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Applied Sciences
, Volume 9 (6) – Mar 13, 2019

/lp/multidisciplinary-digital-publishing-institute/an-improved-a-star-algorithm-considering-water-current-traffic-RNRcAzk1qS

- Publisher
- Multidisciplinary Digital Publishing Institute
- Copyright
- © 1996-2019 MDPI (Basel, Switzerland) unless otherwise stated
- ISSN
- 2076-3417
- DOI
- 10.3390/app9061057
- Publisher site
- See Article on Publisher Site

applied sciences Article An Improved A-Star Algorithm Considering Water Current, Trafﬁc Separation and Berthing for Vessel Path Planning 1 2, 3 3 Chenguang Liu , Qingzhou Mao * , Xiumin Chu and Shuo Xie State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, Wuhan 430079, China; cgliu@whu.edu.cn School of Remote Sensing and Information Engineering, Wuhan University, Wuhan 430079, China National Engineering Research Center for Water Transport Safety, Wuhan University of Technology, Wuhan 430063, China; chuxm@whut.edu.cn (X.C.); xieshuo@whut.edu.cn (S.X.) * Correspondence: qzhmao@whu.edu.cn Received: 14 February 2019; Accepted: 10 March 2019; Published: 13 March 2019 Abstract: A traditional A-Star (A*) algorithm generates an optimal path by minimizing the path cost. For a vessel, factors of path length, obstacle collision risk, trafﬁc separation rule and manoeuvrability restriction should be all taken into account for path planning. Meanwhile, the water current also plays an important role in voyaging and berthing for vessels. In consideration of these defects of the traditional A-Star algorithm when it is used for vessel path planning, an improved A-Star algorithm has been proposed. To be speciﬁc, the risk models of obstacles (bridge pier, moored or anchored ship, port, shore, etc.) considering currents, trafﬁc separation, berthing, manoeuvrability restriction have been built ﬁrstly. Then, the normal path generation and the berthing path generation with the proposed improved A-Star algorithm have been represented, respectively. Moreover, the problem of combining the normal path and the berthing path has been also solved. To verify the effectiveness of the proposed A-Star path planning methods, four cases have been studied in simulation and real scenarios. The results of experiments show that the proposed A-Star path planning methods can deal with the problems denoted in this article well, and realize the trade-off between the path length and the navigation safety. Keywords: ship path planning; A-Star algorithm; berthing; current effect 1. Introduction The autonomous surface vessel (ASV) technologies have grown rapidly and obtained more attention than ever both in military and commercial applications [1]. Among them, path planning in complex navigation area is one of most difﬁcult and challenging topics that need to be developed. The aim of path planning is to ﬁnd the optimal navigation path from the start location to the destination location [2]. Generally, path planning can be divided to two classes, i.e., global path planning and local path planning (real-time) [3]. A global path planner usually calculates paths off-line in static environments with stationary obstacles. In contrast, a local path planner usually calculates paths on-line in dynamic environments with moving obstacles. For the global planning, path length, risk factor, dynamic constraint, etc. should be considered at the same time. Speciﬁcally, current effect and navigation rules should also be included during the path planning for a vessel. In this article, we try to propose a global path planning method to satisfy the requirements of safety and efﬁciency for autonomous surface vessel. For a vessel, especially an autonomous surface vessel, factors of the path length, the obstacle collision risk, the trafﬁc separation rule and the manoeuvrability restriction should be all taken into account when a path is planned for navigation. To solve these problems, a lot Appl. Sci. 2019, 9, 1057; doi:10.3390/app9061057 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 1057 2 of 17 of studies for vessel path planning have been conducted. For instance, Szlapczynsk reported a ship route ﬁnding method with turn penalties and collision avoidance [4]; Kim et al. put forward a grid map-based path planning method with the kinematic constraint of the vessels [5]; Lyu et al. proposed a path-guided hybrid artiﬁcial potential ﬁeld (PGHAPF) method to generate adaptive, collision-free and International Regulations for Preventing Collisions at Sea (COLREGS)-constrained paths in restricted waters [6,7]. However, the majority of path planning methods did not consider the current effect, berthing requirements and trafﬁc separation rule together much. In recent years, Ma et al. proposed a dynamic augmented multi-objective particle swarm optimization (PSO) algorithm to obtain a path considering the path length, smoothness, economical efﬁciency and safety for unmanned surface vehicles [8]. Though this algorithm can get good performance of path planning, the PSO based method usually has the possibility of trapping in a local optima and the berthing requirements were not taken into account. For the berthing problem during path planning, Lee et al. proposed an EEA* algorithm considering hydrodynamic effects, angular constraint and head angle conﬁnement at the goal point [9]. The head angle conﬁnement could be a good attempt to solve the berthing problem, which, however, cannot guarantee the availability of the berthing path accord to the instructions of berthing in [10]. To obey the trafﬁc separation rule for path planning, on the basis of the research of [4], Szlapczynski proposed an improved Evolutionary Algorithms (EA) method considering the Trafﬁc Separation Schemes (TSS) [11]. We think that these attempts for solving the vessel path planning problems are beneﬁcial. However, the demands of the vessel path planning proposed in this article have not been all satisﬁed, e.g., the