09-Optimal search path planning of UUV in battlefeld ambush scene

<img src="/media/202408//1724838579.7562811.png" /><a href="https://doi.org/10.1016/j.dt.2023.03.018">Defence Technology xxx (xxxx) xxx</a><img src="/media/202408//1724838579.7588868.jpeg" />keA1C H I N E S E R O OTS G L O BA L I M PACTContents lists available at <a href="https://www.sciencedirect.com/science/journal/22149147">ScienceDirect</a>Defence Technologyjournal homepage: <a href="http://www.keaipublishing.com/en/journals/defence-technology">www.keaipublishing.com/en/journals/defence-technology</a><table><tr><td colspan="3">Optimal search path planning of UUV in battlefeld ambush sceneWei Feng, Yan Ma<a href="#bookmark1">**</a>, Heng Li<a href="#bookmark1">*</a>, Haixiao Liu, Xiangyao Meng, Mo Zhou Naval Research Institute, Beijing, 100161, China</td></tr><tr><td>a r t i c l e i n f o</td><td rowspan="3"></td><td>a b s t r a c t</td></tr><tr><td>Article history:Received 28 October 2022 Received in revised form 3 February 2023Accepted 22 March 2023 Available online xxx</td><td rowspan="2">Aiming at the practical application of Unmanned Underwater Vehicle (UUV) in underwater combat, this paper proposes a battleﬁeld ambush scene with UUV considering ocean current. Firstly, by establishing these mathematical models of ocean current environment, target movement, and sonar detection, the probability calculation methods of single UUV searching target and multiple UUV cooperatively searching target are given respectively. Then, based on the Hybrid Quantum-behaved Particle Swarm Optimization (HQPSO) algorithm, the path with the highest target search probability is found. Finally, through simulation calculations, the inﬂuence of different UUV parameters and target parameters on the target search probability is analyzed, and the minimum number of UUVs that need to be deployed to complete the ambush task is demonstrated, and the optimal search path scheme is obtained. The method proposed in this paper provides a theoretical basis for the practical application of UUV in the future combat.© 2023 China Ordnance Society. Publishing services by Elsevier B.V. on behalf of KeAi Communications Co. Ltd. This is an open access article under the CC BY-NC-ND license (<a href="http://creativecommons.org/licenses/by-nc-nd/4.0/">http://creativecommons.org/</a> <a href="http://creativecommons.org/licenses/by-nc-nd/4.0/">licenses/by-nc-nd/4.0/</a>).</td></tr><tr><td>Keywords:Battleﬁeld ambushOptimal search path planning UUV path PlanningProbability of cooperative search</td></tr></table>1. Introduction1.1. BackgroundBattleﬁeld ambush is one of the commonly used tactics in submarine combat [<a href="#bookmark1">1</a>]. It refers to the pre-deployment of subma- rine in the sea area (battleﬁeld) where the enemy target ship may appear. After target enters the sea area, the submarine will start to search for the target. Once the target is found, the torpedo is launched to destroy the target. In order to take advantage of weapons instead of forces, UUVs can be used instead of submarines to complete the above tasks. <a href="#bookmark1">Fig. 1</a> is a schematic diagram of the combat scene for UUV battleﬁeld ambush.1.2. Combat scene assumptionThe battleﬁeld ambush combat mode of UUV is mainly divided into two stages.<img src="/media/202408//1724838579.779117.png" />* Corresponding author. ** Corresponding author.E-mail addresses: <a href="mailto:AI_worshipper@163.com">AI_worshipper@163.com</a> (Y. Ma), <a href="mailto:henry11312@163.com">henry11312@163.com</a> (H. Li).Peer review under responsibility of China Ordnance Society1) The cruiseing stage. The UUV is assigned to sail to a certain sea area hundreds of kilometers away at a constant propulsion speed (7.2e28.8 km/h) for shutdown and standby. In this pro- cess, the UUV needs to consider the inﬂuence of ocean current, obstacle collision, and navigation error to safely navigate to the designated standby position in the shortest time.2) Approaching the enemy stage. Suppose that at a certain moment, the UUV captures the intelligence that the enemy ship target is about to cross the battleﬁeld through the relay communication buoy. At this time, the UUV plans a path with the highest search probability through the online path planning system. Then, the UUV starts from the standby position and searches for targets along this path. Once the detection sonar of UUV ﬁnds the target ship, the UUV immediately turns into an autonomous attack state and destroys the ship at the fastest speed. In this stage, the speed of UUV in still water remains unchanged except for the ﬁnal attack.This paper only focuses on the optimal search path planning of UUV in the stage of approaching the enemy.The following describes the UUV battleﬁeld ambush scene in detail with reference to <a href="#bookmark1">Fig. 2</a>.<a href="#bookmark1">Fig. 2</a> is the schematic diagram of UUV in the implementation of battleﬁeld ambush combat task. The rectangular area OABC repre- sents the battleﬁeld that the target ship will cross, which is actually a strait or an international channel. The point D represents the<a href="https://doi.org/10.1016/j.dt.2023.03.018">https://doi.org/10.1016/j.dt.2023.03.018</a>2214-9147/© 2023 China Ordnance Society. Publishing services by Elsevier B.V. on behalf of KeAi Communications Co. Ltd. This is an open access article under the CC BY-NC- ND license (<a href="http://creativecommons.org/licenses/by-nc-nd/4.0/">http://creativecommons.org/licenses/by-nc-nd/4.0/</a>).<table><tr><td>Please cite this article as: W. Feng, Y. Ma, H. Li etal., Optimal search path planning of UUV in battlefeld ambush scene, Defence Technology, <a href="https://doi.org/10.1016/j.dt.2023.03.018">https://doi.org/10.1016/j.dt.2023.03.018</a></td></tr></table><img src="/media/202408//1724838579.8047528.png" />W. Feng, Y. Ma, H. Li et al. Defence Technology xxx (xxxx) xxx<img src="/media/202408//1724838579.8311732.jpeg" />Fig. 1. Schematic diagram of the combat scene for UUV battleﬁeld ambush.<img src="/media/202408//1724838579.8357959.png" />Fig. 2. Schematic diagram of UUV in the implementation of battleﬁeld ambush combat task.standby position of UUV, that is, the initial search position of UUV. The point E is the initial position of the target ship entering the strait. rOAr and r OCr represent the length and width of the strait respectively. Now the following assumptions are made.Hypothesis 1. The threat of obstacles in the channel is not considered, and the movement of ocean current is assumed to be steady.Hypothesis 2. There are enough terrain matching areas distrib- uted within the channel. For this reason, the navigation error during UUV traveling can be corrected by terrain matching assisted navigation.Hypothesis 3. Intelligence information only provides the initial position and speed of the target ship, and the initial heading in- formation is unknown. In addition, the heading and speed of the target ship remain unchanged during the whole process of crossing the battleﬁeld.Hypothesis 4. UUV can successfully destroy the target immedi- ately after discovering the target. It is considered that the target is destroyed when it is found. Therefore, the problems need to be solved in this paper are as follows:(1) How to obtain a path with the highest probability of searching target for UUV when it is known that the target ship is about to enter the target sea area at a certain moment.