Intelligent Collision Avoidance Method for Ships Based on COLRGEs and Improved Velocity Obstacle Algorithm

Xingya Zhao; Yixiong He; Liwen Huang; Junmin Mou; Ke Zhang; Xiao Liu

doi:10.3390/app12188926

Zhao, Xingya;He, Yixiong;Huang, Liwen;Mou, Junmin;Zhang, Ke;Liu, Xiao

2022-09-06 00:00:00

applied sciences Article Intelligent Collision Avoidance Method for Ships Based on COLRGEs and Improved Velocity Obstacle Algorithm 1 , 2 1 , 2 1 , 2 1 , 2 1 , 2 , 1 , 2 , Xingya Zhao , Yixiong He , Liwen Huang , Junmin Mou , Ke Zhang * and Xiao Liu * School of Navigation, Wuhan University of Technology, Wuhan 430063, China Hubei Key Laboratory of Inland Shipping Technology, Wuhan 430063, China * Correspondence: kezhang_rq@whut.edu.cn (K.Z.); lxiao@whut.edu.cn (X.L.) Abstract: Collision prevention is critical for navigational safety at sea, which has developed rapidly in the past decade and attracted a lot of attention. In this article, an improved velocity obstacle (IVO) algorithm for intelligent collision avoidance of ocean-going ships is proposed in various operating conditions, taking into count both a ship’s manoeuvrability and Convention on the International Regulations for Preventing Collisions at Sea (COLREGs). An integrated model combines a three- degree-of-freedom manoeuvring model with ship propeller characteristics to provide a precise prediction of ships in various manoeuvring circumstances. In the given case, what is different to present studies, this improved algorithm allows for decision-making in two ways: altering course and changing speed. The proposed technique is demonstrated in a variety of scenarios through simulation. The ﬁndings reveal that collision-avoidance decision-making can intelligently avoid collisions with the target ships (TSs) in multi-ship situations. Citation: Zhao, X.; He, Y.; Huang, L.; Keywords: intelligent collision avoidance; improved velocity obstacle; COLREGs; variable speed; Mou, J.; Zhang, K.; Liu, X. Intelligent improved MMG Collision Avoidance Method for Ships Based on COLRGEs and Improved Velocity Obstacle Algorithm. Appl. Sci. 2022, 12, 8926. 1. Introduction https://doi.org/10.3390/ Maritime transactions, due to the large capacity and low cost, which play a vital part app12188926 in the global economy have become a larger and more important means of international Academic Editors: Dimitrios S. and domestic transportation. Marine accidents, on the other hand, always represent a Paraforos, Giovanni Randazzo, signiﬁcant risk to societies and the environment in terms of human life and property loss, Anselme Muzirafuti and Stefania environmental damage, and transportation safety [1]. Furthermore, among all sorts of Lanza Marine catastrophes, ship collisions as one of the key contributors would have a negative Received: 20 July 2022 and unanticipated inﬂuence on the stakeholders’ reputation and property [2]. Accepted: 26 August 2022 According to studies, human error accounts for more than 80% of ship collisions [3], Published: 6 September 2022 while violations of International Regulations for Preventing Collisions (COLREGs) by mariners account for more than 56% of major marine accidents [4]. Human error is fre- Publisher’s Note: MDPI stays neutral quently caused by crews failing to observe COLREGs and good seamanship, such as failing with regard to jurisdictional claims in to take steps timely, wrong avoidance action, insufﬁcient risk assessment, and failing to published maps and institutional afﬁl- iations. continue at a safe pace [5]. On the other hand, the COLREGs guidelines do not provide precise operational instructions [6]. How to assist or perhaps replace the ofﬁcer of the watch (OOW) in making decisions to avoid collision is an intriguing subject that more and more researchers are addressing. Many techniques have been presented in recent Copyright: © 2022 by the authors. years to handle this subject, and in 2018, the International Maritime Organization (IMO) Licensee MDPI, Basel, Switzerland. formulate the standard of the degree level for Maritime Autonomous Surface Ship (MASS). This article is an open access article Although the MASS concept has been introduced in recent years, there is still a signiﬁcant distributed under the terms and gap in autonomous navigation. Due to the limitations of present ship navigation equip- conditions of the Creative Commons ment and shortcoming of the algorithm, it is not feasible to completely replace human Attribution (CC BY) license (https:// operators with machines in a short period [7]. Thus, the decision-making for ship- intel- creativecommons.org/licenses/by/ ligent collision avoidance is a complex process, especially in the multi-ship situation. At 4.0/). Appl. Sci. 2022, 12, 8926. https://doi.org/10.3390/app12188926 https://www.mdpi.com/journal/applsci Appl. Sci. 2022, 12, 8926 2 of 22 present, much research and technologies have been developed for the intelligent collision avoidance problem. For decades, the geometric analysis approach, which includes the Distance to the Closest Point of Approaching (DCPA) and the Time to Closest Point of Approaching (TCPA), has advised OOWs and ship captains to conduct effective collision-prevention procedures in sea practice [8]. Then, in the early 1970s the ship domain which refers to the area around a ship that prevents other ships from entering to maintain navigational safety, was introduced. Many researchers have presented multiple deﬁnitions of ship domain over the years, with varied forms and sizes [9,10]. Some researchers evaluated the ship domain of the target ship (TS) [10], while others consider the ship domain of their ship [11]. The velocity obstacle algorithm was presented to give greater support for the OOW in collision prevention [12], which described a set of velocities to determine whether their own velocity fell into the velocity obstacle zone, hence reducing the danger of collision. As a result, a growing number of researchers were focusing on the use of velocity obstacle (VO) algorithms for collision avoidance in a variety of settings, including conﬁned waters [13], and considering rules [14]. Furthermore, nonlinear VO based on ship motion was proposed in [15], and Chen [16] adopted an improved time-discretized non-linear velocity obstacle technique for multi- ship encounter detection. With the advancement of computer technology, the study of vessel collision avoidance has progressed to a new level, and deep reinforcement learning based on COLREGs has been proposed for use in multi-ship collision avoidance [17]. Simultaneously, genetic algorithms for ship collision avoidance and path planning were made public [18,19]. Ni [20], established a ship manoeuvrability-based simulation for ship navigation in collisions, and Wang [21], presented a ship manoeuvrability-based collision avoidance dynamic support system in close-quarters situations. Zhang adopted linear extension algorithms for a multi-ship anti-collision decision in the manner of course alternation and speed reduction [22]. To decide the path planning for autonomous ships, Zhang used a modiﬁed artiﬁcial potential ﬁeld and a modiﬁed velocity obstacle algorithm method [23]. Singh suggested a system that combines ofﬂine optimum path planning with a constrained A* algorithm which employs an artiﬁcial potential ﬁeld to increase online adaptive weighting based on USV manoeuvring reaction time [24,25]. For the study of collision avoidance in the multi-ship situation, many algorithms have been applied in recent years. The paper [26], proposes a unique maximum-course and minimum-speed change approach for decision-making on the basis of a novel Fuzzy-logic algorithm. Furthermore, VO algorithms are provided for collision avoidance based on the target ships whose motions are non-linear and probabilistically predictable [27]. Moreover, the article [28], presents a generalised VO algorithm for ship collision avoidance and designs a collision, which is more reliable and suitable for close-range ship collision avoidance. Considering the ship maneuverability, multi-ship, COLREGs, off-course, and seamanship, Wang [29], designed the multistage optimisation decision model based on the modiﬁed VO method. Despite a lot of research work and achievements being completed on ship collision avoidance, there are still some unignorable shortcomings in the available studies. Most of the current research on decision-making for collision avoidance in the multi-ship situation considers the ship motion model insufﬁciently, especially in the process of the changing speed of ship motion. Little research has been focused on the model in which the ship’s speed and rotation of the propeller are not matched. Moreover, the decision-making of ship-intelligent collision avoidance should be expected to follow COLRGEs and good seamanship, which are guidelines for all vessels. Although humans are good at interpreting navigational regulation, negligence will lead to marine accidents. Therefore, a model combining the advantages of human intelligence and machine intelligence is necessary for ship collision avoidance. The advantages of machine intelligence in computing power are extremely complementary to human intelligence. In other words, this model will not only provide real-time decision-making for collision avoidance but will also take into account the COLREGs and good seamanship. Appl. Sci. 2022, 12, 8926 3 of 22 Considering the characteristics of ship movement, an accurate manoeuvring model is required to deduce future ship motion trends and states for collision avoidance, especially when large ships are involved in adjustable speed operations. In conclusion, this paper aims to design an intelligent collision avoidance model for ships, which combines the COLREGs and good seamanship mentioned above, as well as ship manoeuvrability, based on the advantages of both machine intelligence and human intelligence. Furthermore, the primary research topic for this project is: how can the ship combine machine intelligence and human intelligence to make intelligent collision avoidance decisions? The main contributions of this article are as follows: 1. This paper provides the improved mathematical model group (MMG) based on ship manoeuvrability to deduce the process of own ship (OS) motion, which is also appro- priate for the situation when a ship’s speed and rotation of propeller are not matched. 2. An integrated algorithm for collision avoidance decision-making based on improved VO algorithm and MMG is presented, which comprehensively considers COLREGS, sea practice, and good seamanship. 