risk of an obstacle in the upstream of current should be higher than that in the downstream of current, a berthing path should be generated considering the current effects, and the trafﬁc separation rule should be modelled as a risk of path planning. Grid map-based global path planning algorithms have been applied frequently and widely because their advantages in computation time and avoidance of local optimum [12]. A Dijkstra path planning algorithm based on grid map was ﬁrst proposed in [13], which is to ﬁnd the shortest paths between nodes in a graph. Dijkstra algorithm creates an open list and a closed list to denote the next alternative nodes and the selected nodes, respectively. A node in the open list will be transferred to the closed list when the distance from the start node is less than the best path so far [14]. In consideration of low search efﬁciency of Dijkstra path planning algorithm, an improved algorithm extended from Dijkstra algorithm, i.e., A-Star (A*) algorithm, was introduced by [15]. In contrast to the Dijkstra algorithm, A-Star algorithm transfers a node in the open list to the closed list by introducing the node potential that is the underestimate of the remaining distance to the destination node. A-Star algorithm has higher efﬁciency mainly because the node potential gives the direction to the destination node. A-Star algorithm forms the backbone of vehicle navigation systems, which has been used for robots, road vehicles, unmanned aerial vehicles, humans, etc. [2,14,16–18]. A traditional A-Star path planning algorithm generates an optimal path only by minimizing the path length in a grid map. Improvements for the A-Star algorithm have been done all the time. Bell proposed a Hyperstar algorithm based on A-Star approach for risk averse vehicle navigation [14]. Pochmara et al. improved the A-Star to decrease the computer calculation cost [19]. Fernandes et al. presented an enhanced A-Star algorithm capable of generating smooth and feasible paths considering orientation restrictions [2]. These improvements for the A-Star algorithm have solved several speciﬁc problems and widen the application areas. However, for the path planning of vessels, the A-Star and the A-Star based methods still need to be improved to generate safer and more efﬁcient paths. This study presents an improved A-Star based global path planning method to satisfy the requirements of the economy (path length), the obstacle avoidance, the trafﬁc separation rule, the berthing and the manoeuvrability restriction considering the water current effects at the same time. The remainder of this article is organized as follows. Risks of the obstacles, the trafﬁc separation, the berthing, manoeuvrability restriction are modelled in Section 2. In Section 3, an improved A-Star path planning algorithm considering risk models is proposed. In Section 4, four simulation case studies are carried out to verify the performance of the proposed methods for the normal path planning, Appl. Sci. 2019, 9, 1057 3 of 17 berthing path planning and combined path planning, respectively. In the Section 5, the conclusions and future directions are presented. 2. Risk Modelling During path planning, to ﬁnd a shortest waterway is not the ultimate goal for a vessel because the safety of the vessel usually obtains more attention than the operation efﬁciency. However, existing path planning algorithms usually ignored the impact on the vessel safety when a vessel voyages nearby obstacles, such as bridge pier, shore, other vessels, port, etc., when the water current speed is large enough to be considered, when the trafﬁc separation has been carried out, or when a vessel is for berthing. Hence, it is necessary to build relevant risk models to denote the impacts of these factors. In this section, we propose the four risk models designed for the obstacles, trafﬁc separation, berthing and manoeuvrability, respectively. 2.1. Obstacle Modelling With Current The traditional A-Star algorithm can generate an optimal path with keeping off the obstacle nodes, which shown in Figure 1. From Figure 1, it can be seen that the generated path could be very close to obstacles to guarantee the path shortest. However, if this path is adopted by a sailing vessel as the planning route, the vessel will have a high risk to collide the obstacles. Meanwhile, the current will also increase the collision risk if the current pushes the vessel to approach one of obstacles. Therefore, it is indispensable to consider both keeping an appropriate distance from obstacles and current inﬂuence during path planning. obstacle 11 start node destination node 9 navigable node path generated by traditional A-Star 1 2 3 4 5 6 7 8 9 10 11 12 Figure 1. The grid map and path generated by the traditional A-Star. The obstacles for vessel can be divided into 5 types, namely, bridge pier, shore, vessel, port, and others. Different types of obstacles have different risks for vessel navigation. The risk degree r [m, n] of navigable node N [m] nearby obstacle node O [n] is deﬁned as follows: e c v m m r [m, n] = + , (1) a d where d stands for the Euclidean distance from an navigable node N [m] to the obstacle node O [n]. a stands for the risk coefﬁcient for different types of obstacles. In this article, a is assigned as 5, 5, 3, 4, 3 for bridge pier, shore, vessel, port, and others. v stands for the current velocity. c stands for the m m direction coefﬁcient, which is as follows: cos f f , cos f f < 0 c g c g c = , (2) 0, cos f f 0 c g where, f 2 [0, 2p) is the current direction toward the true North; f 2 [0, 2p) is the node direction c g relative to obstacle node O [n] toward the true North, i.e., the direction of vector N [m] O [n] toward the true North. Appl. Sci. 2019, 9, 1057 4 of 17 A grid map considering the obstacle with current is represented in Figure 2. In Figure 2, the number in a navigable node denotes the risk degree, and the direction and