(2) When the probability of searching target is required to reach a certain value Pf(such as 90%), how to get the optimal path search scheme, that is, how many UUVs need to be arranged in this sea area.1.3. Brief review on path planning of UUVThe problem of underwater path planning for UUV can be divided according to the perception of environmental information. Speciﬁcally, it can be divided into ofﬂine path planning (also known as static path planning) with completely known environmental information and online path planning (also known as dynamic path planning) with unknown environmental information.1.3.1. Research on ofﬁinepath planning methodThe current common ofﬂine path planning methods are mainly divided into the following four types: the ﬁrst one is based on graph search algorithm, and the more typical ones are Dijkstra Algorithm [<a href="#bookmark1">2</a>], A* Algorithm [<a href="#bookmark1">3</a>], Filed D* Algorithm [<a href="#bookmark1">4</a>,<a href="#bookmark1">5</a>],etc.; the second one is based on biological intelligence algorithm, and the more typical ones are Genetic Algorithm(GA) [<a href="#bookmark1">6</a>], Particle Swarm Optimization Algorithm(PSO) [<a href="#bookmark1">7</a>], Ant Colony Algorithm(ACO) [<a href="#bookmark1">8,9</a>], Flower Pollination Algorithm (FPA) [<a href="#bookmark1">10</a>], etc.; the third one is based on sampling-based search algorithm, and the more typical ones are Probabilistic Roadmap Method (PRM) [<a href="#bookmark1">11</a>], Rapidly-exploring Random Tree Method (RRT) [<a href="#bookmark1">12,13</a>], etc.; the fourth one is other algorithms, including Artiﬁcial Potential Field Method [<a href="#bookmark1">14</a>,<a href="#bookmark1">15</a>], Vis- ibility Graph Approach [<a href="#bookmark1">16</a>], Inner Normal Guided Segmentation Algorithm [<a href="#bookmark1">17</a>] and Homotopy Continuation Methods (HCM) [<a href="#bookmark1">18</a>], etc.1.3.2. Research on online path planning methodMany ofﬂine path planning algorithms are also suitable for on- line path planning, such as: A* algorithm, RRT, GA, PSO, and simulated annealing approach [<a href="#bookmark1">19</a>], etc. In addition, there is also a class of methods that are only suitable for online path planning, such as the sliding window algorithm [<a href="#bookmark1">20</a>] and so on.The current common online path re-planning methods can be divided into ﬁve types: the ﬁrst one is the partially reactive path planning method, such as that in Refs. [<a href="#bookmark1">21,22</a>] the second one is the path planning method based on unit decomposition, such as that in Refs. [<a href="#bookmark1">23,24</a>]; the third one is the dynamic fast search random tree method, such as that in Refs. <a href="#bookmark1">[13,25</a>]; the fourth one is the path planning method based on neural network learning, such as that in Refs. [<a href="#bookmark1">26,27</a>]; the ﬁfth one is the path planning method based on stochastic level-set partial differential equations, such as that in Ref. [<a href="#bookmark1">28</a>].In the real marine environment, there are not only static ob- stacles such as islands, but also many dynamic environmental factors, such as spatiotemporal current and moving obstacles [<a href="#bookmark1">29</a>]. This requires UUV to have the ability of dynamic path planning during navigation, so it is of great signiﬁcance to study the online path planning method of UUV.1.4. Motivation and contributionThe research motivation of this paper is to efﬁciently obtain the optimal cooperative search path of UUV in the assumption scene of battleﬁeld ambush combat task, and to provide a theoretical basis for the practical application of UUV in the future. Therefore, this paper establishes the probability calculation mathematical models of single UUV searching target and multiple UUV cooperatively searching target respectively, and obtains the path with the highest target search probability based on the HQPSO algorithm. Aiming at the premature convergence problem of QPSO algorithm, the HQPSO algorithm with stronger global search ability is proposed by adopting the strategies of particle average optimal position selec- tion mutation and particle swarm individual selection mutation. The detailed algorithm design process of HQPSO can refer to our previous research results in Ref. [<a href="#bookmark1">30</a>].<img src="/media/202408//1724838579.856417.png" />W. Feng, Y. Ma, H. Li et al. Defence Technology xxx (xxxx) xxxThe contribution and innovation of the method proposed in this paper are as follows.● Aiming at the battleﬁeld ambush combat scene of UUV, the corresponding simpliﬁed mathematical model is established, and the probability calculation methods of single UUV searching target and multiple UUV cooperatively searching target are given respectively.● This paper studies the cooperative search path planning prob- lem of UUV battleﬁeld ambush combat based on the HQPSO algorithm, discusses the inﬂuence of different UUV parameters and target parameters on the probability of searching target, and obtains the optimal search path scheme under different search probability requirements, which provides a theoretical basis for the reasonable deployment of UUV in the actual combat.1.5. OrganizationThe rest of the paper is organized as follows. Section <a href="#bookmark1">2</a> prepares several mathematical models before planning the optimal search path; Section <a href="#bookmark1">3</a> introduces the UUV optimal search path planning method in detail; Section <a href="#bookmark1">4</a> designs several group of simulation experiments. Conclusions are made in Section <a href="#bookmark1">5</a>.2. Mathematical model preparationTo solve the problem of optimal search path planning for UUV in the battleﬁeld ambush combat scene, the key is to design a cost function with the largest search probability as the optimization objective. Before that, several mathematical models need to be established, including the battleﬁeld ocean current model, the target movement model and the sonar detection model of UUV.2.1. Battleﬁeld ocean current modelIn this paper, it is assumed that the ocean current changes very slowly, that is, the speed and direction of ocean current do not change with time during the whole path search task of UUV.In Ref. <a href="#bookmark1">[31</a>], a numerical equation simulation method of ocean current is proposed, which can not only simulate the ocean current movement more realistically, but also provide an ocean current map with the desired resolution according to the needs of users. Therefore, in the process of ocean current modeling, this method is used to simulate the ocean current movement. The ocean current map obtained by this method is essentially realized by super- imposing multiple viscous Lamb vortices. The movement equation of a single viscous Lamb vortex can be expressed as follows [<a href="#bookmark1">32,33</a>]:<img src="/media/202408//1724838579.87843.png" />where u (r), v(r) and w (r) are the vortex velocity components in horizontal, longitudinal and vertical directions respectively, and k, ξ and r0 are used to describe the vortex intensity, vortex radius, and vortex center position coordinates, respectively. <a href="#bookmark1">Fig. 3</a> is a sche- matic diagram of the ocean current obtained by simulation. The<img src="/media/202408//1724838579.890645.png" />Fig. 3. Schematic diagram of the ocean current obtained by simulation.grid size is 25 × 25 and the resolution is 1 km. It is composed of 50 Lambs superimposed, in which the coordinates of vortex center are randomly generated. k = 54 km/h, ξ = 2 m.2.2. Target movement modelAccording to the Hypothesis 3 in the combat scene, since the speed and position are known, and the course remains unchanged, it can be inferred that the area covered by ΔOEC is the area that the target ship may actually pass through when UUV knows that the target ship will start from point E in <a href="#bookmark1">Fig. 2</a> at a certain moment. Assuming that the course θm of the target ship obeys a uniform distribution, then the distribution range of θm is θm ∈ [θmin, θmax ]. Among them, θmin and θmax respectively represent the minimum and maximum course angle of the target ship in the OXY Cartesian coordinate system, and take the counterclockwise as the positive.Assuming that the speed of the target ship is Vm , and its initial position of entering the channel is (x0, y0), then the position equation of the target at any time is as follows:{ <img src="/media/202408//1724838579.897797.png" /> (2)According to the length IOAI and width IOCI of the strait and the longitudinal displacement of the target initial position IAEI , θmin and θmax can