3. The decision-making of collision avoidance following the manoeuvrability habit of manned ships is provided by combining a dynamic feasible manoeuvring decision interval and an optimisation evaluation function, which will gap reduce the difference of awareness about manoeuvrability in the process of collision avoidance between the intelligent and manned ship. The contents of the research are arranged as follows: The quantitative analysis of COLREGs, improved MMG, and improved Velocity Obstacle (IVO) algorithm are covered in Section 2. The intelligent collision avoidance method based on the collision risk index (CRI) model is covered in Section 3, which introduces dynamically feasible manoeuvring intervals calculated by IVO algorithm and the collision decision determined by optimisa- tion functions. The case study in Section 4 demonstrates the accuracy of MMG and the practicality of the proposed strategy. In Section 5, there is an analysis and discussion on future research. Finally, in Section 6, the conclusions are outlined. 2. Methodology 2.1. Methodological Overview of the Research In this research, the IVO algorithm based on the improved MMG and COLRESGs is presented to obtain the collision avoidance decision. Given that COLREGs are the guide- lines for ship collisions, quantitative analysis of the rules is required for intelligent collision avoidance, which is described in Section 2.2. Considering the deﬁnition of collision avoid- ance, the problem is divided into two sub-problems: “collision detection” and “collision resolution”. “Collision detection” utilises the collision risk index (CRI) model to establish the avoidance timing and avoidance sequence for the TS in the multi-ship circumstance, which is illustrated in Section 3.1. The solution to the collision problem is to obtain the collision avoidance decision-making by the improved MMG illustrated in Section 2.3 and Appl. Sci. 2022, 12, x FOR PEER REVIEW 4 of 23 IVO illustrated in Section 2.4, and the details are elaborated in Section 3. The framework of the methodology can be found in Figure 1. Figure 1. The flowchart of the research methodology. Figure 1. The ﬂowchart of the research methodology. 2.2. Quantitative Analyses of COLREGs COLREGs are guidelines for avoiding ship collisions, and every decision made by the OOW in the face of approaching and outgoing ships is governed by rules permitted. Therefore, understanding and following the rules is the key precondition for preventing ship collisions. Although the rules outline some provisions and divisions of the ship’s movements, there is no explanation of the exact decision-making for avoidance actions. The quantitative analysis of the collision avoidance rules must take precedence, and the ship’s actions should conform to the rules’ constraints. Therefore, for the decision-making of intelligent collision avoidance, the construction of strategies for quantitative analysis of the rules is also essential. Much research was presented about the quantitative analysis of the rules, including fruitful research fields like the ship domain, encounter situation, narrow channel, and so on. This paper focuses on two main components: One is determining the meeting stage of various encounter scenarios, and the other is the adaptation of a ship domain model to one’s vessel. Section 2.2.1 and 2.2.2, respectively, detail the four-stage theory and ship do- main by quantitative analyses of rules. 2.2.1. Four-Stage Theory According to COLRGEs, two power-driven vessels in a hazardous conflict approach one another in four stages if they maintain their speed and course [30]. The four-stage theory is established to expound on the timing to avoid the collision, which involves “no risk of collision (No CR)”, “risk of collision (CR)”, “close-quarter situation (CS)”, and “im- mediate danger stage (ID)” as shown in Figure 2. For stand-on vessels and give-way ves- sels, the timing of taking measures to prevent collisions is distinct in the same encounter- ing situation. Furthermore, actions taken by the vessel, which include altering courses and changing speed, are varying with the development of the stage and two vessels approach- ing each other [31]. In stage 1: the vessel is too far to cause a collision, while both vessels are free to move during this time up until the first point of the CR. Instead of a series of minute changes in course and speed as required by Rule 8 and good seamanship in practice, any necessary course and speed changes must be large enough to be easily apparent to another vessel visually or by radar if the circumstances permit. Being able to observe another vessel’s purpose clearly, ensures that the concerned vessel avoids the potential of misunderstand- ings. Furthermore, if the TS eventually invades the OS’s ship domain on the basis that both ships keep their initial course and speed, the potential collision risk (PCR) exists be- tween the ships. The first time-in-point of collision risk (FTCR) indicates the conclusion of this stage and also leads to the formation of stage 2. Appl. Sci. 2022, 12, 8926 4 of 22 2.2. Quantitative Analyses of COLREGs COLREGs are guidelines for avoiding ship collisions, and every decision made by the OOW in the face of approaching and outgoing ships is governed by rules permitted. Therefore, understanding and following the rules is the key precondition for preventing ship collisions. Although the rules outline some provisions and divisions of the ship’s movements, there is no explanation of the exact decision-making for avoidance actions. The quantitative analysis of the collision avoidance rules must take precedence, and the ship’s actions should conform to the rules’ constraints. Therefore, for the decision-making of intelligent collision avoidance, the construction of strategies for quantitative analysis of the rules is also essential. Much research was presented about the quantitative analysis of the rules, including fruitful research ﬁelds like the ship domain, encounter situation, narrow channel, and so on. This paper focuses on two main components: One is determining the meeting stage of various encounter scenarios, and the other is the adaptation of a ship domain model to one’s vessel. Sections 2.2.1 and 2.2.2, respectively, detail the four-stage theory and ship domain by quantitative analyses of rules. 2.2.1. Four-Stage Theory According to COLRGEs, two power-driven vessels in a hazardous conﬂict approach one another in four stages if they maintain their speed and course [30]. The four-stage theory is established to expound on the timing to avoid the collision, which involves “no risk of collision (No CR)”, “risk of collision (CR)”, “close-quarter situation (CS)”, and “immediate danger stage (ID)” as shown in Figure 2. For stand-on vessels and give- way vessels, the timing of taking measures to prevent collisions is distinct in the same encountering situation. Furthermore, actions taken by the vessel, which include altering Appl. Sci. 2022, 12, x FOR PEER REVIEW 5 of 23 courses and changing speed, are varying with the development of the stage and two vessels approaching each other [31]. Figure 2. The diagram of the four-Stage theory. Figure 2. The diagram of the four-Stage theory. In stage 2: The conflict is increasing, and the giving-way vessel shall take action as In stage 1: the vessel is too far to cause a collision, while both vessels are free to soon as feasible to avoid the collision, as per COLREGs. Meanwhile, the stand-on vessel move during this time up until the ﬁrst point of the CR. Instead of a series of minute changes should ma in intai course n itand s spe speed ed and asco requir urse ed in tby he Rule first s8 tages, and good but us seamansh e all avail ip abl in e m practice, eans to monitor the success of the acts made as necessary by the other vessel. For the giving-way any necessary course and speed changes must be large enough to be easily apparent to vessel, this is the “golden time” to take precautions. On the one hand, to act early is not another vessel visually or by radar if the circumstances permit. Being able to observe another only in tvessel’s he spirit purpose of COLRE clearly Gs, bu , ensur t also es gives that the the sh concerned ip more time vessel to dea avoids l withthe the potential urgency. of On misunderstandings. the other hand, it Furthermor provides th e, e if sta the ndTS -on eventually vessel moinvades re time the to ob OS’s serve ship andomain d check on the basis that both ships keep their initial course and speed, the potential collision risk whether the conflict occurred or not after the collision-avoidance action. With the two (PCR) vesselexists s combetween ing close the tog ships. ether, The the ﬁrst risktime- of a in-point collision of iscollision high, an risk d the (FTCR) stand indicates -on vessel the is conclusion of this stage and also leads to the formation of stage 2. permitted to take whatever actions necessary to avoid a collision after it is determined that the give-way ship is not acting correctly and effectively in accordance with COLREGs. This timing is determined by the situation and ship’s manoeuvrability, and it is best de- fined by the ship captain’s evaluation. CR that is the initial point exists at the confined sea only when 𝑇𝐶𝑃𝐴 ≤ 20 mi in this study [30]. In stage 3: CS is created when both ships are unable to pass each other at a safe dis- tance whichever collision avoidance action by each ship is taken. In stage 4: This is the most perilous stage when facing a situation where a disagree- ment has escalated to the point where it cannot be resolved just by the action of the give- way ship. Both vessels must execute collision-avoidance measures; otherwise, a collision will occur in a short period [31]. In this paper, the model of CRI is utilised to quantify the encounter stage of ships, whose larger value indicates a higher risk, and the detail is shown in Section 3.1. 