length of an arrow denote the current direction and velocity, respectively. A navigable node N m could have k risk degree values [ ] if there exist k obstacle nodes nearby N [m] according to Equation (1). In this case, the maximum one will be selected as the ﬁnal risk degree. Therefore, the ﬁnal risk degree of node N m , r m , is deﬁned [ ] [ ] as follows: r [m] = maxfr [m, 1] , r [m, 2] , ..., r [m, k]g , (3) o o o o In Figure 2, the path is generated by the traditional A-Star algorithm, which chooses an optimal path node without considering the risk degree of the path node. 11 12 12 8 7 8 12 8 9 moored or anchored ship 10 12 4 13 12 9 shore 9 12 5 13 12 6 9 9 9 9 bridge pier 8 12 12 7 13 12 4 3 3 3 3 port 7 4 9 7 14 12 4 1 1 1 1 other obstacles 6 2 5 7 4 9 12 8 3 1 1 1 0 start node 5 1 2 2 2 3 4 4 4 4 4 3 1 destination node 4 2 5 7 4 4 8 12 12 12 12 8 3 navigable node 3 3 8 7 4 13 12 4 path generated by 2 7 7 7 7 8 12 12 12 12 8 7 traditional A-Star current 1 2 3 4 5 6 7 8 9 10 11 12 Figure 2. The grid map considering obstacles with current. 2.2. Trafﬁc Separation Modelling Similar to the lane separation for vehicle driving on the road, a vessel also needs to obey the trafﬁc separation rule in the waterway to decrease the risk of collision and increase the operation efﬁciency. Based on the trafﬁc separation rule, a risk model of trafﬁc separation for a navigable node N [m] is built as: 1 cos (f f ) , 0 < cos (f f ) 1 n r n r r [m] = , (4) 1 a cos f f ,1 cos f f 0 ( ) ( ) n r n r where f 2 [0, 2p) is the navigation direction by rule toward the true North. Assume that the path is deﬁned as fN [1] , ..., N [m] , ...,g, f 2 [0, 2p) at node N [m] is the direction of the path vector N [m 1] N [m] toward the true North. When the navigation direction is consistent with the path vector direction, i.e., 0 r [m] < 1; otherwise, , 1 r [m] 1 + a. The grid map considering trafﬁc s s separation is shown in Figure 3. From Figure 3, the path generated by the traditional A-Star algorithm cannot guarantee the planning path to be consistent with the navigation direction by rule. 11 12 12 8 7 8 12 8 9 10 12 4 13 12 9 moored or anchored ship shore 9 12 5 13 12 6 9 9 9 9 bridge pier 8 12 12 7 13 12 4 3 3 3 3 port 7 4 9 7 14 12 4 1 1 1 1 other obstacles start node 6 2 5 7 4 9 12 8 3 1 1 1 0 destination node 5 1 2 2 2 3 4 4 4 4 4 3 1 navigable node 4 2 5 7 4 4 8 12 12 12 12 8 3 path generated by traditional A-Star 3 3 8 7 4 13 12 4 current 2 7 7 7 7 8 12 12 12 12 8 7 navigation direction boundary line 1 2 3 4 5 6 7 8 9 10 11 12 Figure 3. The grid map considering trafﬁc separation. Appl. Sci. 2019, 9, 1057 5 of 17 2.3. Berthing Modelling Considering Current When a vessel needs to berth at a port or an anchorage in a waterway with a non-ignorable current, the vessel usually berths into the current, which means the vessel motion direction is opposite to the current direction. The reasons for it are summarized as: (1) it will give the vessel the advantage of relatively high speed through the water with a reduced speed over the ground; (2) it can be used to push the vessel alongside [10]. Figure 4 shows the berthing path is generated by the traditional A-Star algorithm. From Figure 4, the vessel motion direction planed by the traditional A-Star algorithm is the same as the current direction, which is not beneﬁcial for the berthing. Hence, if the destination node is a berthing node, the berthing path should be considered. moored or anchored ship 11 12 12 8 7 8 12 8 9 shore 10 12 4 13 12 9 bridge pier 9 12 5 13 12 6 9 9 9 9 port 8 12 12 7 13 12 4 3 3 3 3 other obstacles 7 4 9 7 14 12 4 1 1 1 1 start node 6 2 5 7 4 9 12 8 3 1 1 1 0 destination node 5 1 2 2 2 3 4 4 4 4 4 3 1 navigable node 4 2 5 7 4 4 8 12 12 12 12 8 3 normal path 3 3 8 7 4 13 12 4 berthing path 2 7 7 7 7 8 12 12 12 12 8 7 current 1 2 3 4 5 6 7 8 9 10 11 12 Figure 4. The berthing path generated by traditional A-Star algorithm. The generated path with a berthing destination node consists of the normal path P and the berthing path P . The berth path P should consider the current effect. In contrast to the normal A-Star b b algorithm, the berthing path has no speciﬁc destination node with the berth node as the start node. Therefore, how to generate a berth path without a destination node is a key point to deal with. The berthing risk of a navigable node N [m], r [m], is modelled as follows: be r [m] = [1 + cos (f f )] , (5) n r be where r [m] 0, and c > 0 is a constant coefﬁcient. s = 0, 1, ..., N is the sequence number of node be be N m from the berth node, in which N is the node number of the berthing path. f and f are deﬁned [ ] n r be in Equation (4). The speciﬁc procedure to generate the berthing path will be described in Section 3.2.2. 2.4. Manoeuvrability Restriction Modelling The minimum turning radius of a vessel represents the turning ability of the vessel when following a predeﬁned path. Actually, this ability is restricted to the manoeuvrability of the vessel. If the restriction of turning radius is not set, the generated path could not be followed by the vessel and the risk of collision and stranding could even happen. Assume that P , P , P are the continuous nodes of a path, which is shown in Figure 5. i1 i i+1 i-1 i+1 Figure 5. The restriction of turning radius. Appl. Sci. 2019, 9, 1057 6 of 17 q 2 [0, p) is the included angle between line P P and line P P . l is the shorter length between i1 i i i+1 the length of P P and the length of P P . A circle is built through node P , which is tangent to i1 i i i+1 i1 line P P and line P P . Then the radius of the circle, R , is obtained: i1 i i i+1 i R = l tan . (6) If the minimum turning radius of the vessel R is greater than R , it means that the continuous nodes P , P , P is not available for this vessel, and vice versa. Hence, a risk model for manoeuvrability i1 i i+1 restriction is deﬁned as follows: b 1 + , R > R 0 i r [m] = , (7) 0, R R 0 i where b > 0 is a constant coefﬁcient. 