be calculated. The speciﬁc equation is as follows:<img src="/media/202408//1724838579.9084458.png" />2.3. Sonar detection model of UUVSonar is one of the commonly used equipment for searching underwater target. In the process of UUV searching for target ships, detection sonar is used to collect target information. The detection probability of sonar is a complex function affected by many factors, such as physical environment, signal power, signal-to-noise ratio, etc. The detection probability function Ps of sonar is a function of detection distance rm.There are two common detection probability models [<a href="#bookmark1">34</a>], one is the ideal detection model and the other is the attenuation model. Their expressions can be expressed as follows:<img src="/media/202408//1724838579.919809.png" />W. Feng, Y. Ma, H. Li et al. Defence Technology xxx (xxxx) xxx1, 0 ≤ rm ≤ Ls 0, rm&gt;LsPs (rm) = { Ps (rm) = {(4)—lg(9rm 十 Ls )十 lg Ls 十 1, 0 ≤ rm ≤ Ls 0, rm&gt;Ls(5)where, rm represents the distance between the UUV and the target ship, and Ls is the detection distance of sonar.In order to simplify the problem, this paper chooses the ideal detection model to search for the target ship.3. Optimal search path planning of UUVIn order to judge the advantages and disadvantages of different path search schemes, two methods can be used for evaluation. Method 1: Under the same constraints, compare the maximum probability of searching target; Method 2: Under the premise of successfully searching the target, compare the minimum cost of searching target (such as time expectation). The purpose of this paper is to maximize the probability of UUV searching for the target ship. Therefore, Method 1 is selected as the criterion for path evaluation.Deﬁne the search probability Pss as the cost function for searching path. Pss is deﬁned as follows: When 0 ≤ t ≤ , for any feasible search path scheme δ of UUV, the cumulative search probability at time t is Pss(δ , t), then the probability of UUV detecting the target at least once in the time period [0 , t] isPss (δ, t) = P { The target is detected at least once before time t} = The time to first detect the target ≤ t}(6)It is very difﬁcult to accurately obtain the analytical solution of Pss. Monte Carlo method can be used for approximate calculation to obtain the results.3.1. Optimal search path planning for single UUV3.1.1. The search probability calculation method for single UUVFor the case of single UUV searching target, the approximate calculation process of Monte Carlo method is as follows: Firstly, a large number of movement trajectories are generated randomly according to the movement model of target ship. Secondly, the behavior of UUV searching target is simulated. Finally, the search probability Pss is obtained by means of probability statistics.Assume that the UUV starts searching targets within the time period of t ∈ [0,Tmax ]. Since it is a continuous behavior for the UUV searching targets, this behavior is divided into Ns discrete events with a search interval of Δts . Then at time t, for any search path, the calculation formula of the search probability Pss is as follows:t(i) ≤ t ≤ t(i 十 1), 1 ≤ i ≤ Ns t(Ns ) &lt; t ≤ Tmax0 ≤ t &lt; t(1)0,Pss (t(i)), Pss (t(Ns )),( Pss (t)= { ((7)NumTPss (t(i)) = j<img src="/media/202408//1724838579.934794.png" />T(i,j) × 100%(1≤ i≤ Ns, 0≤ t(i)≤ Tmax ) (8)Ps (rm (1,j)),i = 1i &lt;1 ≤ NsPcs (i,j)= {Ps (rm(i,j)).(1—Pcs (i—1,j))十Pcs (i—1,j),(9)where, t(i) represents the moment of the ith search. Ns represents the total number of searches. Pcs (i,j) is the search probability of the jth simulated target after the ith search. Num T represents the total number of target trajectories obtained by simulation. Ps (rm) is the detection probability function of sonar. rm(i,j) represents the dis- tance between the UUV and the jth target in the ith search.According to the ideal detection probability function of sonar, Eq. <a href="#bookmark1">(7)</a> can be simpliﬁed asPss (t)=<img src="/media/202408//1724838579.950647.png" /> × 100%(0≤ t ≤ Tmax ) (10)where, Nf(t) represents the number of targets searched at time t.3.1.2. Steps of planning search path for single UUVIn order to solve the premature problem of QPSO algorithm in the process of convergence, mutation operation can be performed on the basis of QPSO algorithm. After the mutation operation of average optimal position and the selection mutation operation of particle swarm, Hybrid Quantum-behaved Particle Swarm Optim- zation(HQPSO) is obtained. The detailed algorithm design process of HQPSO can refer to Ref. [<a href="#bookmark1">30</a>].After clarifying the cost function of the search path, the HQPSO algorithm can be used to solve the optimal search path. The speciﬁc steps of the path planning method are as follows.Step 1. Preprocess the path planning environment. According to the initial position of the target ship, NumT target trajectories are randomly generated.Step 2. Read the battleﬁeld environment information. It includes the initial position of UUV for battleﬁeld ambushing, the sailing speed in still water and the sonar detection radius of UUV, the target ship speed, the data of ocean current ﬁeld, the maximum search time, and the search time interval, etc. Among them, the maximum search time Tmax can be calculated according to the following equation:Tmax = <s> </s> = <s> </s> (11)|CE| |OA|Vm | cos θmin|Step 3. Set the algorithm parameters. It includes the number of population, the dimension of optimization variables, the maximum number of iterations and so on.Step 4. Particle encoding and population initialization are used to obtain a series of initial search paths. In this process, as the desti- nation of the search path planning becomes a moving target ship, the setting of control points is no longer equidistantly distributed according to the UUV starting point. Therefore, the process of particle encoding and population initialization is changed, and the new particle encoding and population initialization process is implemented according to the following method.It can be seen from <a href="#bookmark1">Fig. 2</a> that assuming the initial speed of the target ship is Vm, and the maximum combined speed of the UUV is Vrmax. In the most ideal case, when the UUV and the target ship are facing each other, the shortest time Tmin for the UUV to meet the target ship is as follows:<img src="/media/202408//1724838579.960737.png" />W. Feng, Y. Ma, H. Li et al. Defence Technology xxx (xxxx) xxx<img src="/media/202408//1724838579.96641.png" />where, Vrmax is the maximum combined speed, Vcmax is the maximum ocean current speed, and Vsw is the UUV sailing speed in still water.Therefore, the longest distance Lmax of UUV sailing along the X- axis isLmax = Tmin . |Vrmax | = Tmin.|Vcmax+ Vsw | (13)Due to the backtracking situation of UUV in the process of searching for the target, ﬁve control points are set in all the search paths planned in this article. Among them, the ﬁrst four control points are equidistantly distributed, and the abscissa of the last control point is not ﬁxed, but is regarded as a variable to be opti- mized. According to the principle of the HQPSO algorithm, the dimension of the particle is Dd = 6 at this time. Assuming that there are N particles, the coding of the ith particle xi is as follows:<img src="/media/202408//1724838579.9733791.png" />where, xi,5 is the abscissa of the ﬁfth control point andyi,1-yi,5 is the ordinate of the ﬁrst to ﬁfth control points. Correspondingly, the variable range of the ith particle should satisfy the following requirements:{ <img src="/media/202408//1724838579.977024.png" /> ;; <img src="/media/202408//1724838579.9823492.png" /> x ; (i = 1; 2; … ; N;j = 1; 2; … 5) (15)Therefore, the initialization of the individual particle swarm can be realized by a random number generator according to Eq. <a href="#bookmark1">(15)</a>.Step 5. The HQPSO algorithm is used to update the population.Step 6. Calculate the cost of each search path according to the cost function deﬁned by Eq. <a href="#bookmark1">(10)</a>, and