2.2.2. Ship Domain Ship domain is the region around a ship that the OOW or Captain wants to keep free of obstructions and other vessels, and it is thought to be inviolable. According to good seamanship in practice and COLREGs, the ship must keep a safe distance when approach- ing various types of power-driven ships or static obstructions [32]. In terms of collision avoidance, the safety distance is equivalent to the ship domain. In this work, two different ship domains are used which include the left eccentric ellipse model (Figure 3a) for the head-on or overtaking situation and the left eccentric circle model (Figure 3b) for the cross- ing situation. Appl. Sci. 2022, 12, 8926 5 of 22 In stage 2: The conﬂict is increasing, and the giving-way vessel shall take action as soon as feasible to avoid the collision, as per COLREGs. Meanwhile, the stand-on vessel should maintain its speed and course in the ﬁrst stages, but use all available means to monitor the success of the acts made as necessary by the other vessel. For the giving-way vessel, this is the “golden time” to take precautions. On the one hand, to act early is not only in the spirit of COLREGs, but also gives the ship more time to deal with the urgency. On the other hand, it provides the stand-on vessel more time to observe and check whether the conﬂict occurred or not after the collision-avoidance action. With the two vessels coming close together, the risk of a collision is high, and the stand-on vessel is permitted to take whatever actions necessary to avoid a collision after it is determined that the give-way ship is not acting correctly and effectively in accordance with COLREGs. This timing is determined by the situation and ship’s manoeuvrability, and it is best deﬁned by the ship captain’s evaluation. CR that is the initial point exists at the conﬁned sea only when TCPA 20 mi in this study [30]. In stage 3: CS is created when both ships are unable to pass each other at a safe distance whichever collision avoidance action by each ship is taken. In stage 4: This is the most perilous stage when facing a situation where a disagreement has escalated to the point where it cannot be resolved just by the action of the give-way ship. Both vessels must execute collision-avoidance measures; otherwise, a collision will occur in a short period [31]. In this paper, the model of CRI is utilised to quantify the encounter stage of ships, whose larger value indicates a higher risk, and the detail is shown in Section 3.1. 2.2.2. Ship Domain Ship domain is the region around a ship that the OOW or Captain wants to keep free of obstructions and other vessels, and it is thought to be inviolable. According to good seamanship in practice and COLREGs, the ship must keep a safe distance when approaching various types of power-driven ships or static obstructions [32]. In terms of collision avoidance, the safety distance is equivalent to the ship domain. In this work, two different ship domains are used which include the left eccentric ellipse model (Figure 3a) Appl. Sci. 2022, 12, x FOR PEER REVIEW 6 of 23 for the head-on or overtaking situation and the left eccentric circle model (Figure 3b) for the crossing situation. Figure 3. Ship domain model in different situations: (a) Head-on & overtaking situation; (b) cross Figure 3. Ship domain model in different situations: (a) Head-on & overtaking situation; (b) cross situation. situation. An imaginary ship, which is located on the starboard and in front of its ship, is set up in the centre of the ship domain of both the left eccentric ellipse and eccentric circle model. More safety distance should be kept from the front of the vessel and the starboard side based on the consideration of COLREGs rules and good seamanship. In practice, the suitable range of the ship domain is established by the OOW or captain depending on the situation and the specifics of both the OS and the TS. It is critical for collision avoidance since an incorrect ship domain would increase the probability of collision [33]. In Figure 3, 𝑎 and 𝑏 are the major and short axial lengths of the ellipse respectively; 𝑐 is the distance from the imaginary ship to the fact ship; 𝑅 is the radius of the circle in the cross situation; and 𝐿 and 𝐿 are the lengths of the OS and TS respectively, where 𝑜 𝑇 𝑎 = 2(𝐿 + 𝐿 ) 𝑜 𝑇 𝑏 = 1.5(𝐿 + 𝐿 ) 𝑜 𝑇 (1) 𝑐 = 0.5(𝐿 + 𝐿 ) 𝑜 𝑇 𝑅 = 2(𝐿 + 𝐿 ) 𝑜 𝑇 2.3. Improved MMG Model For intelligent collision avoidance decision-making in the form of altering course or changing speed by a computer system, it is clear that an accurate and exact motion model for ship navigation is required. The focus of this research is on decision-making for colli- sion avoidance in the confined sea when visibility is good. Considering rolling, pitching, and heaving motions have little impact on ship collision, only sway, surge, and yaw are considered in the calm sea. Therefore a mathematical model called MMG with three de- grees of freedom is used to properly deduce OS’s movements [34]: (𝑚 + 𝑚 )𝑣 ̇ + (𝑚 + 𝑚 )𝑢𝑟 = 𝑌 + 𝑌 + 𝑌 𝑦 𝑥 𝐻 𝑃 𝑅 (𝑚 + 𝑚 )𝑢 ̇ − (𝑚 + 𝑚 )𝑣 𝑟 = 𝑋 + 𝑋 + 𝑋 { (2) 𝑥 𝑦 𝐻 𝑃 𝑅 (𝐼 + 𝑖 )𝑟 ̇ = 𝑁 + 𝑁 + 𝑁 zz 𝑧𝑧 𝐻 𝑅 𝑝 where 𝑚 is the total mass of the vessel, 𝑚 and 𝑚 , respectively, indicate the additional 𝑥 𝑦 vertical and horizontal mass. Furthermore, 𝑢 and 𝑣 denote the lateral and longitudinal speed; 𝑢 ̇ and 𝑣 ̇ denote acceleration in their direction; 𝑟 and 𝑟 ̇ are the angular velocity and angular acceleration; 𝐼 and 𝑖 are the moment of inertia and additional moment of zz 𝑧𝑧 inertia respectively; 𝑋 and 𝑌 represent the force or moment in vertical axis directions and Appl. Sci. 2022, 12, 8926 6 of 22 An imaginary ship, which is located on the starboard and in front of its ship, is set up in the centre of the ship domain of both the left eccentric ellipse and eccentric circle model. More safety distance should be kept from the front of the vessel and the starboard side based on the consideration of COLREGs rules and good seamanship. In practice, the suitable range of the ship domain is established by the OOW or captain depending on the situation and the speciﬁcs of both the OS and the TS. It is critical for collision avoidance since an incorrect ship domain would increase the probability of collision [33]. In Figure 3, a and b are the major and short axial lengths of the ellipse respectively; c is the distance from the imaginary ship to the fact ship; R is the radius of the circle in the cross situation; and L and L are the lengths of the OS and TS respectively, where o T a = 2 L + L ( ) > o T b = 1.5(L + L ) o T (1) > c = 0.5(L + L ) o T R = 2(L + L ) o T 2.3. Improved MMG Model For intelligent collision avoidance decision-making in the form of altering course or changing speed by a computer system, it is clear that an accurate and exact motion model for ship navigation is required. The focus of this research is on decision-making for collision avoidance in the conﬁned sea when visibility is good. Considering rolling, pitching, and heaving motions have little impact on ship collision, only sway, surge, and yaw are considered in the calm sea. Therefore a mathematical model called MMG with three degrees of freedom is used to properly deduce OS’s movements [34]: m + m v + (m + m )ur = Y + Y + Y y x H P R (m + m )u m + m vr = X + X + X (2) x y H P R (I + i )r = N + N + N zz zz H R p where m is the total mass of the vessel, m and m , respectively, indicate the additional x y vertical and horizontal mass. Furthermore, u and v denote the lateral and longitudinal . . . speed; u and v denote acceleration in their direction; r and r are the angular velocity and angular acceleration; I and i are the moment of inertia and additional moment of zz zz inertia respectively; X and Y represent the force or moment in vertical axis directions and in horizontal axis directions, respectively; and N respects the turning moment. H, P and R are the external forces acting on the hull, propeller, and rudder respectively. The propeller ’s power must be sufﬁcient to overcome both the hull’s inherent re- sistance and the additional resistance brought on by the propeller ’s operation, which is expressed as follows: T = R + DT (3) where T and DT are the forces of propeller and propeller derating respectively. R is the resistance of the ship hull. In the past, a lot of research was aimed at establishing an accurate ship motion model with the precondition of ﬁxed rotation of the ship propeller, in which the ship’s speed and rotation of propeller are matched. In this state of the ship moving, the thrust is only inﬂuenced by the rotation of the propeller when additional elements are known. However, the research in this paper focuses on the ship motion and collision avoidance decision method under variable speed. In this situation, the rotation of the propeller and the actual speed of the ship are unmatched, which is more complicated than the ship’s motion in a ﬁxed rotation. Thrust is inﬂuenced by more factors besides the rotation of the propeller, such as ship longitudinal and lateral speeds, thrust deduction, coordinate of buoyancy, and so on. Therefore, a new ship thrust model is needed for the MMG model. Appl. Sci. 2022, 12, 8926 7 of 22 In practice, the propeller ’s effective thrust is often expressed using the thrust derating factor as follows: X = 1 t T (4) p p0 where t is the thrust derating coefﬁcient, which is impacted by the ship’s model, as well p0 as the propeller ’s load and size. Furthermore, for a particular ship, t is a variable for p0 alternation of revolution and ship movement. The alternation of revolution would affect the ship’s resistance by causing a shift in the inﬂow and outﬂow of the propellers. the thrust derating coefﬁcient t under the impact of propeller load is illustrated in Equation (5): p0 0.04 + t 0.04 u/u , u u 0 0 p0 t = (5) p0 t , u > u p0 where u is the designated speed and u is the real-time speed. Further, t is the propeller p0 derating factor at u . It is also vital to evaluate the impact of lateral and rotational motion on the ship motion model in order to increase the accuracy of the ship motion prediction model. The function of the thrust derating coefﬁcient by the ship moving can be expressed as follows: 1 t = 1 t + f (6) p p0 where the f is the function of the effect of ship motion, which is determined as follows: f = k b > R k = 0.00023 g L/D 0.028 < t A r g = (B/d)f1.3(1 C ) 3.1l g (7) A b cb l = x /L 100 > cb b = b l R R where x is the vertical coordinate of buoyancy. b denotes the drift angle at the rudder, and c R b is the drift angle at the center of the ship, which can be determined by b = atan(u/v). B and d denote the ship’s width and the ship draft, respectively. L denotes the ship’s length and D is the diameter of the propeller. g and l are the rectiﬁcation coefﬁcients of the r A cb hull and ﬂoating centre coefﬁcient, respectively. C and l denote the block coefﬁcient of b R the ship and drift angle coefﬁcient. The force of the propeller, mainly from the thrust of the propeller, is expressed as follows: T = r(NP) D K (8) where K is the factor of thrust; r is the density of water; D represents the diameter of T p the propeller; and NP denotes the real-time rotation rate of the propeller, which uses to keep and control the ship speed. There are two different styles of ship manoeuvre at sea: ﬁrst is manoeuvring in the manner of only altering course while maintaining a constant revolution per minute (RPM), which nearly invariably occurs in open water. The second is manoevring in the manner of changing speed by reducing or increasing the revolution of the main engine, which almost always occurs in narrow water or high-density trafﬁc seas. In previous