3. Improved A-Star Algorithm In consideration of the limitations of the traditional A-Star algorithm applying in the path planning of vessels, an improved A-Star algorithm is studied in this section based on the risk models built in Section 2. The traditional algorithm is introduced ﬁrst, and then an A-Star algorithm considering risk modelling is presented. 3.1. Traditional A-Star Algorithm A-Star algorithm was ﬁrst proposed and described in detail in [15]. The A-Star algorithm aims to ﬁnd a path from the start node (N [start]) to the destination node (N [dest]) with the smallest cost, e.g., least distance. The ﬂowchart of the traditional A-Star algorithm is denoted in Algorithm 1. Algorithm 1 Traditional A-Star algorithm (N [start] ! N [dest]) 1: Mark N [start] as o penlist. 2: while o penlist 6= ? do 3: Select the open node N i from the o penlist whose value of evaluation function f N i is [ ] ( [ ]) smallest. 4: if N [i] = N [dest] then 5: return “path is found”. 6: else 7: Mark N [i] as closelist. 8: if Successor N [j] of node N [i] does not belong to closelist or obstacle node then 9: Mark N [j] as o penlist and calculate f (N [j]). i i 10: if N [j] belongs to o penlist and f (N [j]) is smaller than f (N [j]) when N [m] was marked i i m as closelist then 11: f (N [j]) = f (N [j]) and set parent node of N [j] as N [i]. 12: end if 13: end if 14: end if 15: end while 16: return “path is not found”. The evaluation function f (N [i]) of N [i] is deﬁned as: Appl. Sci. 2019, 9, 1057 7 of 17 f N i = g N i + h N i , (8) ( [ ]) ( [ ]) ( [ ]) where g (N [i]) is the cost of the path from N [start] to N [i], and h (N [i]) is a heuristic function that estimates the cost of the cheapest path from N [i] to N [dest]. h (N [i]) can be Euclidean distance (Manhattan distance or Chebyshev distance) from N [i] to N [dest], or other functions denoting the cost from N [i] to N [dest] as in [9]. Please note that in contrast to the A-Star algorithm, the Dijkstra algorithm has no heuristic function that can accelerate the rate of ﬁnding an optimal path . The Euclidean distance is selected as h (N [i]) in this article. 3.2. A-Star Algorithm Considering Risk Models If an A-Star algorithm is used to the path planning for vessels. Theoretically, the traditional A-Star algorithm can ﬁnd the shortest path, which is not nevertheless the optimal path for vessels in some scenarios. In this section, we try to consider the path length, the path risk and navigation rules simultaneously during the path planning. According to the destination, a normal path generation and berthing path generation are introduced with a non-berthing destination and berthing destination. 3.2.1. Normal Path Generation of Vessels To satisfy the requirements of considering risks for vessel path planning, model (8) needs to be modiﬁed. Here, g (N [i]) is redesigned as: g (N [i]) = g (N [i]) + t r (N [i]) , (9) n n n where g (N [i]) is the newly deﬁned cost of the path from N [start] to N [i]. t > 0 is a constant n n coefﬁcient. g (N [i]) is deﬁned as: p1 g (N [i]) = d (k) , (10) k=1 where p 1 is the node amount from N [i] to N [start], i.e., N [i], N [i] . parent, N [i] . parent. parent, ..., N [start], in which N. parent represents the parent node of N. d (k) is the Euclidean distance between kth node to (k + 1)th node, where k stands for the node sequence from the start node. To be speciﬁc, assuming that the coordinates of kth node is (x , y ), then k k d (k) is as follows: 2 2 d (k) = (x x ) + (y y ) . (11) k k+1 k k+1 r (N [i]) is the risk function, which is as follows: r N i = r i + r i + r i , (12) ( [ ]) [ ] [ ] [ ] n o s m where r [i], r [i] and r [i] are deﬁned in Equations (3), (4) and (7), respectively. Please note that r = 0 o s m s if trafﬁc separation rule is not carried out in certain waterway. A new evaluation function f (N [i]) is deﬁned as: f (N [i]) = g (N [i]) + b h (N [i]) , (13) n n n n where b > 0 is a constant coefﬁcient. In contrast with (8), b is added to balance the weight between n n g (N [i]) and h (N [i]) . Assuming that the coordinates of N [i] are x , y and the coordinates n n N i N i [ ] [ ] of N [dest] are x , y , then h (N [i]) is as follows: N[dest] N[dest] 2 2 h (N [i]) = x x + y y . (14) N[i] N[dest] N[i] N[dest] Appl. Sci. 2019, 9, 1057 8 of 17 The proposed A-Star algorithm of the normal path generation is presented as Algorithm 2. Algorithm 2 Normal path generation with the improved A-Star algorithm (N [start] ! N [dest]) 1: Mark N [start] as o penlist. 2: if the destination node is not the berthing one then 3: if trafﬁc separation rule is not carried out then 4: r = 0 at any time. 5: end if 6: while o penlist 6= ? do 7: Select the open node N [i] whose value of evaluation function f (N [i]) is smallest. 8: if N [i] = N [dest] then 9: return “path P is found”. 10: else 11: Mark N [i] as closelist. 12: if Successor N [j] of node N [i] does not belong to closelist or obstacle node then 13: Mark N [j] as o penlist. 14: Calculate r [j], r [j] and r [j] by (3), (4) and (7), respectively. Then, r (N [j]) = r [j] + o s m n o r [j] + r [j]. s m 15: if N [j] belongs to o penlist and f (N [j]) is smaller than f (N [j]) when N [m] was n n m i i marked as closelist then 16: f (N [j]) = f (N [j]) and N [j] . parent = N [i]. n n 17: end if 18: end if 19: end if 20: end while 21: return “path P is not found”. 