calculate the individual optimal extremum and the global optimal extremum of each particle.Step 7. Recalculate the individual optimal extremum and global optimal extremum of each particle for the updated population.Step 8. Repeat Step 5 to Step 7 until the maximum number of iterations is reached.Step 9. End the iteration. Output the coordinates of optimal control point and generate the optimal search path. The algorithm ﬂow chart of planning search path for single UUV is shown in <a href="#bookmark1">Fig. 4</a>.3.2. Optimal search path planning for multiple UUVBecause a single UUV often cannot ensure that the target is completely searched, it is necessary to study the path planning problem of multiple UUVs for collaborative search.3.2.1. The search probability calculation method for multiple UUVFor the situation where multiple UUVs are searching for targets at the same time, due to the problem of repeated searching for targets in the collaborative search of multiple UUVs, the collabo- rative search probability Pcom cannot be directly obtained by simply accumulating the search probability of each single UUV.Assuming that there are n UUVs setting out to search for the target ship at the same time. NumT target trajectories are randomly generated. At the moment of t, if the ith UUV searches for the jth target, then record Ct(i, j) = 1, otherwise record Ct(i, j) = 0. The deﬁnition of the marking function H(t, j) is as follows:<img src="/media/202408//1724838579.994901.jpeg" />Fig. 4. The algorithm ﬂow chart of planning search path for single UUV.<img src="/media/202408//1724838579.99958.png" />(16)The above equation indicates that when all UUVs do not search for the jth target, the value ofH(t, j) is 1; and when at least one UUV searchs for the jth target, the value of H(t, j) is 0. If Nr(t) is the total number of unsearched targets at the moment of t, then Nr(t) can be calculated by the following equation:<img src="/media/202408//1724838580.0143979.png" /> (17)Therefore, the cooperative search probability in the case of multiple UUVs is as follows:<img src="/media/202408//1724838580.03986.png" />3.2.2. Steps of planning search path for multiple UUVThe steps of planning search path for multiple UUV are similar to that of single UUV, but the difference lies in the process of the coding and initialization of the particle swarm. In the search path planning of multiple UUVs, the variable range of the control points should be determined according to the number of UUVs. The gen- eral idea is as follows: Firstly, the search area is divided into equal areas according to the number of UUVs, then the change range of the path control point position is determined according to the divided area corresponding to each UUV, and ﬁnally the search range of each dimension variable of the particle in the HQPSO al- gorithm is obtained. The particle initialization process in the case of multiple UUVs will be described in detail with reference to <a href="#bookmark1">Fig. 5</a>. <a href="#bookmark1">Fig. 5</a> is the area division diagram when three UUVs search for<img src="/media/202408//1724838580.0553222.png" />W. Feng, Y. Ma, H. Li et al. Defence Technology xxx (xxxx) xxx<img src="/media/202408//1724838580.070593.jpeg" />Fig. 5. Area division diagram when three UUVs search for targets.targets. It can be seen that in the case of searching target by three UUVs, the battleﬁeld area is divided into three triangles of equal area, where D and F are the three division-points on the line segment OC respectively. Connect CE, DE, FE, and OE respectively to get the straight lines l 1el4. The equation of the ith line li is as follows:li : y = k ix þ bi (19)According to the setting rules of the control points in a single UUV, the abscissa of the ﬁrst four control points are equidistantly distributed. Then their corresponding linear equations is: x = xi, i = 1, 2, 3, 4. Therefore, the coordinates of any intersection point can be obtained according to the two linear equations. In <a href="#bookmark1">Fig. 5</a>, y1,min(1) and y1,max(1) are respectively the intersection coordinates of line x = x1 and lines l1 : y = k1x þ b1, l2 : y = k2x þ b2. They respectively correspond to the upper and lower limits of the ordinate for the ﬁrst control point that determines the searched path of UUV1. Ac- cording to the same method, the variation range of the ordinate for each control point can be solved in turn. In this way, the initiali- zation of the ﬁrst four dimensional variables in the particle can be realized. According to the speciﬁc combat scene in this paper, the value range of the ﬁfth-dimensional variable of all particles is: 0≤xi,5≤35.6. As for the ordinate variable of the ﬁfth control point corresponding to the ith UUV search path (the sixth-dimensional variable in the particle), the variation range can be obtained ac- cording to the following equation:{ ;;<img src="/media/202408//1724838580.160774.png" /> i{<img src="/media/202408//1724838580.193841.png" />i<img src="/media/202408//1724838580.249357.png" />m;<img src="/media/202408//1724838580.2603562.png" />x(1(1);)y;;;<img src="/media/202408//1724838580.266148.png" />x(2(2);)y;;;<img src="/media/202408//1724838580.292035.png" />a<img src="/media/202408//1724838580.309005.png" />(<img src="/media/202408//1724838580.350015.png" />;<img src="/media/202408//1724838580.382263.png" />iy;;<img src="/media/202408//1724838580.4124959.png" />a(()4}) } (20)Through the above processing, the initialization process of the HQPSO algorithm in the case of multiple UUVs can be realized. Compared with the single UUV, the following algorithm steps are the same except that the calculation formula of cost function needs to be changed to Eq. <a href="#bookmark1">(18)</a>. For this reason, no further explanation is given here.4. Simulation and result analysis4.1. Results and analysis of search path planning for single UUVInitial simulation conditions: The initial position of the UUV battleﬁeld ambush is (0, 50). The initial position of the target ship is (100, 50). The sailing speed of UUV in still water Vsw = 10.8 km/h. The ship speed Vm = 33.33 km/h. The number of target movement trajectories NumT = 1000. The time interval of sonar detection Δts = 50 s. The maximum search time Tmax = 12,100 s. The detection radius of sonar Ls = 5 km. The number of particle populationN = 100. The particle dimension Dd = 6. The maximum number of iterations MaxDT = 100, Pa = Pb = 0.5. The simulation calculation is run independently for 50 times,and the average search probability, the maximum search probability and the standard deviation of the search probability are recorded respectively.The path planning results are shown in <a href="#bookmark1">Figs. 6 and 7.Fig. 6</a> shows the optimal search path of UUV. Comparing to <a href="#bookmark1">Fig. 3</a>, it can be found that when sailing along this path, the UUV happens to be advancing in the direction of the maximum ocean current speed. This shows that UUV chooses to approach the target as fast as possible when searching for the target. The purpose of this navigation is to reduce the spread range of the target and increase the density of the target in per unit area, so that the UUV has a greater probability of searching the target. In addition, the “ *” in the ﬁgure indicates the location of the target when it was discovered, and it can be seen that the area where the target was searched is very concentrated. This shows that due to the low navigation speed of the UUV, the time for the target to cross the detection range is very short, which makes it difﬁcult for the UUV to ﬁnd the target again in the remaining search time range. <a href="#bookmark1">Fig. 7</a> is the convergence graph of the average search probability with the number of iterations obtained after 50 independent runs. It can be seen that the search probability increases with the number of iterations and ﬁnally converges to a stable solution. The above results show that the HQPSO algorithm always takes the maximum target search probability as the opti- mization objective in the optimization process, which can ﬁnd the path with the largest search probability for UUV. This also veriﬁes the effectiveness of the path planning method based on the HQPSO algorithm.In order to further clarify the inﬂuence of different parameter conditions on the search probability, the following simulations are carried out on conditions of different navigation speeds for UUV in still water, different detection radius of UUV sonar, different navi- gation speeds of target ship, and different initial position of target ship.4.1.1. Inﬁuence of different UUV navigation speed on searchprobabilitySuppose that the navigation speeds Vm for UUV in still water are respectively 7.2 km/h, 10.8 km/h, 14.4 km/h, 18.0 km/h, 21.6 km/h, 25.2 km/h, 28.8 km/h, and other simulation conditions are the same as those in <a href="#bookmark1">subsection 4.1.