move is from point E to F, the first choice of the next-waypoint is point G since there is no deviation between EF and FG; then, points H and I are suboptimal. The final objective function f3 addresses the higher probability of discovering the target, which guides the robot to cells close to the target. The f3 is depicted as max f 3 c i = p D T c i − p D T m i n p D T m a x − p D T m i n , c i ∈ C a v a $\mathrm{max}\ {f}_{3}\left({c}_{i}\right)=\frac{{p}_{DT}\left({c}_{i}\right)‐{p}_{DTmin}}{\ {p}_{DTmax}‐{p}_{DTmin}},\quad {c}_{i}\in {C}_{ava}$ (8)where pDTmin and pDTmax denote minimum and maximum pDT, respectively. In the case shown in Figure 6c, the next step from point J is preferably point K since point K has the maximum probability of discovering the target. Based on the above considered three criteria, the cell selection strategy is formulated as a multi-objective optimisation problem. In this research study, the typical weighted metric method is used for cell performance assessment. The ideal solution of single objective functions are f 1 ∗ = 1 , f 2 ∗ = 0 , f 3 ∗ = 1 ${f}_{1}^{\ast }=1,{f}_{2}^{\ast }=0,{f}_{3}^{\ast }=1$ . Then, the final multi-objective function is expressed as min h c i = ∑ m = 1 M w m f m c i − f m ∗ 2 , c i ∈ C a v a $\mathrm{min}\ h\left({c}_{i}\right)=\sum\limits _{m=1}^{M}{w}_{m}{\left\vert {f}_{m}\left({c}_{i}\right)‐{f}_{m}^{\ast }\right\vert }^{2},\quad {c}_{i}\in {C}_{ava}$ (9)where wm is corresponding weight, M = 3. As the heuristic of the proposed SAR-A* method, the h is a comprehensive assessment of unvisited waypoints based on the proposed three criteria f1, f2, f3. The SAR-A* method prioritises cells in Cava with minimum h value to balance the mentioned criteria. Framework of the SAR-A* method The SAR-A* coverage path planner adopts the A* algorithm as the framework. A* algorithm is a common graph traversal and path search algorithm [31]. The applied A* algorithm framework can guarantee that there are no repetition cells in the generated path. The A* framework is used to manage the process of iteratively determining the optimal next waypoint without repetition and missing. In the A* framework, two lists, that is, the openList and closeList are defined to record visited cells and unvisited cells. The main idea of the A* framework is to take appropriate cells from the openList and put visited cells into the closeList. Thus, with the A* framework, the task area is covered by visiting all cells. Based on the A* framework, the pseudo-code of the proposed method is illustrated in Algorithm 1. The inputs are the waypoint set Call and the probability of discovering the target set is Pl. Functions h, fpass and fdist are used to compute objective function values, waypoints passed and distance between two waypoints, respectively. For complete coverage, the main loop is the process of determining the optimal next-waypoint from availableList shown in lines 7–8. The criteria, also the heuristic, is the multi-objective function h defined in Equation (9). While the availableList is empty, the closest passing waypoint of the minimum objective function value is selected as shown in lines 10–11. Finally, the path Pa is generated until the openList is empty. Algorithm 1. The SAR-A* method Require: waypoints of map Call probability of discovering the target pDT Ensure: Path Pa 1: openList ← Call 2: Pa ← Ø 3: current ← c1 4: while Pa ≠ Call do 5: parent ← current 6: availableList ← current.neighbours ∩ openList 7: if availableList ≠ Ø then 8: current ← argmin h(availableList) 9: else if openList ≠ Ø then 10: currenth ← argmin f3(openList) 11: current ← argmin fdist(fpass(current, currenth)) 12: end if 13: openList ← openList\current 14: return Pa ← parent 15: end while The major steps of the SAR-A* method are depicted in the flow diagram shown in Figure 7. The proposed algorithm starts with decomposing the search area into a large number of hexagon cells, and complete coverage is attained by traversing all of the cells. Next, the target presence probability and detection capability are deduced, which are used as the basis for the search. Based on the multi-objective function, the main loop of the approach is the A* algorithm framework, which combines models and the objective function into the AUV path planning. With the A* framework, the availableList is introduced to extract neighbours of the current cell as candidate cells to be visited next. The criteria to determine the next cell is the optimal next-waypoint selection strategy. The loop is terminated while all cells are visited. FIGURE 7Open in figure viewerPowerPoint Flow diagram of the path planning process using the SAR-A* method Computational complexity The computational complexity of the proposed SAR-A* method is fairly low. The computational complexity of the first two steps, that is, task area decomposition and the target presence probability map construction, is O 2 × N c $\mathcal{O}\left(2\times {N}_{c}\right)$ . To reduce the candidate cells involved in calculating the objective function, the process of computing availableList requires O N c × 5 × ( α + 1 ) $\mathcal{O}\left({N}_{c}\times 5\times (\alpha +1)\right)$ computational cost, where α denotes the number of unvisited cells. While calculating and comparing objective function values of candidate cells, the computational complexity is O N c × ( 7 × β + 1 ) $\mathcal{O}\left({N}_{c}\times (7\times \beta +1)\right)$ , where β denotes the number of candidate cells in availableList. The price of ordering the availableList is O ( log β ) $\mathcal{O}(\mathrm{log}\,\beta )$ . To sum up, the SAR-A* method has computational complexity O N c 2 $\mathcal{O}\left({{N}_{c}}^{2}\right)$ . 4 SIMULATION AND EXPERIMENT RESULTS To gain further insight into the proposed method, comparison studies with both classical and recent methods are given in simulation. Additionally, the field experiment was conducted in a lake trial. The analysis and discussion of the results are given. Simulation study In this simulation, a search area of 5000 × 2500 m, is decomposed into hexagon grids as shown in Figure 8a. An AUV equipped with a SSS then performs the coverage task by iteratively determining the optimal next-waypoint. The range of the SSS is WS = 200 m. The radius of hexagon cells is r = 100 m. FIGURE 8Open in figure viewerPowerPoint The task area is decomposed into hexagon cells. The SSS detection capability in the area is non-uniform. (a) The task area in a plane view. The area in light blue and in wheat, respectively, have good and bad SSS detection capability. (b) The exact value of SSS detection capability in each cells In the comparison study, the performance of the SAR-A* method is compared with the LM algorithm [21], the random planner (Rnd) [32] and Zacchini's