22: end if 3.2.2. Berthing Path Generation of Vessels As previously mentioned, when the destination node is the berthing one, a specific path planning rule should be obeyed in order to guarantee the safety of vessels. The algorithm of generating berthing path P described in Section 2.3 is presented in Algorithm 3, in which N berth is the starting node of P . [ ] b b The evaluation function for berthing path generation, f (N [i]), is deﬁned as: f (N [i]) = g (N [i]) + b h (N [i]) , (15) b b b b where b > 0 is a constant coefﬁcient. h (N [i]) = 0 at any time since the destination node is not b b known before the berthing path generation ﬁnishes. g (N [i]) is as follows: g (N [i]) = g (N [i]) + t r (N [i]) , (16) b b b where g (N [i]) = 0 at any time, t > 0 is a constant coefﬁcient. r (N [i]) is as follows: b b r (N [i]) = r [i] + r [i] + r [i] . (17) b o be m Please note that the trafﬁc separation rule is not considered for the berthing path generation, i.e., r = 0 at any time, because a vessel in the berthing process always ﬁnds the path that is against the current ignoring the trafﬁc separation rule temporarily. Appl. Sci. 2019, 9, 1057 9 of 17 Combining (15), (16) and (17), f N i can be denoted as: ( [ ]) f (N [i]) = r [i] + r [i] + r [i] . (18) b o be m Please note that t = 1 and b = 1 are both eliminated in (18) since g (N [i]) and h (N [i]) are b b b constant to 0. Algorithm 3 Berthing path generation algorithm (N [berth] ! N [dest]) 1: Mark N [dest] as o penlist. 2: if the destination node is the berthing one then 3: Set the predeﬁned number of berthing path nodes as q . 4: while o penlist 6= ? do 5: Select the open node N [i] whose value of evaluation function f (N [i]) is smallest. 6: Set N [i] as the berth node and N [i] ! N [dest] as the berth path P . Count the number of the path P nodes, which is denoted as q . b b 7: if q >= q then 8: “path P is found”. 9: else 10: Mark N i as closelist. [ ] 11: if Successor N [j] of node N [i] does not belong to closelist or obstacle node then 12: Mark N [j] as o penlist. 13: Calculate r [j], r [j] and r [j] by (3), (4) and (7), respectively. Then, r (N [j]) = o be m b i r [j] + r [j] + r [j]. o be m 14: if N [j] belongs to o penlist and f (N [j]) is smaller than f (N [j]) when N [m] was i b i b m marked as closelist then 15: f (N [j]) = f (N [j]) and N [j] . parent = N [i]. b b i 16: end if 17: end if 18: end if 19: end while 20: return “path P is not found”. 21: end if 3.2.3. Combined Path Generation of Vessels In most of path planning scenarios, the normal path and the berthing path both need to be generated. Therefore, it should be considered that how to combine the normal path and the berthing path. The main problem is that the manoeuvrability risk r [m] deﬁned in (7) could be high, which will increase the risk of collision or stranding when the connection node of the normal path and the berthing path is inappropriate. For instance, in Figure 6, the sharp turn could not be tracked well by the vessel. The distance between two adjacent nodes, l , is usually relevant to the minimum radius of the vessel, R . In this article, we set l = R . 0 0 The combined path P consists of the normal path P and the berthing path P , i.e., P = P + P . c n c n b b To avoid a sharp turn at node N [berth], i.e., R > R , where N [berth] is the ﬁrst node of P , a new 0 berth b node can be added between N [berth] and the second node of P . As shown in Figure 7, a node is added to eliminate the sharp turn and make the generated path smooth enough be followed. The algorithm of adding node or smoothing the combined path is denoted in Algorithm 4. Appl. Sci. 2019, 9, 1057 10 of 17 Algorithm 4 Smoothing the combined path algorithm (P = P + P ) c n 1: Deﬁne the last node and the last but one node of P = fN [n1] , ..., N [ne] , N [berth]g as N [berth] and N [be] respectively, and deﬁne the second node of P = fN [berth] , N [b1] , ...g as N [b1], which is shown in Figure 7. 2: Calculate r [berth] by (7). 3: if r [berth] == 0 then 4: return “added node does not need. “ 5: end if 6: if r [berth] > 0 then 7: The turn of N [ne] ! N [berth] ! N [b1] is too sharp to track by the vessel. 8: Try to ﬁnd a node from eight neighbourhood of N except node N [b1] and node N [ne] that berth will be added to P . The coordinates of N [berth] is denoted as x , y . b N[berth] N[berth] 9: count = 0. 10: for i = 1 : 1 do 11: for j = 1 : 1 do 12: A candidate node N [c] x + i, y + j is selected to be veriﬁed. N[berth] N[berth] 13: if N [c] 6= N [berth] and N [c] 6= N [b1] and N [c] is not an obstacle node and N [c] is not beyond the boundary of the grid map. then 14: Calculate r [c] of N [berth] ! N [c] ! N [b1] and r [berth] of N [ne] ! N [berth] ! m1 m1 N [c] by (7). 15: Calculate r [c] of N [c] ! N [berth] ! N [b1] and r [berth] of N [ne] ! N [c] ! m2 m2 N [berth] by (7). 16: if r [c] == 0 and r [berth] == 0 then m1(2) m1(2) 17: Add N [c] x + i, y + j to addnodelist. N berth N berth [ ] [ ] 18: Calculate the danger risk r [c] = r [c] + r [c], where r [c] is calculated by add m1(2) o1(2) o1(2) (3). 19: f [c] = 1(2) and add f [c] to f l aglist. add add 20: count = count + 1. 21: end if 22: end if 23: end for 24: end for 25: if count > 0 then 26: Obtain the danger risk list r [c ] = r [c ] + r [c ] for node list N [c ] (n = 1, 2, ..., count). n m n o n n add 27: Select the node with the smallest value of r from r [c ] (n = 1, 2, ..., count), i.e., N [c ], n min add add and set N add = N c . [ ] [ ] min 28: if f [c ] == 1 then add min 0 0 29: Change P to P = fN [berth] , N [add] , N [b1] , ...g and P = P + P . b c n b b 30: else 0 0 31: Change P to P = fN [n1] , ..., N [add] , N [berth]g and P = P + P . n c b n n 32: end if 33: return “added node is found. “ 34: else 35: return “added node cannot be found. “ 36: end if 37: end if Appl. Sci. 2019, 9, 1057 11 of 17 11 12 12 8 7 8 12 8 9 10 12 4 13 12 9 sharp turn 9 12 5 13 12 6 9 9 9 9 moored or anchored ship 8 12 12 7 13 12 4 3 3 3 3 shore 7 4 9 7 14 12 4 1 1 1 1 bridge pier port 6 2 5 7 4 9 12 8 3 1 1 1 0 other obstacles 5 1 2 2 2 3 4 4 4 4 4 3 1 start node 4 2 5 7 4 4 8 12 12 12 12 8 3 destination node 3 3 8 7 4 13 12 4 navigable node 2 7 7 7 7 8 12 12 12 12 8 7 normal path berthing path current 1 2 3 4 5 6 7 8 9 10 11 12 Figure 6. The combined path with a sharp turn added node moored or anchored ship 11 12 12 8 7 8 12 8 9 shore 10 12 4 13 12 9 bridge pier 9 12 5 13 12 6 9 9 9 9 port 8 12 12 7 13 12 4 3 3 3 3 9 9 9 9 other obstacles 7 4 9 7 14 12 4 1 1 1 1 N[b1] N[add] start node 6 2 5 