</a> The simulation calculation under each condition is run independently for 50 times, and the average search probability, the maximum search probability and the standard de- viation of the search probability under each simulation condition are recorded respectively. The statistical results are shown in <a href="#bookmark1">Table 1.</a><img src="/media/202408//1724838580.433419.png" />Fig. 6. Optimal search path of UUV when Vsw = 10.8 km/h, Vm = 33.33 km/h, Ls = 5 km.<img src="/media/202408//1724838580.44298.png" />W. Feng, Y. Ma, H. Li et al. Defence Technology xxx (xxxx) xxx<img src="/media/202408//1724838580.445549.png" /><img src="/media/202408//1724838580.449387.jpeg" />Fig. 7. Convergence diagram of average search probability when Vsw = 10.8 km/h, Fig. 9. Convergence diagram of average search probability when Vm = 28.8 km/h.Vm = 33.33 km/h, Ls = 5 km.<a href="#bookmark1">Table 1</a> shows the corresponding search probabilities underdifferent navigation speeds for UUV in still water. It can be seenfrom the table that as the navigation speed for UUV in still water increases successively, the search probability increases signiﬁ- cantly. When the navigation speed in still water changes from 7.2 to 28.8 km/h, the average search probability increases by 18.5%. <a href="#bookmark1">Fig. 8</a> is the diagram of optimal search path when Vm = 28.8 km/h, and <a href="#bookmark1">Fig. 9</a> is the convergence diagram of average search probability when Vm = 28.8 km/h. By comparing <a href="#bookmark1">Fig. 6</a> with <a href="#bookmark1">Fig. 8</a>, it can be seen that when the UUV sailing speed in still water is Vm = 28.8 km/ h, the UUV also approaches the target at the fastest speed ﬁrst.However, in <a href="#bookmark1">Fig. 8</a>, a roundabout search section appears in the path of UUV, and the area where the target is searched is also concentrated and scattered on this section in path. This shows that when the speed of UUV increases, the time range for the UUV to detect the target becomes longer, which improves the probability of the target being searched. <a href="#bookmark1">Fig. 9</a> corresponds to the convergence of the average search probability at this time. It shows that when Vm = 28.8 km/h the average search probability of the target ﬁnally converges to 33.4%.<img src="/media/202408//1724838580.465493.png" />Fig. 8. Optimal search path of UUV when Vm = 28.8 km/h.4.1.2. Inﬁuence of different UUV sonar detection radius on searchprobabilitySuppose that the detection radius Ls of UUV are respectively 2 km, 3 km, 4 km, 5 km, 6 km, 7 km, 8 km, and other simulation conditions are the same as those in <a href="#bookmark1">subsection 4.1.</a> The simulation calculation under each condition is run independently for 50 times, and the average search probability, the maximum search proba- bility and the standard deviation of the search probability under each simulation condition are recorded respectively. The statistical results are shown in <a href="#bookmark1">Table 2</a>.<a href="#bookmark1">Table 2</a> shows the corresponding search probability under different detection radius of UUV sonar. It can be seen from the table that as the detection radius of UUV sonar increases succes- sively, the search probability increases. <a href="#bookmark1">Fig. 10</a> shows the optimal search path of UUV when the sonar detection distance is 8 km. Comparing with <a href="#bookmark1">Fig. 6</a>, it can be seen that the two search paths are basically the same, which illustrates the stability of the HQPSO al- gorithm for searching the optimal solution. However, in <a href="#bookmark1">Fig.10</a>, due to the increase of sonar detection distance, the coverage area of each detection becomes larger, so the probability of the target being searched is also greater. <a href="#bookmark1">Fig. 11</a> corresponds to the convergence of the average search probability at this time. When Ls = 8 km, the average search probability of the target ﬁnally converges to 25.9% stably.4.1.3. Inﬁuence of different target navigation speed on search probabilitySuppose that the navigation speeds Vm of target are respectively 33.33 km/h, 37.04 km/h, 40.74 km/h, 44.44 km/h, 48.15 km/h, 51.85 km/h, 55.56 km/h, and other simulation calculation under each condition is run independently for 50 times, and the average search probability, the maximum search probability and the stan- dard deviation of the search probability under each simulation condition are recorded respectively. The statistical results are shown in <a href="#bookmark1">Table 3</a>.<a href="#bookmark1">Table 3</a> shows the corresponding search probability under different sailing speeds of target. It can be seen from the table that as the navigation speed of target increases, the target search proba- bilityof UUV decreases. <a href="#bookmark1">Fig.12</a> shows the optimal search path whenTable 1The relationship between navigation speed and search probability of UUV in still water.<table><tr><td>Vsm /(km.h-1)</td><td>7.2</td><td>10.8</td><td>14.4</td><td>18.0</td><td>21.6</td><td>25.2</td><td>28.8</td></tr><tr><td>Average search probability/%</td><td>14.9</td><td>16.2</td><td>18.7</td><td>21.4</td><td>23.5</td><td>27.1</td><td>33.4</td></tr><tr><td>Standard deviation of probability</td><td>0.003</td><td>0.002</td><td>0.003</td><td>0.003</td><td>0.005</td><td>0.004</td><td>0.002</td></tr><tr><td>Maximum search probability/%</td><td>15.4</td><td>16.6</td><td>19.3</td><td>21.8</td><td>24.4</td><td>27.8</td><td>33.8</td></tr></table><img src="/media/202408//1724838580.482223.png" />W. Feng, Y. Ma, H. Li et al. Defence Technology xxx (xxxx) xxxTable 2The relationship between search radius and search probability of UUV.<table><tr><td>Ls/km</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td></tr><tr><td>Average search probability/%</td><td>7.7</td><td>10.6</td><td>13.1</td><td>16.2</td><td>19.7</td><td>22.0</td><td>25.9</td></tr><tr><td>Standard deviation of probability</td><td>0.005</td><td>0.004</td><td>0.002</td><td>0.003</td><td>0.007</td><td>0.004</td><td>0.006</td></tr><tr><td>Maximum search probability/%</td><td>8.5</td><td>11.4</td><td>13.5</td><td>16.9</td><td>21.0</td><td>22.7</td><td>26.6</td></tr></table><img src="/media/202408//1724838580.4893398.jpeg" />Fig. 10. Optimal search path of UUV when Ls = 8 km.<img src="/media/202408//1724838580.492921.jpeg" />Fig. 11. Convergence diagram of average search probability when Ls = 8 km.<img src="/media/202408//1724838580.499778.png" />Fig. 12. Optimal search path of UUV when Vm = 55.56 km/h.<img src="/media/202408//1724838580.5041149.jpeg" />Fig. 13. Convergence diagram of average search probability when Vm = 55.56 km/h.the navigation speed of target is 55.56 km/h. Comparing with <a href="#bookmark1">Fig. 6</a>, it can be seen that the search path of the UUV becomes shorter at this time, and the area where target is searched appears near the starting point of the UUV. The main reason for this is that when the navigation speed of the target increases, on the one hand, it will make the navigation distance longer at the same time and thedispersion range of the target larger, which will lead to the decrease of the target density in per unit area; on the other hand, it will make the time of the target crossing the sonar detection range shorter, so the probability of the target being searched will be lower. <a href="#bookmark1">Fig. 13</a> corresponds to the convergence of the average search probabilityTable 3The relationship between navigation speed of target and search probability.