studies, the ship’s speed was rarely taken into account in the ship motion model, with just the ship’s course being taken into account. Moreover, the study using ship models that are affected by changes in direction and speed almost exclusively focuses on small boats. The difference of characteristics of propellers between tiny ships and large ships is huge. Furthermore, the ship in the variable-speed process has more inﬂuencing components than the ship in the variable-course motion process, and the interaction between each factor is more sophisticated. Furthermore, minor speed changes have little effect on the ship’s collision avoidance in practice, which is why captains and pilots prefer to manage the telegraph rather than using a small variation of the main engine revolution to modify the speed. Telegraphs are used to control the ship’s speed, which includes ten different Appl. Sci. 2022, 12, 8926 8 of 22 presupposed speciﬁc NPs. Propulsion particulars about a bulk carrier called “Huayang dream” are illustrated in Table 1, where the gap of RPM between different telegraphs is relatively large in order to accommodate a large vessel speed change. Table 1. Propulsion particulars about RPM and speed of ship “Huayang dream”. Item Telegraph RPM (r/min) Speed (knot) Nav. ahead 92 14 Full ahead 85 12 Half ahead 70 10.5 Ahead Slow ahead 48 5 Dead slow ahead 35 3 Stop 0 0 Dead slow astern 35 3 Slow astern 48 5 Astern Half astern 70 10.5 Full astern 85 12 Those telegraphs are applied to adjust the speed to manoeuvre the ship to avoid collision by OOW and pilots in practice. If NP is the initial rotation rate of the propeller of a telegraph at the time t , and NP is the ﬁnal rotation rate of the propeller of a telegraph at the time t , the rotation rate of the propeller at a time t can be determined by the following: NP , t > t > 2 2 NP , t < t t 1 1 NP = (9) NP (t t ) k, t < t< t , NP >NP 1 1 1 2 1 2 NP + (t t ) k, t < t < t , NP < NP 1 1 1 2 1 2 where k is the meaning of a change of rotation rate, which was deduced to be around 0.25 rotation per second based on the researcher ’s experience as the ship’s captain/ofﬁcers and the model projection. The Runge–Kutta method and improved MMG should be used to deduce the future state of the ship by combining the initial state of OS, such as speed, course, and position. The details of simulation results about improved MMG are provided in Section 4.1. 2.4. Improved Velocity Obstacle The VO method deﬁnes a collection of velocity sets in which a collision occurs if the velocity falls inside one of them. Assuming that two ships, V and V , represent the velocity of the OS and the TS, respectively. V is the relative velocity of OS caused by TS, OT then this can be expressed in the following way: ! ! ! V = V V (10) OT O T As displayed in Figure 4, the white circle region, the TS’s ship domain, deﬁnes all of the TS’s possible positions when a collision occurs [28]. Once V falls into the Relative OT collision cone (RCC), the collision between ships exists. On the contrary, there is no risk of collision existing between ships. The absolute collision cone (ACC, purple region) is presented by RCC along V , which implies that the V dropping into ACC is equivalent T O to V falling into RCC. In other words, the collision between ships exists when V falls OT O into the ACC, which can be shown as Equation (11) considering that the VO algorithm is time-varying: ! ! (t) (t) (t) V 2 VO = RCC V (11) o T where operation is the Minkowski addition. However, since it ignores the characteristics of ship manoeuvring and the particular navigation environment, typical VO cannot be Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 23 The Runge–Kutta method and improved MMG should be used to deduce the future state of the ship by combining the initial state of OS, such as speed, course, and position. The details of simulation results about improved MMG are provided in Section 4.1. 2.4. Improved Velocity Obstacle The VO method defines a collection of velocity sets in which a collision occurs if the velocity falls inside one of them. Assuming that two ships, 𝑉 and 𝑉 , represent the ve- 𝑂 𝑇 locity of the OS and the TS, respectively. 𝑉 is the relative velocity of OS caused by TS, 𝑂𝑇 then this can be expressed in the following way: ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ 𝑉 = 𝑉 − 𝑉 (10) 𝑂𝑇 𝑂 𝑇 As displayed in Figure 4, the white circle region, the TS’s ship domain, defines all of the TS’s possible positions when a collision occurs [28]. Once 𝑉 falls into the Relative 𝑂𝑇 collision cone (RCC), the collision between ships exists. On the contrary, there is no risk of collision existing between ships. The absolute collision cone (ACC, purple region) is presented by RCC along 𝑉 , which implies that the 𝑉 dropping into ACC is equivalent 𝑇 𝑂 to 𝑉 falling into RCC. In other words, the collision between ships exists when 𝑉 falls 𝑂𝑇 𝑂 into the ACC, which can be shown as Equation (11) considering that the VO algorithm is time-varying: ⃗⃗⃗⃗⃗(⃗⃗𝑡⃗ ) ⃗⃗⃗⃗⃗⃗(⃗⃗𝑡⃗ ) (t) (11) 𝑉 ∈ VO = RCC ⊗ 𝑉 𝑜 𝑇 where operation ⊗ is the Minkowski addition. However, since it ignores the characteris- Appl. Sci. 2022, 12, 8926 9 of 22 tics of ship manoeuvring and the particular navigation environment, typical VO cannot be directly applied to ship collision avoidance even though it is very effective and simple directly applied to ship collision avoidance even though it is very effective and simple in in identifying the presence of a collision. Therefore, an IVO is presented, which combines identifying the presence of a collision. Therefore, an IVO is presented, which combines typical VO, MMG, and COLREGs. typical VO, MMG, and COLREGs. Figure 4. The diagram of the VO method. Figure 4. The diagram of the VO method. The ship’s movement is non-linear due to the large inertia of its properties, especially during signiﬁcant changes in course and speed [34]. Therefore, an IVO model based on the The ship’s movement is non-linear due to the large inertia of its properties, especially consideration of the ship-manoeuvring motion characteristics is proposed to quantify the during significant changes in course and speed [34]. Therefore, an IVO model based on VO set in the encounter situation, which includes telegraph and course being able to be the consideration of the ship-manoeuvring motion characteristics is proposed to quantify manoeuvred by the ship. the VO set in the encounter situation, which includes telegraph and course being able to The typical VO set is shown as the red region in Figure 5a, which is linear and appropriate for dynamic collision avoidance methods under ideal conditions. The outer be manoeuvred by the ship. and inner of the circle denote the maximum and minimum speeds of the ship, respectively, The typical VO set is shown as the red region in Figure 5a, which is linear and appro- which are based on the manoeuvrability parameter of the propeller. In practice, the priate for dynamic collision avoidance methods under ideal conditions. The outer and maximum and minimum speeds are maintained by the telegraph of “Full. ahead” (F) of the ship inner ando “Dead f the slow circl ahead” e deno (D t), e rth espectively e maximum , and the and other minim speed for um collision speedavoidance s of the ship, respectively, can be controlled by the “Half. ahead” (H) and “Slow. ahead” (S). Except for these, other telegraphs are seldom used to avoid collisions based on good seamanship since their slow speeds make it impossible to maintain the course. Therefore, an IVO algorithm shown in Figure 5b is provided combined with the four telegraphs and a typical VO, which can be used to avoid collision based on the ship’s manoeuvrability. Four sectors denote the VO set of the different telegraphs, which can be expressed as followed: VO = ACC T > F F VO = ACC T H H (12) VO = ACC T S S VO = ACC T D D where shows mathematical expression. T , T , T , and T denote telegraph F, H, S, F H S D and D, respectively. VO (purple area), VO (yellow area), VO (blue area), and VO (red F H S D area) denote the VO sets of telegraph F, H, S, and D, respectively. Appl. Sci. 2022, 12, x FOR PEER REVIEW 10 of 23 which are based on the manoeuvrability parameter of the propeller. In practice, the max- imum and minimum speeds are maintained by the telegraph of “Full. ahead” (F) of the ship and “Dead slow ahead” (D), respectively, and the other speed for collision avoidance can be controlled by the “Half. ahead” (H) and “Slow. ahead” (S). Except for these, other telegraphs are seldom used to avoid collisions based on good seamanship since their slow speeds make it impossible to maintain the course. Therefore, an IVO algorithm shown in Figure 5b is provided combined with the four telegraphs and a typical VO, which can be used to avoid collision based on the ship’s manoeuvrability. Four sectors denote the VO set of the different telegraphs, which can be expressed as followed: 𝑉 𝑂 = 𝐴𝐶𝐶 ⊕ 𝑇 𝐹 𝐹 𝑉 𝑂 = 𝐴𝐶𝐶 ⊕ 𝑇 𝐻 𝐻 { (12) 𝑉 𝑂 = 𝐴𝐶𝐶 ⊕ 𝑇 𝑆 𝑆 𝑉 𝑂 = 𝐴𝐶𝐶 ⊕ 𝑇 𝐷 𝐷 where ⊕ shows mathematical expression. 𝑇 , 𝑇 , 𝑇 , and 𝑇 denote telegraph F, H, S, 𝐹 𝐻 𝑆 𝐷 Appl. Sci. 2022, 12, 8926 10 of 22 and D, respectively. 