7 4 9 12 8 3 1 1 1 0 8 3 3 3 destination node 5 1 2 2 2 3 4 4 4 4 4 3 1 N[ne] N[berth] navigable node 4 2 5 7 4 4 8 12 12 12 12 8 3 normal path 3 3 8 7 4 13 12 4 7 1 1 1 berthing path 2 7 7 7 7 8 12 12 12 12 8 7 current 1 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 Figure 7. The combined path to deal with a sharp turn. 4. Case Study To verify the effectiveness of the proposed improved A-Star path planning method, simulation experiments are conducted in this section. Firstly, a normal path is generated by the proposed A-Star algorithm in consideration of obstacle, trafﬁc separation and manoeuvrability risk, and the performance of path planning is compared between the proposed A-Star method and traditional A-Star method. Then, the berthing path planning is realized by the proposed A-Star algorithm, and the berthing path is evaluated. The combination of the normal path and the berthing path is conducted by the proposed A-Star algorithm and the performance of the combined path is compared with the uncombined path. Finally, the proposed improved A-Star method is used to generate the path in a real scenario in Yangtze waterway. 4.1. Case 1: Normal Path Planning During the normal path planning, the obstacle risk and the manoeuvrability restriction risk are always considered, and it depends on the trafﬁc rule that whether the trafﬁc separation risk is considered or not. In this case, all these types of risks are all taken into consideration. 4.1.1. Setup To model the obstacle risk, related model parameters need to be determined. It is known that the current velocity near the shore is smaller than one far away from the shore. Hence, v m/s at node ( ) N [m] is set as: 0.4d , d 4l s s n v = , (19) 2, d > 4l s n where d (m) is the vertical distance from node N [m] to the nearest shore node. Other parameters are shown in Table 1. The vessel starts from the start node N [start] (3, 3) to the destination node (32, 21). Appl. Sci. 2019, 9, 1057 12 of 17 Table 1. Parameter initialization. Parameters Value node range of x axis [1, 38] node range of y axis [1, 25] grid length l 30 m minimum radius of vessel R 27 m a 100 t 0.2 b 0.5 4.1.2. Results The normal path generation performance with a traditional A-Star algorithm and an improved A-Star algorithm for vessels is shown in Figure 8. From Figure 8, it can be seen that the traditional A-Star algorithm generates the optimal path only by taking the shortest path length as the standard. On the contrary, the improved A-Star algorithm considers the path length, obstacle risk, the trafﬁc separation rule and manoeuvrability restriction simultaneously. Speciﬁcally, the path length of the traditional A-Star algorithm and the improved A-Star algorithm is 34.0 and 38.1, respectively. Generally, the improved A-Star algorithm will generate a longer path than the traditional A-Star algorithm. The sum of path node obstacle risk for the traditional A-Star algorithm and the improved A-Star algorithm is 150 and 22, respectively. The path generated by the traditional A-Star algorithm has much higher risk to collide with obstacles. For instance, the vessel is too close to the bridge piers and other vessels, which could increase the collision risks enormously in Figure 8. The path generated by the improved A-Star algorithm has taken into account the trafﬁc separation rule, which decreases the risk of collision with other navigating vessels. Moreover, the manoeuvrability risk is also considered in the improved A-Star algorithm to avoid the vessel failing to track the generated path. In summary, the improved A-Star algorithm can improve the navigating safety with slightly increasing the path length. path by improved A-Star path by traditional A-Star 24 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 9 9 7 23 2 2 2 2 2 2 2 3 8 7 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 9 9 3 22 0 0 0 0 0 0 1 2 5 7 4 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 10 9 3 21 0 0 0 0 0 0 0 1 2 2 2 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 1 3 6 9 9 9 9 9 6 2 20 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 2 3 4 4 4 4 4 4 3 1 0 1 1 3 3 3 3 3 3 2 1 19 0 0 0 0 0 0 1 1 2 2 2 1 0 0 0 1 2 4 9 121212121212 8 3 1 0 0 1 1 1 1 1 1 1 0 18 0 0 0 0 0 0 1 2 5 7 4 2 0 0 0 1 2 5 14 12 4 1 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 1 3 9 7 2 0 0 1 1 2 4 9 121212121212 8 3 1 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 3 9 7 2 0 1 2 3 4 3 3 4 4 4 4 4 4 3 1 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 1 2 5 7 4 2 1 2 4 9 12 8 3 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 14 0 0 0 0 0 0 1 1 2 2 2 1 1 2 5 14 12 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 1 0 0 0 1 2 5 14 12 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 1 2 2 2 1 1 2 5 14 12 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 1 2 5 7 4 2 1 2 5 14 12 4 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 10 0 0 0 0 0 0 1 3 9 7 2 1 2 5 14 12 4 1 0 0 0 0 0 0 1 2 3 4 4 4 4 4 3 1 0 0 9 0 0 0 0 0 0 1 3 9 7 2 1 2 5 14 12 4 1 0 0 1 1 1 1 2 4 9 1212121212 8 3 1 0 8 0 0 0 0 0 0 1 2 5 7 4 2 1 2 4 9 12 8 3 1 0 1 2 3 4 3 2 5 14 12 4 1 0 7 0 0 0 0 0 0 1 1 2 2 2 1 0 1 2 3 4 3 1 0 1 2 4 9 12 8 3 4 9 1212121212 8 3 1 0 6 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 2 4 9 14 12 4 2 3 4 4 4 4 4 3 1 0 0 5 0 0 0 0 0 0 0 1 2 2 2 1 0 0 0 0 0 0 0 1 4 8 13 12 8 3 1 1 1 1 1 1 1 1 0 0 0 4 0 0 0 0 0 0 1 2 5 7 4 2 0 0 0 0 0 0 0 2 5 13 12 8 3 1 0 0 0 0 0 0 0 0 0 0 0 3 2 2 2 2 2 2 2 3 8 7 2 2 2 2 2 2 2 2 2 3 8 12 8 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526272829303132333435363738 Figure 8. Normal path generation with a traditional A-Star algorithm and an improved A-Star algorithm for vessels (The deﬁnition of the graphic symbols in this ﬁgure is the same as the deﬁnition in Figure 7). Appl. Sci. 2019, 9, 1057 13 of 17 4.2. Case 2: Berthing Path Planning 4.2.1. Setup As previously mentioned, a vessel should berth into the current when the vessel needs to anchor or berth at a port. In this case, the vessel starts from the node of N [start] (6, 3) to the port whose node is N [dest] (32, 21). The node number of berthing path P is set to 6. The other parameters are the same as Table 1. The performance of the berthing path planning is presented as Figure 9. 