<table><tr><td>Vm /(km.h-1)</td><td>33.33</td><td>37.04</td><td>40.74</td><td>44.44</td><td>48.15</td><td>51.85</td><td>55.56</td></tr><tr><td>Average search probability/%</td><td>16.2</td><td>15.5</td><td>15.0</td><td>14.6</td><td>14.1</td><td>13.8</td><td>13.5</td></tr><tr><td>Standard deviation of probability</td><td>0.003</td><td>0.004</td><td>0.002</td><td>0.003</td><td>0.004</td><td>0.003</td><td>0.004</td></tr><tr><td>Maximum search probability/%</td><td>16.9</td><td>16.1</td><td>15.5</td><td>15.2</td><td>14.8</td><td>14.4</td><td>14.1</td></tr></table>Table 4The relationship between initial positions of target and search probability.<table><tr><td>Initial positions of target</td><td>(100,0)</td><td>(100,25)</td><td>(100,50)</td><td>(100,75)</td><td>(100,100)</td></tr><tr><td>Average search probability/%</td><td>17.1</td><td>16.4</td><td>16.2</td><td>16.6</td><td>17.3</td></tr><tr><td>Standard deviation of probability</td><td>0.005</td><td>0.007</td><td>0.003</td><td>0.002</td><td>0.004</td></tr><tr><td>Maximum search probability/%</td><td>17.7</td><td>17.1</td><td>16.9</td><td>17.0</td><td>17.8</td></tr></table><img src="/media/202408//1724838580.516414.png" />W. Feng, Y. Ma, H. Li et al. Defence Technology xxx (xxxx) xxx<img src="/media/202408//1724838580.5206022.png" />Fig. 14. Optimal search path of UUV when the initial position of the target is (100, 75).<img src="/media/202408//1724838580.52475.png" />Fig. 15. Optimal search path of UUV when the initial position of the target is (100,100).<img src="/media/202408//1724838580.5283048.png" />Fig. 16. Optimal search paths in the case of two UUVs.<a id="bookmark1"></a>at this time. When Vm = 55.56 km/h, the average search probability of the target is 13.5%.4.1.4. Inﬁuence of different target initial positions on searchprobabilitySuppose that the initial positions of target are respectively (100,0), (100,25), (100,50), (100,75), (100,100), and other simula- tion conditions are the same as those in <a href="#bookmark1">subsection 4.1.</a> The simu- lation calculation under each condition is run independently for 50 times, and the average search probability, the maximum search probability and the standard deviation of the search probability<img src="/media/202408//1724838580.53348.jpeg" />Fig. 17. Convergence diagram of search probabilities in the case of two UUVs.<img src="/media/202408//1724838580.536875.png" />Fig. 18. Optimal search paths in the case of three UUVs.<img src="/media/202408//1724838580.543515.png" />Fig. 19. Convergence diagram of search probabilities in the case of three UUVs.under each simulation condition are recorded respectively. The statistical results are shown in <a href="#bookmark1">Table 4</a>.<a href="#bookmark1">Table 4</a> shows the corresponding search probability under different initial positions of target. It can be seen from the table that the different initial positions of target have little inﬂuence on the search probability, and the maximum probability difference be- tween them is only 1.1%. <a href="#bookmark1">Figs. 14 and 15</a> are the optimal search paths of UUV when the target initial position coordinates are (100,75) and (100,100) respectively. It can be found that the two<img src="/media/202408//1724838580.549982.png" />W. Feng, Y. Ma, H. Li et al. Defence Technology xxx (xxxx) xxx<img src="/media/202408//1724838580.575406.jpeg" /><img src="/media/202408//1724838580.608522.jpeg" />Fig. 20. Optimal search paths in the case of six UUVs.<img src="/media/202408//1724838580.712883.jpeg" />Fig. 21. Convergence diagram of search probabilities in the case of six UUVs.paths are basically similar, and the average search probability dif- ference is only 0.7%.Through the analysis of the inﬂuence of the above different conditions on the UUV target search probability, the following conclusions can be obtained.(1) The higher the navigation speed of UUV in still water, the higher the probability of target being searched.(2) The larger the detection radius of UUV sonar, the higher the probability of target being searched.(3) The higher the navigation speed of target, the lower the probability of target being searched by UUV.(4) When the initial position of the target is different, the probability of the target being searched remains basically unchanged.4.2. Results and analysis of search path planning for multiple UUVIn order to solve the second problem raised in the combat scene assumption: When the probability of the target being searched reaches a certain value (for example, 90%), how many UUVs need to be deployed in the sea area? In this paper, the search path planning problem in the case of multiple UUVs is simulated and analyzed.Initial simulation conditions: The initial position of target ship is (100, 50). The sailing speed of UUV in still water Vsw = 10.8 km/h. The ship speed Vm = 33.33 km/h. The number of target movement trajectories NumT = 1000. The time interval of sonar detection ΔTs = 50 s. The maximum search time Tmax = 12,100 s. The detection radius of sonar Ls = 5 km. The number of particleFig. 22. Optimal search paths in the case of seven UUVs.<img src="/media/202408//1724838580.7473862.png" />Fig. 23. Convergence diagram of search probabilities in the case of seven UUVs.population N = 100. The particle dimension Dd = 6. The maximum number of iterations MaxDT = 100, Pa = Pb = 0.5. The simulation calculation is run independently for 50 times. The initial position of UUV is set according to the number of UUVs, and the search area is divided according to the principle of equal area. For example, the initial positions when two UUVs are arranged are (0,75), (0,25), and the initial positions when three UUVs are arranged are (0,16.6), (0,50), (0,83.3). Other situations can be deduced by analogy.<a href="#bookmark1">Figs. 16</a>e<a href="#bookmark1">23</a> respectively correspond to the simulation results when 2 UUVs, 3 UUVs, 6 UUVs and 7 UUVs are arranged. <a href="#bookmark1">Table 5</a> records the average collaborative search probability, the maximum collaborative search probability, and the standard de- viation of collaborative search probability that calculated under different UUV numbers.It can be seen from <a href="#bookmark1">Table 5</a> that with the number of UUVs increasing, the probability of collaborative search increases by more than 10%. When six UUVs are arranged, the UUVs can search for the target ship with a probability close to 90%. When seven UUVs are arranged, it is basically ensured that the UUVs can fully search for the target ship. As can be seen from <a href="#bookmark1">Figs.16,18, 20 and 22</a>, the total range covered by the search path becomes larger with the number of UUVs increasing. Each search path does not interfere with each other. This maximizes the coverage of the target distri- bution space and ensures that the probability of the target being detected increases with the number of UUVs. From <a href="#bookmark1">Figs. 17, 19, 21</a> <a href="#bookmark1">and 23</a>, it can be seen that the average cooperative search proba- bility increases with the number of iterations, and it can eventually<img src="/media/202408//1724838580.78431.png" />W. Feng, Y. Ma, H. Li et al. Defence Technology xxx (xxxx) xxxTable 5The relationship between UUV quantities and search probability.<table><tr><td>UUV quantities</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td></tr><tr><td>Average collaborative search probability/%</td><td>33.8</td><td>48.4</td><td>62.7</td><td>75.1</td><td>87.9</td><td>99.4</td></tr><tr><td>Standard deviation of collaborative search probability</td><td>0.005</td><td>0.013</td><td>0.007</td><td>0.006</td><td>0.004</td><td>0.002</td></tr><tr><td>Maximum collaborative search probability/%</td><td>34.2</td><td>49.6</td><td>63.6</td><td>76.