COLREGs-Compliant Distributed Stochastic Search Algorithm for Multi-Ship Collision Avoidance

Abstract

The increasing complexity of maritime traffic imposes growing demands on the safety and rationality of ship-collision-avoidance decisions. While most existing research focuses on simple encounter scenarios, autonomous collision-avoidance strategies that comply with the International Regulations for Preventing Collisions at Sea (COLREGs) in complex multi-ship environments remain insufficiently investigated. To address this gap, this study proposes a novel collision-avoidance framework that integrates a quantitative COLREGs analysis with a distributed stochastic search mechanism. The framework consists of three core components: encounter identification, safety assessment, and stage classification. A cost function is employed to balance safety, COLREGs compliance, and navigational efficiency, incorporating a distance-based weighting factor to modulate the influence of each target vessel. The use of a distributed stochastic search algorithm enables decentralized decision-making through localized information sharing and probabilistic updates. Extensive simulations conducted across a variety of scenarios demonstrate that the proposed method can rapidly generate effective collision-avoidance strategies that fully comply with COLREGs. Comprehensive evaluations in terms of safety, navigational efficiency, COLREGs adherence, and real-time computational performance further validate the method’s strong adaptability and its promising potential for practical application in complex multi-ship environments.

1. Introduction

1.1. Research Background

Maritime transport accounts for more than 90% of global trade, serving as an indispensable component of the world economy due to its operational efficiency, cost-effectiveness, and relatively favorable environmental impact [1]. However, with the increasing size and speed of vessels, maritime traffic congestion has intensified, leading to an increase in ship collision incidents that pose significant safety concerns. Accident analyses reveal that approximately 80% of these collisions are attributable to navigators’ operational errors and misinterpretations of the COLREGs [2]. Consequently, improving collision-avoidance capabilities and minimizing human error have emerged as critical priorities. In recent years, recent advances in artificial intelligence and big data analytics have provided innovative solutions for intelligent collision-avoidance systems, which are increasingly recognized as pivotal enablers in the transition toward autonomous and smart shipping paradigms [3].

As maritime traffic density continues to escalate, encounter scenarios involving multiple vessels have grown markedly more complex. Conventional single-vessel-centric collision-avoidance strategies exhibit limited effectiveness in addressing the dynamic and interactive risks inherent in multi-ship environments [4]. To achieve safer and more efficient decision-making, there is an urgent need for a distributed coordination approach that integrates both the local navigation status and global avoidance cooperation. In multi-ship encounter situations, collision-avoidance decisions must not only ensure navigational safety but also strictly adhere to the COLREGs and the principles of good seamanship [5]. Against this backdrop, this study focuses on a distributed COLREGs-compliant collision-avoidance method for multi-ship environments, aiming to enhance the autonomous decision-making capabilities of intelligent vessels and improve the overall safety and coordination of intelligent maritime systems.

1.2. Summary and Analysis of Related Research

Several review studies have systematically summarized the fundamental concepts and key technologies of ship collision avoidance [6,7,8,9]. Driven by rapid advances inn artificial intelligence, decision-making theories originally developed for unmanned systems have been increasingly applied to maritime navigation.

Over the past decades, significant progress has been made in intelligent ship collision avoidance, with single-ship proactive strategies being the most representative approach. These methods typically adopt the own-ship perspective to plan avoidance maneuvers for one or more target vessels. A representative approach is the Velocity Obstacle (VO) method [10], which calculates unsafe velocity sets based on the relative motion of nearby ships, thereby enabling the determination of collision-free velocity options. An enhanced variant, Optimal Reciprocal Collision Avoidance (ORCA) [11], demonstrates superior geometric computational efficiency and real-time performance. The Artificial Potential Field (APF) method [12,13] simulates attractive and repulsive forces to guide ships toward goals while avoiding obstacles. However, it tends to fall into local minima in complex environments, limiting its applicability. Model Predictive Control (MPC) [14] incorporates multi-step prediction and constraint optimization, demonstrating strong robustness under dynamic and uncertain conditions. Recently, Deep Reinforcement Learning (DRL) [15,16] has enabled agents to autonomously learn complex collision avoidance policies with strong adaptability to uncertainty. Additionally, some studies have explored AIS-based trajectory mining and behavior modeling [17] for predicting potential conflicts and supporting decision-making. However, traditional single-ship methods primarily focus on individual optimization, often neglecting the interaction and coordination among vessels. In multi-ship encounter scenarios, such independent decision-making may result in uncoordinated global strategies and even generate new potential conflicts.