𝑉 𝑂 (purple area), 𝑉 𝑂 (yellow area), 𝑉 𝑂 (blue area), and 𝑉 𝑂 𝐹 𝐻 𝑆 𝐷 (red area) denote the VO sets of telegraph F, H, S, and D, respectively. Figure 5. The diagram of the IVO method: (a) Traditional VO set; (b) Non-linear VO set. Figure 5. The diagram of the IVO method: (a) Traditional VO set; (b) Non-linear VO set. 3. Autonomous Collision Avoidance Method 3. Autonomous Collision Avoidance Method 3.1. Collision Risk Index 3.1. Collision Risk Index The CRI [32], is a physical quantity for evaluating the risk of a ship collision, used to The CRI [32], is a physical quantity for evaluating the risk of a ship collision, used to determine the severity of the collision threat between ships, especially in the multi-ship determine the severity of the collision threat between ships, especially in the multi-ship encounter scenario. The two elements of the CRI are the time of collision risk index (TCRI) encounter scenario. The two elements of the CRI are the time of collision risk index (TCRI) and the space of collision risk index (SCRI). The SCRI is the likelihood of two ships colliding and in the space, space wher ofeas collis theion TCRI risk is index the urgency (SCRI) of. collision The SCRI avoidance is the lik measur elihood eso which f two must ships be collid- taken by the two ships at risk of collision. The range of CRI is from 0 to 1, where the risk of ing in space, whereas the TCRI is the urgency of collision avoidance measures which must collision is increasing with the value. The SCRI can be determined as follows: be taken by the two ships at risk of collision. The range of CRI is from 0 to 1, where the risk of collision is increasing with the value. The SCRI can be determined as follows: 1,9(x, y) 2 Domain \ t 2 [0, TCPA] u = (13) st t 𝑡 𝑡 1,∃(𝑥 ,𝑦 ) ∈ 𝐷𝑜𝑚𝑎𝑖 𝑛 ∩ 𝑡 ∈ [0,𝑇𝐶𝑃𝐴 ] 0,8(x, y) 2 / Domain \ t 2 [0, TCPA] 𝑢 = { (13) 𝑠𝑡 𝑡 𝑡 0,∀(𝑥 ,𝑦 ) ∉ 𝑎𝑖𝐷𝑜𝑚 𝑛 ∩ 𝑡 ∈ [0,𝑇𝐶𝑃𝐴 ] where u denotes the function of SCRI and (x, y) denotes the position of TS at t. The st where 𝑢 denotes the function of SCRI and (𝑥 ,𝑦 ) denotes the position of TS at 𝑡 . The 𝑠𝑡 t Domain denotes the ship domain of OS at t. The expression of TCRI is given as follows: 𝐷𝑜𝑚𝑎𝑖 𝑛 denotes the ship domain of OS at 𝑡 . The expression of TCRI is given as follows: >1, TCS 0 1, 𝑇𝐶𝑆 ≤ 0 3.03 TCS 𝑇𝐶𝑆 u = (14) (1 ) , 0 < TCS < t tT 0 t 3.03 𝑢 = {(1 − ) , 0 < 𝑇𝐶𝑆 < 𝑡 (14) 𝑡𝑇 0 0, TCS t 0, 𝑇𝐶𝑆 ≥ 𝑡 where u means the function of TCRI. Time to the close-quarters situation (TCS) is the tT where 𝑢 means the function of TCRI. Time to the close-quarters situation (TCS) is the 𝑡𝑇 duration from the current moment to the ﬁrst time-in-point of the close-quarters situation duration from the current moment to the first time-in-point of the close-quarters situation (FTCS), whereas the t is the duration from the ﬁrst time-in-point of collision risk (FTCR) (FTCS), whereas the 𝑡 is the duration from the first time-in-point of collision risk (FTCR) to FTCS. When the OS takes the most effective collision avoidance action to make other ships merely pass on the edge of the ship domain is referred to as the FTCS. Regardless of the OS’s actions, if this time elapses, the TS will enter the ship domain of the OS. Thus, the CRI model can be established by combining the TCRI and SCRI as follows: u = u u (15) T st tT 3.2. Dynamic Feasible Manoeuvring Interval The feasible manoeuvring decision interval, which contains all the decision-making for collision avoidance of ships, can be provided by the IVO algorithm based on the improved MMG. If both ships maintain their original course and speed, as shown in Figure 6, a risk of collision between ships is possible. The C and C are the minimal course changes of the 0 1 feasible manoeuvring decision to the port and starboard side respectively at the present Appl. Sci. 2022, 12, x FOR PEER REVIEW 11 of 23 to FTCS. When the OS takes the most effective collision avoidance action to make other ships merely pass on the edge of the ship domain is referred to as the FTCS. Regardless of the OS’s actions, if this time elapses, the TS will enter the ship domain of the OS. Thus, the CRI model can be established by combining the TCRI and SCRI as follows: 𝑢 = 𝑢 ⋅ 𝑢 (15) 𝑇 𝑠𝑡 𝑡𝑇 3.2. Dynamic Feasible Manoeuvring Interval The feasible manoeuvring decision interval, which contains all the decision-making for collision avoidance of ships, can be provided by the IVO algorithm based on the im- Appl. Sci. 2022, 12, 8926 11 of 22 proved MMG. If both ships maintain their original course and speed, as shown in Figure 6, a risk of collision between ships is possible. The 𝐶 and 𝐶 are the minimal course 0 1 changes of the feasible manoeuvring decision to the port and starboard side respectively timing. The moving target will enter (not enter) the OS’s domain if the angle is lower at the present timing. The moving target will enter (not enter) the OS’s domain if the angle (gr is lower eater) (g than reater) this. than this. Figure 6. Feasible manoeuvring decision interval. Figure 6. Feasible manoeuvring decision interval. However, under the constraint of COLRGEs, some decisions should be discarded However, under the constraint of COLRGEs, some decisions should be discarded in in marine practice. Figure 7 illustrates collision avoidance decision-making under the marine practice. Figure 7 illustrates collision avoidance decision-making under the COLREGs constraint in a variety of encounter scenarios. The dotted line indicates the side COLREGs constraint in a variety of encounter scenarios. The dotted line indicates the side of the course followed by the COLREGs, with Figure 7a indicating an overtaking situation, of the course followed by the COLREGs, with Figure 7a indicating an overtaking situation, Figure 7b indicating a head-on situation, and Figure 7c indicating a cross situation. The Figure 7b indicating a head-on situation, and Figure 7c indicating a cross situation. The course is followed by the rules illustrated by the dotted line. Thus, the dynamically feasible course is followed by the rules illustrated by the dotted line. Thus, the dynamically feasi- manoeuvring interval of OS is offered for preventing collision action based on the MMG ble manoeuvring interval of OS is offered for preventing collision action based on the model, COLREGs, and IVO algorithm. The interval made at the time t is described as MMG model, COLREGs, and IVO algorithm. The interval made at the time 𝑡 is described follows, which includes all feasible ship-manoeuvring decisions: as follows, which includes all feasible ship-manoeuvring decisions: 2 3 t t t t 𝑡 𝑡 𝑡 𝑡 C(,𝐶 T,𝑇 , .).,...,.,C(𝐶 , T,𝑇 ) 1 1F 𝐹 𝑚 F𝐹 6 7 t 𝑡 t t t 𝑡 𝑡 𝑡 C(,𝐶 T,𝑇 ,) .,....,., C(𝐶 ,,T𝑇 ) 6 7 1 1H 𝐻 n H𝐻 6 7 𝐷 = (16) 𝑡 𝑡 𝑡 D = 6 7 (16) t(𝐶 t,𝑇 ),...,(𝐶 t ,𝑇 t ) 𝑆 𝑝 𝑆 C , T1 , . . . , C , T 6 7 1 S S 𝑡 𝑡 𝑡 𝑡 4 5 (𝐶 ,𝑇 ),...,(𝐶 ,𝑇 ) [ ] 1 𝐷 𝑞 𝑆 t t t t C , T , . . . , C , T 1 D S where 𝐶 means the altering course which 𝐶 ∈ (−90,90) in the overtaking situation and ∈ (0,90) in the head-on and cross situation. 𝐷 denotes OS’s dynamic feasible where C means the altering course which C 2 (90, 90) in the overtaking situation and DC 2 (0, 90) in the head-on and cross situation. D denotes OS’s dynamic feasible manoeuvring interval at a time t. m, n, p and q denotes the total number of the sets at the telegraph of F, H, S and D, respectively. Furthermore, combined with the model of CRI, the dynamic feasible manoeuvring intervals are still appropriate for collision avoidance in the multi-ship scenario. There are generally two types of collision avoidance for TSs in the multi-ship scenario. One is that OS should take measures to avoid collision for one TS where the value of CRI is greater than the threshold. Although, a variety of TSs are in danger for OS, where the value of CRI of some TSs is too low to take collision avoidance at this moment. Therefore, in this situation, OS just avoids collision for the one TS at the same time. The other is that OS should take measures to avoid collision for two or more ships at once where their values of CRI are all 𝛥𝐶 Appl. Sci. 2022, 12, 8926 12 of 22 greater than the threshold. Therefore, the dynamically feasible manoeuvring interval of TSs is the intersection of the manoeuvring interval of each TS, which is shown as Equation (17): Appl. Sci. 2022, 12, x FOR PEER REVIEW 12 of 23 t t t t t t D = D \ D D D \ D (17) 1 2 i m1 where m denotes the total number of TSs with the higher value of CRI and D denotes the manoeuvring interval at a time 𝑡 . 𝑚 、𝑛 、𝑝 and 𝑞 denotes the total numberi of the sets dynamically feasible manoeuvring interval of TS . at the telegraph of 𝐹 , 𝐻 , 𝑆 and 𝐷 , respectively. i Figure 7. Decisions in different encountering situations: (a) Overtaking situation; (b) Head-on situ- Figure 7. Decisions in different encountering situations: (a) Overtaking situation; (b) Head-on ation; (c) cross situation. situation; (c) cross situation. Furthermore, combined with the model of CRI, the dynamic feasible manoeuvring in- 3.3. Optimisation Evaluation Function tervals are still appropriate for collision avoidance in the multi-ship scenario. There are gen- According to the provided algorithm, a variety of decisions in the dynamic feasible ma- erally two types of collision avoidance for TSs in the multi-ship scenario. One is that OS noeuvring interval should be made at any given time, which can safely avoid obstructions should take measures to avoid collision for one TS where the value of CRI is greater than and ships. The two criteria of economy and timeliness are also taken into consideration the threshold. Although, a variety of TSs are in danger for OS, where the value of CRI of when choosing collision avoidance decision options for ships as evaluation indicators on some TSs is too low to take collision avoidance at this moment. Therefore, in this situation, the basis of safety. OS just avoids collision for the one TS at the same time. The other is that OS should take The ship’s timeliness can be measured by how long it takes to complete the collision measures to avoid collision for two or more ships at once where their values of CRI are all avoidance action. There is less risk of collision during an encounter stage when the time greater than the threshold. Therefore, the dynamically feasible manoeuvring interval of TSs value is short. The function of timeliness can be shown as follows: is the intersection of the manoeuvring interval of each TS, which is shown as Equation (17): Dt 𝑡 𝑡 𝑡 𝑡 (i) 𝑡 𝑡 𝐷 = 𝐷 ∩ 𝐷 ⋅⋅⋅ 𝐷 ⋅⋅⋅ 𝐷 ∩ 𝐷 (17) 1 f =2 𝑖 𝑚 −1 𝑚 (18) max(Dt) where 𝑚 denotes the total number of TSs with the higher value of CRI and 𝐷 denotes where Dt means the period consumed by the collision avoidance action of i and f denotes the dynam (i)ically feasible manoeuvring interval of . the timeliness. The economy mostly pertains to the ship’s fuel usage during collision avoidance, 3.3. Optimisation Evaluation Function which is primarily shown in the steering amplitude, which can be determined as follows: According to the provided algorithm, a variety of decisions in the dynamic feasible manoeuvring interval should be made at any given time, which can safely avoid obstruc- f = Dc (19) tions and ships. The two criteria of economy and timeliness are also taken into considera- tion when choosing collision avoidance decision options for ships as evaluation indicators where Dc means the amplitude of altering course, and the lower the amplitude, the less on the basis of safety. energy the ship avoidance operation costs. The ship’s timeliness can be measured by how long it takes to complete the collision In conclusion, the optimisation evaluation function is calculated decision-making in avoidance action. There is less risk of collision during an encounter stage when the time terms of ship timeliness and economy, which can be expressed as follows: value is short. The function of timeliness can be shown as follows: F = a f + b f (20) Δ𝑡 2 (𝑖 ) 𝑓 = (18) 𝑚𝑎𝑥 (Δ𝑡 ) where a and b represent the weights of timeliness and economy indicators, respectively, where Δ𝑡 means the period consumed by the collision avoidance action of 𝑖 and 𝑓 and a + b (= 𝑖 ) 1. 