</td><td>88.6</td><td>100.0</td></tr></table>converges to a stable solution. This shows that the HQPSO algo-rithm is also effective and feasible in the case of multiple UUVs in the optimal path planning problem with the largest collaborative search probability as the goal. According to the simulation results, if the probability of the target being searched reaches Pf = 90%, the optimal search scheme is to arrange seven UUVs. In this case, as long as the target ship enters the battleﬁeld, it must be searched by the UUV.5. ConclusionsBased on the combat scene assumption of battleﬁeld ambush for UUV, this paper establishes the probability calculation mathemat- ical models of single UUV searching target and multiple UUV cooperatively searching target respectively, and obtains the path with the highest target search probability based on the HQPSO algorithm. The method in this paper can provide an optimal search path scheme that meets the probability requirements of the target being searched, which will provide a theoretical basis for the practical application of UUV in the future warfare.The conclusions obtained through the simulation and quanti- tative analysis in this paper are highly consistent with the practical application experience, which indirectly proves the effectiveness and applicability of the path planning method proposed in this paper from the perspective of application.Although the method proposed in this article has a certain guiding effect on the actual warfare, there are still some researches that need to be improved and perfected in the future.(1) Path planning in three-dimensional space UUV should also have vertical maneuverability and detection capabilities during navigation. It can realize the battleﬁeld surveys in three-dimensional space through vertical movement. For this reason, the path planning problem in three-dimensional space should be further considered in the follow-up research.(2) Collaborative path planning for multiple UUV under the condition of space-time synchronization. The collaborative path planning models of single UUV and multiple UUVs established in this paper are limited to the optimal search path problem in a given combat scene.In the future warfare of Underwater Unmanned system, more complex optimization allocation of time and space resources will be involved, which requires multiple UUVs to be able to perform path planning in complex combat scene. Therefore, in the follow-up research, we should further study the collaborative path planning for multiple UUV under the condition of time and space synchro- nization in complex scene.Declaration of competing interestThe authors declare that they have no known competing ﬁnancial interests or personal relationships that could have appeared to inﬂuence the work reported in this paper.References[1] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref1">Blouin Stphane. Adaptive multi-sensor biomimetics for unsupervised sub-</a><a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref1">marine hunt (AMBUSH): early results. Proc SPIE-Int Soc Opt Eng 2014:9248</a>.[2] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref2">Abdulkadir S IFadzli, AjamalAA S, et al. Indoor global path planning based on</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref2">critical cells using Dijkstra algorithm. J Theor Appl Inf Technol 2015;79(3):</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref2">358</a>.[3] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref3">Sun Wei, Lv Yunfen, Tang Hongwei, et al. Mobile robot path planning based on</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref3">an improved A</a>*<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref3">algorithm. J Hunan Univ (Soc Sci) 2017;44(4):94</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref3">101</a>.[4] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref4">Paden Bap MYong SZ, et al. A survey of motion planning and control tech-</a><a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref4">niques for self-driving urban vehicles. IEEE Transactions on Intelligent Vehi-</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref4">cles 2016;1(1):33</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref4">55</a>.[5] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref5">Ferguson Dstentz A. Using interpolation to improve path planning: the Field</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref5">D</a>*<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref5">algorithm. J Field Robot 2006;23(2):79</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref5">101</a>.[6] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref6">Mohanta J, Cparhi D, Rpatel SK. Path planning strategy for autonomous mobile</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref6">robot navigation using Petri-GA ptimisation. Comput Electr Eng 2011;37(6):</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref6">1058</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref6">70</a>.[7] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref7">Zeng NZhang HChen Yet al. Path planning for intelligent robot based on</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref7">switching local evolutionary PSO algorithm. Assemb Autom 2016;36(2):</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref7">120</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref7">6</a>.[8] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref8">Feng Wrao Zwang Z. Research on the application of Ant Colony algorithm in</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref8">underwater path planning. International Symposium on Advances in Electrical</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref8">Electronics and Computer Engineering Guangzhou; 2016. p. 46</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref8">50. 2016</a>.[9] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref9">Xiao-ming YOU, Liu Sheng, Jin-qiu LV. Ant colony algorithm based on dynamic</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref9">search strategy and its application on path planning of robot. Control Decis</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref9">2017;32(3):552</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref9">6</a>.[10] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref10">Zhou Yongquan. An improved ﬂower pollination algorithm for optimal un-</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref10">manned undersea vehicle path planning problem. Int J Pattern Recogn Artif</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref10">Intell 2016;30(4):1659010</a>.[11] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref11">Yang LIU, Zhangwei-guo, Guang-wen LI. Study on path planning based on</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref11">improved PRM method. Appl Res Comput 2012;1:104</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref11">6</a>.[12] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref12">Wang Quan. Global path planning method based on RRT and its application.</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref12">Chang Sha: National University of Defense Science and technology; 2014</a>.[13] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref13">Jin-ze SONG, Dai Bin, Shan En-zhong, et al. An improved RRT path planning</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref13">algorithm. Acta Electron Sin 2010;38(b02):225</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref13">8</a>.[14] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref14">Montiel Oorozco-Rosas Usepúlveda R. Path planning for mobile robots using</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref14">Bacterial Potential Field for avoiding static and dynamic obstacles. Expert Syst</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref14">Appl 2015;42(12):5177</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref14">91</a>.