In contrast to single-ship strategies, centralized coordination approaches focus on enhancing consistency and global optimality in multi-ship collision avoidance. These methods generally assume that all ships are coordinated by a central authority, formulating the problem as a multi-objective optimization task and coordinating ship actions from a global perspective to achieve an overall optimal solution. Liu et al. [18] proposed a hybrid collision-avoidance algorithm that integrates Particle Swarm Optimization and Bacterial Foraging. Li et al. [19] formulated a multi-ship receding horizon optimization method aimed at minimizing the heading deviation and time cost in collision-avoidance maneuvers. Han et al. [20] optimized ship collision avoidance using an enhanced VO model and employed Non-dominated Sorting Genetic Algorithm II (NSGA-II) to derive safe cooperative maneuvers under safety constraints. Although centralized multi-objective optimization offers theoretically optimal global decisions, it relies on the strong assumption that all ships strictly follow centralized commands—an assumption often impractical at sea. Furthermore, simultaneously solving for all ships’ decision variables results in a rapid escalation of dimensionality and computational complexity as the number of vessels increases, thereby rendering real-time global optimization increasingly challenging. More importantly, globally optimal solutions may not align with individual ships’ locally optimal preferences. Without stringent external enforcement and coordination mechanisms, such solutions are unlikely to be reliably implemented by all vessels [7].

In recent years, the increasing complexity of multi-ship dynamic encounters has underscored the significance of distributed cooperative collision-avoidance algorithms. Several formulations have been proposed for distributed cooperative collision avoidance, among which one notably formulates the problem as a Distributed Constraint Optimization Problem (DCOP), allowing each autonomous agent to independently compute its collision-avoidance decisions within a decentralized framework. This reduces reliance on centralized control and enhances system autonomy and robustness [4]. The mainstream DCOP solution algorithms include the Synchronous Branch and Bound (SyncBB) [21], the Dynamic Programming Optimization Protocol (DPOP) [22], and the Asynchronous Forward Bounding (AFB) [23]. Various distributed coordination algorithms have been successfully applied to ship collision avoidance. Among them, the Distributed Tabu Search Algorithm (DTSA) [24] and the Distributed Local Search Algorithm (DLSA) [25] enable coordinated heading adjustment among multiple ships through synchronized local search mechanisms. Rong et al. [26] further proposed a density-based collision risk-assessment method, integrated within the DLSA framework to solve multi-ship collision-avoidance strategies. However, in dense maritime environments, such synchronous search mechanisms often require frequent communication rounds, making them prone to delays and synchronization bottlenecks, which limits their practical applicability. To address this issue, Zhang et al. [27] proposed the Distributed Stochastic Algorithm (DSA), which has demonstrated excellent convergence and robustness in DCOP problems such as graph coloring and task scheduling. In recent years, DSA has been applied to multi-ship collision-avoidance scenarios, demonstrating high coordination efficiency and real-time adaptability, highlighting its strong potential in complex and dynamic maritime environments [28,29]. In addition, recent studies have investigated another formulation where the Alternating Direction Method of Multipliers (ADMM) for distributed collision avoidance. [30,31]. Although ADMM has demonstrated satisfactory performance in certain ship-avoidance scenarios, it requires each agent to solve a complex nonlinear continuous optimization problem with constraints, leading to higher computational costs and slower convergence rates in dynamic multi-ship environments. In comparison with DSA, these limitations constrain its applicability in time-sensitive collision-avoidance tasks [7].

Table 1. The summary of the literature review.