1 denotes the timeliness. The smaller the evaluation function F, the better the decision option. When the weight The economy mostly pertains to the ship’s fuel usage during collision avoidance, a of timeliness is greater, the ship takes less time in avoidance decision, which means which is primarily shown in the steering amplitude, which can be determined as follows: that the potential danger in the whole process of collision avoidance is decreased. When 𝑓 = Δ𝑐 (19) 𝑇𝑆 Appl. Sci. 2022, 12, 8926 13 of 22 the weight b of the economy is greater, the steering amplitude is greater, which means the safety distance between ships is larger. But the time of ship avoidance increases, and the potential danger also increases. Therefore, it satisﬁes different situation requirements to adjust the corresponding weights of timeliness and economy indicators according to different scenarios. 3.4. Collision Avoidance Decision A model is proposed for the challenging subject of multi-ship collision avoidance, which integrates the improved MMG model and COLREGs. These decisions encompass Appl. Sci. 2022, 12, x FOR PEER REVIEW 14 of 23 altering course and changing speed, and the ﬂowchart of decision-making for collision avoidance is shown in Figure 8. Figure 8. Decision-making of collision avoidance. Figure 8. Decision-making of collision avoidance. 4. Case Study Step 1 is the input module which is used to collect the data of initial ship motion by the ship’s current navigation equipment, the M means the number of TS. 4.1. Simulation of Improved MMG Model Step 2 is the risk detection module, which uses a typical VO algorithm to judge whether After finishing the MMG factor modification, a panama maximum size bulk carrier or not the collision risk exists by the typical VO algorithm. is designated as the simulated ship, with detailed information as indicated in Table 2. In- Step 3 is the CRI module, which is used to determine the timing and priority of the itially, the speed and course are 12.5 knots and 000°, respectively. TSs avoidance, based on the value of CRI between OS and TS. Step 4 is to obtain the dynamically feasible manoeuvring decisions, D , based on the Table 2. Principal particulars and manoeuvrability characteristics of OS. IVO, COLREGs, and MMG. Parameters Values Parameters Values Step 5 is to identify the ship’s ﬁnal manoeuvring decision by the optimisation evalua- tion function. Length Overall/m 225 Propeller number 1 Breadth/m 32.5 Propeller advance/m 4.738 4. Case Study Draft/m 14.5 Acreage of Rudder/m 56.88 4.1. Simulation of Improved MMG Model Block coefficient 0.8715 Propeller blade number 4 After ﬁnishing the MMG factor modiﬁcation, a panama maximum size bulk carrier is Propeller diameter/m 6.8 Propeller type Fixed pitch designated as the simulated ship, with detailed information as indicated in Table 2. Initially, Propeller pitch/m 2.81 blade ratio 0.91 the speed and course are 12.5 knots and 000 , respectively. As illustrated in Figure 9a, turning one circle takes 927 s, and the final speed is 5.75 knots (2.96 m/s), a 54 percent drop in speed. The advance and transfer of the ship are, respectively, 785 m (3.48 L) and 430 m (1.91 L), where L represents the length overall (LOA) of OS. Figure 9b shows that the first decrease in speed is rapid and that the speed at the last trend is constant. When comparing the decrease in speed, transfer, advance, and other factors under the turning circle in the real world, the precision meets the colli- sion avoidance criterion at sea. Figure 10 shows that under different RPMs of the propel- ler, the real ship speed and the simulation speed in MMG are closer, and the parameter setting of the MMG model has fulfilled the requirements of ship collision avoidance. Appl. Sci. 2022, 12, 8926 14 of 22 Table 2. Principal particulars and manoeuvrability characteristics of OS. Parameters Values Parameters Values Length Overall/m 225 Propeller number 1 Breadth/m 32.5 Propeller advance/m 4.738 Draft/m 14.5 Acreage of Rudder/m2 56.88 Block coefﬁcient 0.8715 Propeller blade number 4 Propeller diameter/m 6.8 Propeller type Fixed pitch Propeller pitch/m 2.81 blade ratio 0.91 As illustrated in Figure 9a, turning one circle takes 927 s, and the ﬁnal speed is 5.75 knots (2.96 m/s), a 54 percent drop in speed. The advance and transfer of the ship are, respectively, 785 m (3.48 L) and 430 m (1.91 L), where L represents the length overall (LOA) of OS. Figure 9b shows that the ﬁrst decrease in speed is rapid and that the speed at the last trend is constant. When comparing the decrease in speed, transfer, advance, and other factors under the turning circle in the real world, the precision meets the collision avoidance criterion at sea. Figure 10 shows that under different RPMs of the propeller, the Appl. Sci. 2022, 12, x FOR PEER REVIEW 15 of 23 Appl. Sci. 2022, 12, x FOR PEER REVIEW 15 of 23 real ship speed and the simulation speed in MMG are closer, and the parameter setting of the MMG model has fulﬁlled the requirements of ship collision avoidance. Figure 9. Simulation of turning circle. (a) ship trajectory at starboard 35°; ( b) tendency of speed. Figure 9. Simulation of turning circle. (a) ship trajectory at starboard 35°; (b) tendency of speed. Figure 9. Simulation of turning circle. (a) ship trajectory at starboard 35 ; (b) tendency of speed. Figure 10. Difference between real ship and simulation ship. Figure 10. DifferFigure ence between 10. Diffr erence eal ship betwee and simulation n real ship ship. and simulation ship. Four turning circles are illustrated in Figure 11a, which is taken by different telegraphs. Four turning circles are illustrated in Figure 11a, which is taken by different telegraphs. The gap between different turning circles is small because the ship’s speed has minimal in- The gap between different turning circles is small because the ship’s speed has minimal in- fluence on the advance of the ship’s turn in actuality. When the speed tends to be steady, a fluence on the advance of the ship’s turn in actuality. When the speed tends to be steady, a ship controlled by telegraph from D to F at 1200 s. The tendency of speed change is in align- ship controlled by telegraph from D to F at 1200 s. The tendency of speed change is in align- ment with the practice of navigation in reality. Therefore, the accuracy of improved MMG ment with the practice of navigation in reality. Therefore, the accuracy of improved MMG conforms to the requirement of intelligent collision avoidance. conforms to the requirement of intelligent collision avoidance. Appl. Sci. 2022, 12, 8926 15 of 22 Four turning circles are illustrated in Figure 11a, which is taken by different telegraphs. The gap between different turning circles is small because the ship’s speed has minimal inﬂuence on the advance of the ship’s turn in actuality. When the speed tends to be steady, a ship controlled by telegraph from D to F at 1200 s. The tendency of speed change is in Appl. Sci. 2022, 12, x FOR PEER REVIEW 16 of 23 alignment with the practice of navigation in reality. Therefore, the accuracy of improved MMG conforms to the requirement of intelligent collision avoidance. Figure 11. Simulation of the different telegraphs. (a) Turing circle at different telegraph; (b) Speed Figure 11. Simulation of the different telegraphs. (a) Turing circle at different telegraph; (b) Speed change from D to F. change from D to F. 4.2. Simulation Result 4.2. Simulation Result Two groups of case studies are conducted to demonstrate the feasibility of the pro- Two groups of case studies are conducted to demonstrate the feasibility of the proposed posed method. Every case study includes four TSs, which are in different encountering method. Every case study includes four TSs, which are in different encountering situations. situations. 4.2.1. Scenario 1 4.2.1. Scenario 1 Initial information of relative ship about the multi-ships scenario case 1 is illustrated in Initial information of relative ship about the multi-ships scenario case 1 is illustrated Table 3. In this case, according to COLRGEs, OS has a responsibility to move out of the way in Table 3. In this case, according to COLRGEs, OS has a responsibility to move out of the with TS in a crossing position and TS in a head-on situation, and TS in the overtaking 1 2 3 way with TS1 in a crossing position and TS2 in a head-on situation, and TS3 in the overtaking situation. The dynamic feasible manoeuvring interval is shown in Figure 12, where the situation. The dynamic feasible manoeuvring interval is shown in Figure 12, where the black black colour, red colour, green colour, and blue colour denote the D telegraph, S telegraph, colour, red colour, green colour, and blue colour denote the D telegraph, S telegraph, H H telegraph, and F telegraph, respectively. telegraph, and F telegraph, respectively. Table 3. Initial information of the ships for simulation in scenario 1. Ship List Position (mile) Course ( ) Speed (knot) LOA (m) TCS CRI OS (0, 0) 000 12.5 225 - - TS1 (2, 2) 300 8.0 225 725 0.202 TS2 (0, 4.4) 181 6.0 299 623 0.035 TS3 (0.2, 2.0) 000 5.5 225 743 0.393 TS4 (2, 0) 000 6.5 245 0 0 OS takes measures to avoid collision by altering course 37 to the starboard side at the “H” telegraph, and Figure 13a shows the ship’s path to avoid the current collision for multi-ships, which is shown in different colours. However, OS will introduce additional collision risk with TS while avoiding collision for TS , TS and TS . Thus, OS should 4 1 2, 3 take caution to alter course to the starboard side for the collision avoidance of TS , and the relative distance between the OS and the TS is shown in Figure 13b, in which P , P , P 1 2 3, and P are represented as the minimum distance of relative TS, respectively. Figure 13d Figure 12. Dynamic feasible manoeuvring interval of scenario 1. Appl. Sci. 2022, 12, x FOR PEER REVIEW 16 of 23 Figure 11. Simulation of the different telegraphs. (a) Turing circle at different telegraph; (b) Speed change from D to F. 4.2. Simulation Result Two groups of case studies are conducted to demonstrate the feasibility of the pro- posed method. Every case study includes four TSs, which are in different encountering situations. Appl. Sci. 2022, 12, 8926 16 of 22 4.2.1. Scenario 1 Initial information of relative ship about the multi-ships scenario case 1 is illustrated in Table 3. In this case, according to COLRGEs, OS has a responsibility to move out of the represents the OS speed change during alternation operation. Figure 13c demonstrates the way with TS1 in a crossing position and TS2 in a head-on situation, and TS3 in the overtaking general tendency of DCPA between OS and TS. Under the guidance of the telegraph, the Appl. Sci. 2022, 12, x FOR PEER REVIEW 17 of 23 situation. The dynamic feasible manoeuvring interval is shown in Figure 12, where the black ship’s speed gradually decelerates, initially rapidly, then slowly