[15] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref15">Chen YLuo GMei Yet al. UAV path planning using artiﬁcial potential ﬁeld</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref15">method updated by optimal control theory. Int J Syst Sci 2016;47(6):1407</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref15">20</a>.[16] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref16">Xu Sijun, Cao Qiying. A visibility graph based path planning algorithm for</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref16">modile robot. Computer Applications and Software 2011;28(3):220</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref16">2</a>.[17] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref17">Sun Zheng, Shao Zhu-Feng, Li Hui. An eikonal equation based path planning</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref17">method using polygon decomposition and curve evolution. Defence Tech-</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref17">nology 2020;16:1001</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref17">18</a>.[18] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref18">Vazquez-Leal H, Marin-Hernandez A, Khan Y, et al. Exploring collision-free</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref18">path planning by using homotopy continuation methods. Appl Math Com-</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref18">put 2013;219:7514</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref18">32</a>.[19] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref19">Miao HHui, Tian Yu-Chu. Dynamic robot path planning using an enhanced</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref19">simulated annealing approach. Appl Math Comput 2013;222:420</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref19">37</a>.[20] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref20">Zhang Chungang, Xi Yugeng. Robot path planning based on rolling window</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref20">when global environment is unknown. Science in China(Series E). 2001;31(1):</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref20">51</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref20">91</a>.[21] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref21">Berger Cyrille, Wzorek Mariusz, et al. Area coverage with heterogeneous UAVs</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref21">using scan patterns. IEEE International Symposium on Safety, Security, and</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref21">Rescue Robotics (SSRR) 2016:342</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref21">9</a>.[22] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref22">Rudol Pwzorek Mdoherty P. Vision-based pose estimation for autonomous</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref22">indoor navigation of micro-scale unmanned aircraft systems. IEEE Interna-</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref22">tional Conference on Robotics and Automation (ICRA); 2010. p. 1913</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref22">20.</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref22">2010</a>.[23] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref23">Tsiotras Pjung Dbakolas E. Multi-resolution Hierarchical pathplanning for</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref23">small UAVs using wavelet decompositions. J Intell Rob Syst 2012;66(4):</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref23">505</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref23">22</a>.[24] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref24">Jung Dratti Jtsiotras P. Real-time implementation and validation of a new</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref24">hierarchical path planning scheme of UAVs via hardware-in-theloop simula-</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref24">tion. J Intell Rob Syst 2009;54(1</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref24">3):163</a>.[25] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref25">Wang Jiankun, Max Q.-H. Meng. Optimal path planning using generalized</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref25">voronoi graph and multiple potential functions. IEEE Trans Ind Electron</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref25">2020;67(12):10621</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref25">30</a>.[26] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref26">Yu Jinglun, Su Yuancheng, Liao Yifan. The path planning of mobile robot by</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref26">neural networks and hierarchical reinforcement learnings. Front Neurorob</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref26">2020;14:63</a>.[27] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref27">Lin Changjian, HongjianWang, Yuan Jianya, et al. Research on UUV obstacle</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref27">avoiding method based on recurrent neural networks. Complexity 2019;2019:</a><img src="/media/202408//1724838580.851109.png" />W. Feng, Y. Ma, H. Li et al.<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref27">1</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref27">16</a>.[28] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref28">Deepak N, Subramani Quantum J, Wei Pierre, Lermusiaux FJ. Stochastic time-</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref28">optimal path-planning in uncertain, strong, and dynamic ﬂows. Comput</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref28">Methods Appl Mech Eng 2018;333:218</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref28">37</a>.[29] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref29">Subramani Deepak N, Lermusiaux Pierre FJ. Risk-optimal path planning in</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref29">stochastic dynamic environments. Comput Methods Appl Mech Eng</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref29">2019;353:391</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref29">415</a>.[30] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref30">Ma Y, Feng W, Mao Z, et al. Path planning of UUV based on HQPSO algorithm</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref30">with considering the navigation error. Ocean Eng 2022;244:110048</a>.[31] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref31">Garau Balvarez Aoliver G. Path planning of autonomous underwater vehicles</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref31">in current ﬁelds with complex spatial variability: an A</a>*<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref31">approach. IEEE</a>Defence Technology xxx (xxxx) xxx<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref31">International Conference on Robotics and Automation 2005:194</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref31">8</a>.[32] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref32">Feng W, Zhang JY, Wang Z. A time-optimal path planning method based on</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref32">quantum-behaved particle swarm optimization in ocean environment. J Nav</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref32">Univ Eng 2017;29:72</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref32">7</a>.[33] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref33">Ma Y, Mao Z, Wang T, et al. Obstacle avoidance path planning of unmanned</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref33">submarine vehicle in ocean current environment based on improved</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref33">ﬁrework-ant colony algorithm. Comput Electr Eng 2020;87:106773</a>.[34] <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref34">Cho JH, Kim J, Slim JS, et al. Optimal acoustic search path planning based on</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref34">genetic algorithm in continuous path system. Int J Offshore Polar Eng</a> <a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref34">2007;17(3):1</a>e<a href="http://refhub.elsevier.com/S2214-9147(23)00081-8/sref34">5</a>.