Type	Reference	Technique	Multi-Ship Encounter	Computational Overhead	COLREGs Compliance
Single-ship Strategies Method	[10,11]	Velocity obstacle method	No	Low	Partially
	[12,13]	Artificial potential field method	No	Low	No
	[14]	Model predictive	No	High	Yes
	[15,16]	DRL in continuous action space	No	High	No
	[17]	AIS historical data mining	No	High	Partially
Centralized Coordination Method	[18]	Hybrid swarm method	Yes	High	No
	[19]	Receding horizon optimization	Yes	High	Yes
	[20]	Multi-objective evolutionary method	Yes	High	No
Distributed Coordination Method	[24]	Distributed tabu search algorithm	Yes	Low	No
	[25,26]	Distributed local search algorithm	Yes	Low	No
	[28,29]	Distributed stochastic search algorithm	Yes	Low	No
	[30,31]	Alternating direction method of multipliers	Yes	High	Partially
	This study	DSA-based considering the COLREGs	Yes	Low	Yes

summarizes the main categories of collision-avoidance methods. As shown, each approach has inherent limitations: single-ship strategies prioritize individual optimality but lack systematic consideration of multi-ship interactions and coordination mechanisms, often resulting in conflict escalation. Centralized coordination approaches offer global optimization potential but rely heavily on centralized control architectures, which are ill-suited for real-world maritime environments where vessels operate autonomously. Furthermore, their computational complexity increases rapidly with the number of participating ships, compromising real-time performance and practical feasibility. In contrast, distributed cooperative approaches exhibit superior scalability, autonomy, and robustness. Notably, DSA-based methods exhibit high coordination efficiency and rapid responsiveness. However, most existing studies lack explicit modeling and compliance mechanism design for COLREGs, thereby limiting their ability to ensure compliant and interpretable decision-making in real-world applications. Therefore, there is a urgent need for a decision-making framework that integrates COLREGs compliance with distributed optimization capabilities to enable safe, efficient, and controllable multi-ship cooperation in complex and dynamic environments.

1.3. The Motivation for the Study

The rationality and coordination of collision-avoidance decision-making in multi-ship scenarios remain critical challenges in the field of intelligent maritime navigation. This study addresses these challenges by proposing a distributed decision-making framework that facilitates cooperation among multiple ships while ensuring strict adherence to the COLREGs. The objective is to enable intelligent navigation systems to achieve safe, compliant, and efficient collision avoidance. The main contributions of this paper are summarized as follows:

• A quantitative modeling framework for COLREGs is proposed, encompassing encounter situation identification, collision risk assessment, and phase division of the collision-avoidance process. This framework establishes precise mapping between regulatory provisions and the collision-avoidance behaviors of autonomous ship agents, thereby enhancing their comprehension of navigational rules while improving the interpretability and compliance of their decision-making processes;
• A comprehensive collision avoidance cost function integrating multiple factors is constructed, considering collision risk, COLREGs compliance, and navigation efficiency. A dynamic weight-adjustment mechanism based on relative distance is introduced to more accurately reflect the varying threat levels posed by each encountered vessel in a multi-ship environment, providing a unified and regulation-aware evaluation criterion for distributed optimization;
• A distributed multi-ship cooperative collision-avoidance decision-making framework is developed under a decentralized communication structure. Ship agents cooperate through local information exchange and probabilistic strategy updates. This method has the ability to escape local optima, effectively reduces communication frequency, and is suitable for intelligent cooperative navigation in large-scale autonomous ship systems.

The rest of the paper is structured as follows. Section 2 introduces the quantitative modeling approach for COLREGs and good seamanship. Section 3 describes the architecture of the distributed stochastic search algorithm and formulates the composite collision-avoidance cost function. Section 4 presents and analyzes the experimental results. Section 5 concludes the study and outlines directions for future research.

2. Quantitative Analysis of COLREGs

To align this study more closely with realistic maritime scenarios, the following reasonable assumptions were adopted:

• All ships are assumed to navigate in open waters under calm sea conditions and good visibility. The navigational environment excludes fixed obstacles such as islands, buoys, and other stationary obstructions, focusing solely on interactions with surrounding dynamic target ships;
• All ships are equipped with Automatic Identification System (AIS) transceivers that enable real-time transmission and reception of both static and dynamic navigational data. AIS also allows each vessel to broadcast its navigational intent, thereby facilitating information interoperability and sharing among ships.