and steadily, as seen in colour, red colour, green colour, and blue colour denote the D telegraph, S telegraph, H Figure 13d. telegraph, and F telegraph, respectively. Table 3. Initial information of the ships for simulation in scenario 1. Ship List Position (mile) Course (° ) Speed (knot) LOA (m) TCS CRI OS (0, 0) 000 12.5 225 - - TS1 (2, 2) 300 8.0 225 725 0.202 TS2 (0, 4.4) 181 6.0 299 623 0.035 TS3 (0.2, 2.0) 000 5.5 225 743 0.393 TS4 (2, 0) 000 6.5 245 0 0 OS takes measures to avoid collision by altering course 37° to the starboard side at the “H” telegraph, and Figure 13a shows the ship’s path to avoid the current collision for multi- ships, which is shown in different colours. However, OS will introduce additional collision risk with TS4 while avoiding collision for TS1, TS2, and TS3. Thus, OS should take caution to alter course to the starboard side for the collision avoidance of TS4, and the relative distance between the OS and the TS is shown in Figure 13b, in which P1, P2, P3, and P4 are represented as the minimum distance of relative TS, respectively. Figure 13d represents the OS speed change during alternation operation. Figure 13c demonstrates the general tendency of DCPA between OS and TS. Under the guidance of the telegraph, the ship’s speed gradually Figure 12. Dynamic feasible manoeuvring interval of scenario 1. Figure dece12. leraDynamic tes, initialfeasible ly rapidl manoeuvring y, then slowlyinterval and stea of discenario ly, as seen 1. in Figure 13d. Figure 13. Simulation results of scenario 1: (a) Trajectory of OS and TSs; (b) Distance between OS Figure 13. Simulation results of scenario 1: (a) Trajectory of OS and TSs; (b) Distance between OS and different TSs; (c) DCPA between OS and different TSs; (d) Speed of OS. and different TSs; (c) DCPA between OS and different TSs; (d) Speed of OS. 4.2.2. Scenario 2 4.2.2. Scenario 2 Table 4 illustrates the ship’s initial information about the multi-ship scenario case 2. Table 4 illustrates the ship’s initial information about the multi-ship scenario case 2. For the large ship of TS1, TS2, and TS4, OS is the give-way vessel taking manoeuvres to For the large ship of TS , TS , and TS , OS is the give-way vessel taking manoeuvres 1 2 4 avoid the collision. However, OS is the stand-on vessel in the encountering situation with to avoid the collision. However, OS is the stand-on vessel in the encountering situation Appl. Sci. 2022, 12, 8926 17 of 22 Appl. Sci. 2022, 12, x FOR PEER REVIEW 18 of 23 with TS . The dynamic feasible manoeuvring decision interval calculated by the IVO algorithm is illustrated in Figure 14, where the “D” telegraph, “S” telegraph, “H” telegraph, TS3. The dynamic feasible manoeuvring decision interval calculated by the IVO algorithm and “F” telegraph are represented by the black colour, red colour, green colour, and blue is illustrated in Figure 14, where the “D” telegraph, “S” telegraph, “H” telegraph, and “F” colour, respectively. telegraph are represented by the black colour, red colour, green colour, and blue colour, respectively. Table 4. Initial information of the ships for simulation in scenario 2. Table 4. Initial information of the ships for simulation in scenario 2. Ship list Position (mile) Course ( ) Speed (knot) LOA (m) TCS (s) CRI Ship list Position (mile) Course (° ) Speed (knot) LOA (m) TCS (s) CRI OS (0, 0) 000 12.5 225 - - OS (0, 0) 000 12.5 225 - - TS1 (1, 5.0) 200 14.5 225 451 0.009 TS1 (1, 5.0) 200 14.5 225 451 0.009 TS2 (0, 5.5) 175 14.5 299 569 0.001 TS2 (0, 5.5) 175 14.5 299 569 0.001 TS3 (1.0, 3.5) 145 9.0 225 410 0.124 TS3 (−1.0, 3.5) 145 9.0 225 410 0.124 TS4 (0.3, 1.5) 000 8.0 245 852 0.525 TS4 (0.3, 1.5) 000 8.0 245 852 0.525 Figure 14. Dynamic feasible manoeuvring interval of scenario 2. Figure 14. Dynamic feasible manoeuvring interval of scenario 2. Figure 15a shows the ship’s motion trajectory when the OS avoids collision by alter- Figure 15a shows the ship’s motion trajectory when the OS avoids collision by altering ing course 35° to the starboard side. The ship’s relative distance curves between TS1, TS2, course 35 to the starboard side. The ship’s relative distance curves between TS , TS , TS , 1 2 3 TS3, and TS4 are shown in Figure 15b, which reach the lowest point in P1, P2, P3, and P4. and TS are shown in Figure 15b, which reach the lowest point in P , P , P , and P . The 4 1 2 3 4 The minimum relative distance is greater than the safety value, which means the TS is not minimum relative distance is greater than the safety value, which means the TS is not invading the ship domain. Figure 15c shows the differences in DCPA between this ship invading the ship domain. Figure 15c shows the differences in DCPA between this ship and and the TS, which shrinks at first and then increases when the course is altered. As shown Appl. Sci. 2022, 12, x FOR PEER REVIEW 19 of 23 the in Figure TS, which 15d, tshrinks he tendenc aty ﬁrst of OS and spee then d is incr variabl eases e in when the who the le pro course cedure is o alter f altering ed. As shown in course and then tends to be stable when finishing the altering action. Figure 15d, the tendency of OS speed is variable in the whole procedure of altering course and then tends to be stable when ﬁnishing the altering action. Figure 15. Cont. Figure 15. Simulation results of scenario 2: (a) Trajectory of OS and TSs; (b) Distance between OS and different TSs; (c) DCPA between OS and different TSs; (d) Speed of OS. 5. Analysis and Discussion 5.1. Results and Discussion The core algorithm for avoiding collisions, which combines the improved MMG and VO algorithms, was proposed in the preceding section. Two groups of case studies are of- fered to demonstrate the effectiveness of the basic algorithm in various encounter circum- stances in this paper. The results and discussion are as follows. In the situation of a multi-vessel encounter, the ship’s avoidance action should not only evaluate the vessels with whom the ship is at risk of collision with, but should also take into account whether this action will generate a new danger of collision with other TS. In Sce- nario 1, the action taken by OS to avoid collision with TS1, TS2, and TS3, will impede the safe movement of TS4; In Scenario 2, when taking measures to present the TS in danger, the ships must take into account the TS3, although the risk of collision does not exist. In some circumstances, changing speed can successfully lessen the amplitude of the alternation of course, which is highly advantageous for confined seas. In Scenario 1, only three decisions by altering course to the starboard side to avoid collision for all the TSs, and the minimum degree (41°) in the manner of only altering course are greater than the minimum degree (37°) in the manner of both alterations of course and speed. However, in Scenario 2, only altering course is the optimal decision for collision avoidance. Thus, alternation of speed is a benefit for collision avoidance in some special circumstances. The dynamic feasible manoeuvring interval is shown in Figure 12 The outcome demonstrates that decision-making at distinct telegraphs may be used to avoid collision in common scenarios. The evaluation function is used to determine the optimal decision based on timeliness and economy. From Figures 13b and 15b, the relative distance be- tween OS and TS shows that TS is not invaded its ship domain. In all case investigations, the minimum distance between OS and TS at various deci- sion-making points is more than 2(𝐿 + 𝐿 ), indicating that TS does not invade OSs ship 1 2 domain. Furthermore, the resulting DCPA is significantly bigger than the safety Appl. Sci. 2022, 12, x FOR PEER REVIEW 19 of 23 Appl. Sci. 2022, 12, 8926 18 of 22 Figure 15. Simulation results of scenario 2: (a) Trajectory of OS and TSs; (b) Distance between OS Figure 15. Simulation results of scenario 2: (a) Trajectory of OS and TSs; (b) Distance between OS and different TSs; (c) DCPA between OS and different TSs; (d) Speed of OS. and different TSs; (c) DCPA between OS and different TSs; (d) Speed of OS. 5. Analysis and Discussion 5. Analysis and Discussion 5.1. Results and Discussion The core algorithm for avoiding collisions, which combines the improved MMG 5.1. Results and Discussion and VO algorithms, was proposed in the preceding section. Two groups of case studies The core algorithm for avoiding collisions, which combines the improved MMG and are offered to demonstrate the effectiveness of the basic algorithm in various encounter VO algorithms, was proposed in the preceding section. Two groups of case studies are of- circumstances in this paper. The results and discussion are as follows. fered to dIn em the on situation strate th of e a efmu feclti-vessel tivenessencounter of the b,athe sic ship’s algoriavoidance thm in va action riousshould encounot nter circum- only evaluate the vessels with whom the ship is at risk of collision with, but should also stances in this paper. The results and discussion are as follows. take into account whether this action will generate a new danger of collision with other TS. In the situation of a multi-vessel encounter, the ship’s avoidance action should not only In Scenario 1, the action taken by OS to avoid collision with TS , TS , and TS , will impede 1 2 3 evaluate the vessels with whom the ship is at risk of collision with, but should also take into the safe movement of TS ; In Scenario 2, when taking measures to present the TS in danger, accouthe nt ships wheth must er ttake his a into ctio account n will the gen TS era , talthough e a newthe darisk ngeof r o collision f collisi does on w not ith exist. other TS. In Sce- In some circumstances, changing speed can successfully lessen the amplitude of the nario 1, the action taken by OS to avoid collision with TS1, TS2, and TS3, will impede the safe alternation of course, which is highly advantageous for conﬁned seas. In Scenario 1, only movement of TS4; In Scenario 2, when taking measures to present the TS in danger, the ships three decisions by altering course to the starboard side to avoid collision for all the TSs, must take into account the TS3, although the risk of collision does not exist. and the minimum degree (41 ) in the manner of only altering course are greater than the In some circumstances, changing speed can successfully lessen the amplitude of the minimum degree (37 ) in the manner of both alterations of course and speed. However, in Scenario 2, only altering course is the optimal decision for collision avoidance. Thus, alternation of course, which is highly advantageous for confined seas. In Scenario 1, only alternation of speed is a beneﬁt for collision avoidance in some special circumstances. three decisions by altering