4.2.2. Results In Figure 9, the path generated by the A-Star algorithm without considering the berthing path is hard to berth to the port because the vessel cannot be manipulated well if the current direction is consistent with the vessel sailing direction. When the berthing procedure is taken into account, the berthing path P and the berthing node N [berth] are obtained ﬁrstly according to Algorithm 3, speciﬁcally, the coordinates of N berth are 37, 18 . Then, the normal path P can be generated with [ ] ( ) the destination node N [dest] = N [berth]. Finally, P and P are combined to become the complete path P . normal path considering berthing berthing path considering berthing path not considering berthing 24 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 9 9 7 23 2 2 2 2 2 2 2 3 8 7 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 9 9 3 22 0 0 0 0 0 0 1 2 5 7 4 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 10 9 3 21 0 0 0 0 0 0 0 1 2 2 2 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 1 3 6 9 9 9 9 9 6 2 20 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 2 3 4 4 4 4 4 4 3 1 0 1 1 3 3 3 3 3 3 2 1 19 0 0 0 0 0 0 1 1 2 2 2 1 0 0 0 1 2 4 9 12 12 12 12 12 12 8 3 1 0 0 1 1 1 1 1 1 1 0 18 0 0 0 0 0 0 1 2 5 7 4 2 0 0 0 1 2 5 14 12 4 1 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 1 3 9 7 2 0 0 1 1 2 4 9 12 12 12 12 12 12 8 3 1 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 3 9 7 2 0 1 2 3 4 3 3 4 4 4 4 4 4 3 1 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 1 2 5 7 4 2 1 2 4 9 12 8 3 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 14 0 0 0 0 0 0 1 1 2 2 2 1 1 2 5 14 12 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 1 0 0 0 1 2 5 14 12 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 1 2 2 2 1 1 2 5 14 12 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 1 2 5 7 4 2 1 2 5 14 12 4 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 10 0 0 0 0 0 0 1 3 9 7 2 1 2 5 14 12 4 1 0 0 0 0 0 0 1 2 3 4 4 4 4 4 3 1 0 0 9 0 0 0 0 0 0 1 3 9 7 2 1 2 5 14 12 4 1 0 0 1 1 1 1 2 4 9 12 12 12 12 12 8 3 1 0 8 0 0 0 0 0 0 1 2 5 7 4 2 1 2 4 9 12 8 3 1 0 1 2 3 4 3 2 5 14 12 4 1 0 7 0 0 0 0 0 0 1 1 2 2 2 1 0 1 2 3 4 3 1 0 1 2 4 9 12 8 3 4 9 12 12 12 12 12 8 3 1 0 6 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 2 4 9 14 12 4 2 3 4 4 4 4 4 3 1 0 0 5 0 0 0 0 0 0 0 1 2 2 2 1 0 0 0 0 0 0 0 1 4 8 13 12 8 3 1 1 1 1 1 1 1 1 0 0 0 4 0 0 0 0 0 0 1 2 5 7 4 2 0 0 0 0 0 0 0 2 5 13 12 8 3 1 0 0 0 0 0 0 0 0 0 0 0 3 2 2 2 2 2 2 2 3 8 7 2 2 2 2 2 2 2 2 2 3 8 12 8 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 Figure 9. Berthing path generation with an improved A-Star algorithm considering berthing and an A-Star algorithm not considering berthing for vessels (The deﬁnition of the graphic symbols in this ﬁgure is the same as the deﬁnition in Figure 7). 4.3. Case 3: Combined Path Planning 4.3.1. Setup When combining normal path P and berthing path P , a sharp turn could encounter in the n b combination node, which will result in that the vessel cannot track the path because of the restriction of vessel’s manoeuvrability. In this case, the vessel needs to berth in an anchorage from node (10.11) to node (17, 13). The parameters are the same as Table 1 except the node range of x axis changes to 28. The comparison of path planning with considering smoothing the path or without considering smoothing the path is presented in Figure 10. Appl. Sci. 2019, 9, 1057 14 of 17 4.3.2. Results As shown in Figure 10, a sharp turn appears when smoothing the path is not considered. The sharp turn is eliminated when the smoothing strategy denoted by Algorithm 4 is applied. When the number of candidates for the added node are more than one, a candidate node with the least risk r add deﬁned in Algorithm 4 is selected. path considering smoothing path not considering smoothing 24 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 9 9 7 23 7 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 9 9 3 22 7 4 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 10 9 3 21 2 2 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 1 3 6 9 9 9 9 9 6 2 20 0 0 0 0 0 0 1 2 3 4 4 4 4 4 4 3 1 0 1 1 3 3 3 3 3 3 2 1 19 2 2 1 0 0 1 2 4 9 12 12 12 12 12 12 8 3 1 0 0 1 1 1 1 1 1 1 0 18 7 4 2 0 0 1 2 5 14 12 4 1 0 0 0 0 0 0 0 0 0 0 17 7 2 0 1 1 2 4 9 12 12 12 12 12 12 8 3 1 0 0 0 0 0 0 0 0 0 0 16 7 2 1 2 3 4 3 3 4 4 4 4 4 4 3 1 0 0 0 0 0 0 0 0 0 0 0 added node 15 7 4 2 2 4 9 12 8 3 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 14 2 2 1 2 5 14 12 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 0 0 1 2 5 14 12 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 2 2 1 2 5 14 12 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 7 4 2 2 5 14 12 4 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 10 7 2 2 5 14 12 4 1 0 0 0 0 0 0 1 2 3 4 4 4 4 4 3 1 0 0 9 7 2 2 5 14 12 4 1 0 0 1 1 1 1 2 4 9 12 12 12 12 12 8 3 1 0 8 7 4 2 2 4 9 12 8 3 1 0 1 2 3 4 3 2 5 14 12 4 1 0 7 2 2 1 1 2 3 4 3 1 0 1 2 4 9 12 8 3 4 9 12 12 12 12 12 8 3 1 0 6 0 0 0 0 1 1 1 1 0 1 2 4 9 14 12 4 2 3 4 4 4 4 4 3 1 0 0 5 2 2 1 0 0 0 0 0 0 1 4 8 13 12 8 3 1 1 1 1 1 1 1 1 0 0 0 4 7 4 2 0 0 0 0 0 0 2 5 13 12 8 3 1 0 0 0 0 0 0 0 0 0 0 0 3 7 2 2 2 2 2 2 2 2 3 8 12 8 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Figure 10. Combined path generated by an improved A-Star algorithm considering or not considering smoothing (The deﬁnition of the graphic symbols in this ﬁgure is the same as the deﬁnition in Figure 7). 