Before designing a multi-ship distributed collision-avoidance decision-making system, it is essential to fully consider the constraints and guidance of the International Regulations for Preventing Collisions at Sea (COLREGs) and good seamanship on vessel behavior in two-ship encounter scenarios. This ensures that decision-making in multi-ship situations aligns with the guidance of the COLREGs and remains consistent with practical navigation. In the context of the collision-avoidance process, the decision-making task for each involved target vessel under the COLREGs can be distilled into two key aspects:

• What type of encounter situation (e.g., head-on, crossing, overtaking) and what is the current stage (e.g., close-quarter situation, imminent danger, etc.) that exists between the own ship and the target ship?
• After identifying their respective roles as either the give-way vessel or the stand-on vessel, what collision-avoidance actions should be taken to comply with the relevant avoidance requirements specified in the COLREGs?

By summarizing insights from the relevant literature [5,32,33] and the key provisions of the COLREGs (as presented in

Table 2. Mapping of relevant COLREGs to modules in collision avoidance decision-making.

Decision-Making Algorithm Module	The Relevant Provisions Supporting the Quantification Method of Rules
Situation awareness	Part B: Rule 5, 7
Encounter situation identification	Part B: Rule 7, 13, 14, 15; Part C: Rule 21, 22
Role recognition and responsibility assessment	Part B: Rule 2, 8, 16, 17
Collision avoidance action generation	Part B: Rule 2, 8, 13, 14, 15, 16, 17

), this study establishes a rule-based quantitative analysis framework from three perspectives: ship encounter situation identification, collision risk assessment, and classification of collision-avoidance stages. Appendix A provides detailed descriptions of the ship motion parameters and symbols used in this study.

2.1. Ship Encounter Situation

2.1.1. Ship Encounter Situation Identification

The COLREGs do not provide explicit definitions for multi-ship encounter scenarios. Consequently, in practice, such scenarios are usually simplified as a combination of multiple two-ship encounters. Empirical evidence indicates that inconsistent encounter judgments frequently result in poor coordination or collisions. The COLREGs classify possible two-ship encounter situations into three categories—overtaking (Rule 13), head-on (Rule 14), and crossing (Rule 15)—and provide corresponding criteria for their identification. In previous studies, the relative bearing angle Q_mn of the target ship with respect to the own ship was commonly used as the basis for encounter situation determination. For example, in study [34], when the bearing of the target ship falls within the range of [0°, 5°] or [355°, 360°], the situation is classified as a head-on encounter. However, this approach is inconsistent with the actual provisions of the COLREGs. As illustrated in Figure 1, although the bearings of the four target ships all fall within the above-defined “head-on area,” their actual encounter types differ: ship 1 represents a typical head-on situation, ship 2 is overtaking, and ships 3 and 4 are crossing. These discrepancies highlight that relying solely on one-way relative bearing for encounter classification can lead to significant misjudgment.

In this study, to achieve more accurate encounter situation identification, both Q_mn and Q_nm are considered in the judgment process. By comprehensively analyzing the bidirectional relative bearing information and determining the navigational responsibilities between ships, the typical encounter situations are refined into six categories: (1) head-on, (2) overtaking, (3) being overtaken, (4) small-angle crossing, (5) large-angle crossing, and (6) stand-on ship in a crossing situation. Figure 2 and

Table 3. Encounter situation identification model detailed description.