course to the starboard side to avoid collision for all the TSs, The dynamic feasible manoeuvring interval is shown in Figure 12 The outcome demon- and the minimum degree (41°) in the manner of only altering course are greater than the strates that decision-making at distinct telegraphs may be used to avoid collision in common minimum degree (37°) in the manner of both alterations of course and speed. However, scenarios. The evaluation function is used to determine the optimal decision based on in Scen timeliness ario 2, and only economy altering . Fr om course Figur ies s t 13 hb e and opt15 im b, al the deci relative sion distance for collis between ion avo OSid and ance. Thus, TS shows that TS is not invaded its ship domain. alternation of speed is a benefit for collision avoidance in some special circumstances. In all case investigations, the minimum distance between OS and TS at various The dynamic feasible manoeuvring interval is shown in Figure 12 The outcome decision-making points is more than 2(L + L ), indicating that TS does not invade OSs 1 2 demonstrates that decision-making at distinct telegraphs may be used to avoid collision ship domain. Furthermore, the resulting DCPA is signiﬁcantly bigger than the safety thresh- in common scenarios. The evaluation function is used to determine the optimal decision old. In light of the aforementioned case studies, it is apparent that the presented collision avoidance algorithm is highly adaptable and credible in dealing with ship collision risk in based on timeliness and economy. From Figures 13b and 15b, the relative distance be- a variety of typical encounter situations. Moreover, changing the speed of the telegraph tween OS and TS shows that TS is not invaded its ship domain. can be suitable for the most complicated situation. As a result, the fundamental algorithm In all case investigations, the minimum distance between OS and TS at various deci- is very advantageous for the ship in conﬁned waters. sion-making points is more than 2(𝐿 + 𝐿 ), indicating that TS does not invade OSs ship The main contributions of this paper 1 ar2 e drawn in two aspects based on simulation domai results: n. Fur On therm the one orehand, , the the res impr ulti oved ng DC MMG PA is mor is e sifeasible gnificantly and valuable biggerwhen than con- the safety sidering the inﬂuence of ship manoeuvrability by changing speed. On the other hand, an intelligent collision avoidance model for ships is provided based on the IVO algorithm and improved MMG. In this paper, the decision-making including altering course and changing Appl. Sci. 2022, 12, 8926 19 of 22 speed takes into account COLREGs, ship manoeuvrability, and sea practice, which are reliable and useful for collision avoidance in multi-ship collision. 5.2. Comparison Analysis In this research, decision-making for intelligent collision avoidance of ships based on ship manoeuvring process deduction is proposed. This method can efﬁciently resolve ship collisions in a variety of multi-ship encounter scenarios. In the past, many method- ologies and techniques for the collision-free problem were presented by various relevant scholars [16,20,25,28,31,35,36], however, this paper offers unique diversity and superiority compared to the previous studies. A lot of related studies pay more attention to the collision avoidance for a single ship or open sea, and ignore the multi-ship scenario and conﬁned sea [31,36,37]. Compared to a single ship, collision avoidance is quite complicated in multi-ship circumstances. Although some studies [26–29] have provided some algorithms and models for the multi- ship situation which consider the ship motion model insufﬁciently, most of current research pays less attention on the model in which the ship’s speed and rotation of propeller are not matched. The core model provides two beneﬁts for decision-making of collision avoidance. One is using the CRI model to identify the most dangerous ship out of a group of ships with varying degrees of collision risk. Another is that the decision-making by the proposed algorithm will not create additional collision risk, which will not impede the safe navigation of other TSs. In addition, few studies consider the characteristics of ship motion in the variable speed of the ship. Therefore, it is highly necessary to deduce the next state of OS for collision avoidance. In this research, an improved MMG model is introduced to derive the ships’ manoeuvre motion process, which reduces the gap between the collision-free algorithm and practical applications. Furthermore, the COLREGs and good seamanship are also taken into account for the proposed method for intelligent collision avoidance. In summary, this paper presents an intelligent collision avoidance method based on an IVO algorithm, which utilises alternation of course and changing of speed to present collision for the ship. Many factors are taken into account, including the improved MMG model, COLRGEs, and good seamanship. Moreover, this model can resolve the conﬂict between ships in complicated circumstances, which is also effective for multi-ship en- counter situations. 6. Conclusions This research offers an intelligent ship collision avoidance model based on IVO and improved MMG, which take full account of COLREGs and good seamanship. In order to avoid multi-ship collision, a CRI model is established on the basis of SCRI and TCRI. The dynamic feasible manoeuvring interval is determined by combining with IVO and COLRGEs, which include all the decision-making for collision avoidance of ships. On this basis, the decision of optimisation is determined by the optimisation evaluation function, which includes altering course and changing speed. In addition, a series of scenarios including various encounter circumstances with multi-ships are shown to demonstrate the efﬁcacy and viability of this methodology. Although the proposed methodology has some achievements, it also has some limi- tations. The telegraph about stern telegraphs and “Stop” is not considered in the speed alternation since ship manoeuvres at low speeds are difﬁcult to predict and the lowest speed of the vessel should be maintained to keep the course in the navigation for safety. Furthermore, collision avoidance only considers changing both speed and course at the same moment, but in the real-world situation, additional judgments must be made. In order to improve the dependability and efﬁcacy of the proposed strategy in a real- world environment, we plan to consider the ship motion model at low speeds to avoid collision in future work. Furthermore, adjusting the speed and course at different times and changing the course more than once is considered for future development. Appl. Sci. 2022, 12, 8926 20 of 22 Author Contributions: Conceptualization, L.H. and Y.H.; Methodology, X.Z. and Y.H.; formal analysis, X.Z. and X.L.; investigation X.Z. and Y.H.; Writing original draft, X.Z. and Y.H.; Writing review & editing, X.L. and K.Z.; Visualization, X.Z.; Supervision, Y.H. and J.M.; funding acquisition, L.H. and Y.H. All authors have read and agreed to the published version of the manuscript. Funding: This work is partially supported by the National Key Research and Development Program (Grant number: 2019YFB1600603) and the National Natural Science Foundation of China (Grant number: 52071249). This research was ﬁnancially supported by “Study on Decision-making Method of Ship Autonomous Navigation in Inland Typical Waterways”(Item Number B2021519). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of Interest. Nomenclature Symbols Deﬁnitions Symbols Deﬁnitions L length of the OS I moment of inertia o zz L length of the TS i additional moment of inertia T zz a major axial lengths of the ellipse X vertical axis directions b short axial lengths of the ellipse Y horizontal axis directions c distance from the imaginary ship to the fact ship elative course N turning moment m total mass of the vessel H hull m additional vertical mass P propeller m additional horizontal mass R rudder u lateral speed T force of propeller u lateral acceleration DT force of propeller derating v longitudinal speed t thrust derating coefﬁcient p0 v longitudinal acceleration u designated speed r angular velocity u real-time speed r angular acceleration t propeller derating factor at u p0 B ship’s width f function of the effect of ship motion d ship draft x vertical coordinate of buoyancy L ship’s length b drift angle at the rudder D diameter of propeller b drift angle at the ship’s center g rectiﬁcation coefﬁcient of the hull Minkowski addition l ﬂoating centre coefﬁcient F Full. ahead cb C block coefﬁcient H Half. ahead l drift angle coefﬁcient. S Slow. ahead K factor of thrust D Dead slow ahead r density of water; T Full. ahead D diameter of the propeller T Half. ahead p H NP real-time rotation rate of the propeller T Slow. ahead V velocity of the OS T Dead slow ahead V velocity of the TS (x, y) position of TS at t u function of SCRI Domain ship domain of OS at t st u function of TCRI. C altering course tT References 1. Chen, P.; Huang, Y.; Mou, J.; van Gelder, P.H.A.J.M. Ship Collision Candidate Detection Method: A Velocity Obstacle Approach. Ocean Eng. 2018, 170, 186–198. [CrossRef] 2. Li, J.; Wang, H.; Zhao, W.; Xue, Y. Ship’s Trajectory Planning Based on Improved Multiobjective Algorithm for Collision Avoidance. J. Adv. Transp. 2019, 2019, 4068783. [CrossRef] 3. Eleftheria, E.; Apostolos, P.; Markos, V. Statistical Analysis of Ship Accidents and Review of Safety Level. Saf. Sci. 2016, 85, 282–292. [CrossRef] 4. Statheros, T.; Howells, G.; McDonald-Maier, K. 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Intelligent Collision Avoidance Method for Ships Based on COLRGEs and Improved Velocity Obstacle Algorithm

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ISSN: 2076-3417
DOI: 10.3390/app12188926
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Abstract

Journal

Applied Sciences – Multidisciplinary Digital Publishing Institute

Published: Sep 6, 2022

Keywords: intelligent collision avoidance; improved velocity obstacle; COLREGs; variable speed; improved MMG

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Intelligent Collision Avoidance Method for Ships Based on COLRGEs and Improved Velocity Obstacle Algorithm

Intelligent Collision Avoidance Method for Ships Based on COLRGEs and Improved Velocity Obstacle Algorithm

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Intelligent Collision Avoidance Method for Ships Based on COLRGEs and Improved Velocity Obstacle Algorithm

Intelligent Collision Avoidance Method for Ships Based on COLRGEs and Improved Velocity Obstacle Algorithm

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