4.4. Case 4: Path Planning in Real Scenario A real scenario, Wuhan waterway in Yangtze River, is selected to verify the effectiveness of the improved A-Star method in this case. In this scenario, a vessel is planned to navigate from one side of the river to the other side of the river in a bridge zone, which is shown in Figure 11. The navigable area for crossing bridge is indicated by the two beacon vessels, i.e., the middle area between the beacon vessel is navigable. To apply the proposed path planning method, the grid length, ln, is set as 50 m, and the minimum radius of vessel, R , is set as 45 m. The latitude and longitude coordinates of the start 0 00 0 00 point is (30 36 02.6 N, 114 19 23.4 E), and the latitude and longitude coordinates of the destination 0 00 0 00 point is (30 37 35.9 N, 114 19 36.0 E). Please note that the trafﬁc separation rule is not applied by the navigation rule in this waterway. Sailing vessels in this scenarios are not recognized as the obstacles because only moored or anchored vessels are considered as obstacles for the global path planning in this article. The other parameters are set the same as Case 1. The paths planned with the proposed A-Star algorithm, which consists of the normal path and the berthing path, and the traditional A-Star algorithm are shown in Figure 11, respectively. From Figure 11, it can be seen that the traditional A-Star has high risk to collide with the bridge and cannot guarantee the safety of berthing. To verify that a heuristic algorithm, e.g., A-Star algorithm, has a higher efﬁciency than a non-heuristic algorithm. The computation time records for A-Star algorithm with the improved A-Star algorithm and the search algorithm without heuristic guide for this case are 935 ms and 1127 ms, respectively. It means that the algorithm without heuristic guide actually has a higher efﬁciency. The path smoothed with the uniform B-spline curve of order 3 for the improved Appl. Sci. 2019, 9, 1057 15 of 17 A-Star algorithm is shown Figure 12. The path in Figure 12 can be set as a safety path that can be followed by a vessel. current current beacon vessel beacon vessel Path by traditional A-Star Normal path by improved A-Star Berthing path by improved A-Star Start Point Destination Point Figure 11. The path planned by the proposed A-Star algorithm in Wuhan waterway of Yangtze river. Smoothed Path Start Point Destination Point current current beacon vessel beacon vessel Figure 12. The smoothed path planned by the proposed A-Star algorithm in Wuhan waterway of Yangtze river. 5. Conclusions and Future Research According to the defects of the traditional A-Star algorithm when it is used for vessel path planning, an improved A-Star algorithm has been proposed in this article. Firstly, the risk models of obstacles (bridge pier, moored or anchored ship, port, shore, etc.) with currents, trafﬁc separation, Appl. Sci. 2019, 9, 1057 16 of 17 berthing, manoeuvrability restriction have been built. Then, the procedure of the normal path generation and the berthing path generation with the proposed A-Star algorithm have been depicted, respectively. Moreover, the problem of combining the normal path and the berthing path has been also solved. Four cases have been studied to verify the effectiveness of the proposed A-Star algorithm. The simulation and real experiment results show that the proposed A-Star path planning methods can avoid the high risk of obstacle collision, obey the separation rule and meet the requirements of berthing for vessels. In a word, the proposed algorithms can realize the trade-off between the path length and the navigation safety. In future work, wind and water depth can be also considered during vessel path following, because wind will affect the ship navigation safety and economical efﬁciency, and shallow water will increase the vessel stranding risk. Author Contributions: Conceptualization, C.L. and Q.M.; Methodology, C.L.; Software, C.L.; Validation, C.L and S.X.; Writing-Original Draft Preparation, C.L.; Writing-Review and Editing, C.L., Q.M. and X.C.; Visualization, X.C. and S.X.; Funding Acquisition, C.L, Q.M. and X.C. Funding: This research is supported by the China Postdoctoral Science Foundation (2018M632923), the Wuhan Science and Technology Project (2017010201010132), and the Fundamental Research Funds for the Central Universities (2042017KF0235). Conﬂicts of Interest: The authors declare no conﬂict of interest. References 1. Wuthishuwong, C.; Traechtler, A.; Bruns, T. 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Dijkstra, E.W. A note on two problems in connexion with graphs. Numer. Math. 1959, 1, 269–271. [CrossRef] 14. Bell, M.G.H. Hyperstar: A multi-path Astar algorithm for risk averse vehicle navigation. Transp. Res. Part B Methodol. 2009, 43, 97–107. [CrossRef] 15. Hart, P.E.; Nilsson, N.J.; Raphael, B. A formal basis for the heuristic determination of minimum cost paths. IEEE Trans. Syst. Sci. Cybern. 1968, 4, 100–107. [CrossRef] Appl. Sci. 2019, 9, 1057 17 of 17 16. Rahman, S.M.M.; Al-Arif, A.H.M.; Ferdous, I.; Nizami, M.S.H. Path planning for robotic boats in a rescue system. Int. J. Comput. Appl. 2012, 45, 50–57. 17. Sathyaraj, B.M.; Jain, L.C.; Finn, A.; Drake, S. Multiple UAVs path planning algorithms: A comparative study. Fuzzy Optim. Decis. Mak. 2008, 7, 257–267. [CrossRef] 18. Yao, J.; Lin, C.; Xie, X.; Wang, A.J.A.; Hung, C.C. Path planning for virtual human motion using improved A* star algorithm. In Proceedings of the Seventh International Conference on Information Technology: New Generations (ITNG), Las Vegas, NV, USA, 1 July 2010; pp. 1154–1158. 19. Pochmara, J.; Grygiel, W.; Koppa, R.; Kaminski, K. Mobile robot platform for real-time search algorithms. In Proceedings of the 20th International Conference on Mixed Design of Integrated Circuits and Systems, Gdynia, Poland, 10 October 2013; pp. 615–620. 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Applied Sciences – Multidisciplinary Digital Publishing Institute

**Published: ** Mar 13, 2019

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