Encounter Situation				Limitations
PCR exists and TCPA > 0	Head-on			${\mathrm{Q}}_{\mathrm{mn}}\in \left[0 , {5.7}^{°}\right] \cup \left[{354.3}^{°},{ 360}^{°}\right)$, ${\mathrm{Q}}_{\mathrm{nm}}\in \left[0 ,{ 5.7}^{°}\right] \cup \left[{354.3}^{°},{ 360}^{°}\right)$; ${\mathrm{Q}}_{\mathrm{mn}}\in \left[0 , {112.5}^{°}\right]$, ${\mathrm{Q}}_{\mathrm{nm}}\in \left[0 ,{ 112.5}^{°}\right]$; ${\mathrm{Q}}_{\mathrm{mn}}\in \left[{247.5}^{°},{ 360}^{°}\right]$, ${\mathrm{Q}}_{\mathrm{nm}}\in \left[{247.5}^{°},{ 360}^{°}\right]$
	Overtaking	Give-way		${\mathrm{Q}}_{\mathrm{mn}}\in \left[0 ,{ 90}^{°}\right] \cup \left[{270}^{°},{ 360}^{°}\right)$, ${\mathrm{Q}}_{\mathrm{nm}}\in \left[{112.5}^{°},{ 247.5}^{°}\right]$
		Stand-on		${\mathrm{Q}}_{\mathrm{mn}}\in \left[{112.5}^{°}, {247.5}^{°}\right]$, ${\mathrm{Q}}_{\mathrm{nm}}\in \left[0 ,{ 90}^{°}\right] \cup \left[{270}^{°},{ 360}^{°}\right)$
	Crossing	Small-angle	Give-way	${\mathrm{Q}}_{\mathrm{mn}}\in \left[0 , {67.5}^{°}\right]$, ${\mathrm{Q}}_{\mathrm{nm}}\in \left[{247.5}^{°}, {360}^{°}\right)$
			Stand-on	${\mathrm{Q}}_{\mathrm{mn}}\in \left[{247.5}^{°},{ 360}^{°}\right)$, ${\mathrm{Q}}_{\mathrm{nm}}\in \left[0 , {67.5}^{°}\right]$
		Large-angle	Give-way	${\mathrm{Q}}_{\mathrm{mn}}\in \left[{67.5}^{°}, {112.5}^{°}\right]$, ${\mathrm{Q}}_{\mathrm{nm}}\in \left[{247.5}^{°},{ 360}^{°}\right)$
			Stand-on	${\mathrm{Q}}_{\mathrm{mn}}\in \left[{247.5}^{°}, {360}^{°}\right)$, ${\mathrm{Q}}_{\mathrm{nm}}\in \left[{67.5}^{°},{ 112.5}^{°}\right]$
No PCR or TCPA ≤ 0

respectively illustrate the structure of the encounter situation recognition model proposed in this study and the discriminant conditions for various encounter types. The incorporation of bidirectional bearing analysis significantly improves the accuracy of situation recognition and ensures that all ship agents adopt a consistent judgment basis for encounter classification within the distributed decision-making algorithm. This enhances the overall coordination of collision-avoidance behavior and ensures compliance with COLREGs.

The Potential Collision Risk (PCR) introduced in Section 2.2 quantifies the likelihood of a potential collision likelihood. Large and small-angle crossing encounters are differentiated using a bearing threshold of 67.5°. Furthermore, considering that the relative speed between two ships in a head-on situation is typically high and the greater effectiveness of mutual collision-avoidance actions in reducing collision risk, this study extends the angular threshold for head-on encounters by a half-compass point (11.25°) to enhance the safety margin of situation recognition [35]. In addition, two special encounter scenarios not covered by the previously defined categories are considered: starboard-to-starboard and port-to-port approaches. Although their bearings fall outside the half-compass-point range, these scenarios still pose a potential collision risk and are thus classified as head-on encounters due to their similar characteristics.

2.1.2. Collision-Avoidance Responsibilities Under Different Encounter Situations

In open sea environments, large vessels generally operate at economical speeds. As most ships are equipped with fixed-pitch propellers, the combined effects of main engine protection systems and ship inertia prevent them from significantly reducing speed within a short time frame [35]. In addition, such speed changes are difficult for surrounding ships to detect in time, which can lead to ambiguous collision-avoidance intentions and uncoordinated actions between the ships. In contrast, steering maneuvers provide a faster response and convey clearer intent, making it easier for other vessels to understand and coordinate their own actions. According to Rule 8 of the COLREGs, effective course alterations alone constitute a more practical and recognizable form of collision avoidance. Consequently, as this study focuses exclusively on course-changing maneuvers in the decision-making model for open-sea scenarios.

According to Rules 13 to 17 of the COLREGs, when PCR exists between two ships, the responsibilities of the give-way ship and the stand-on ship are clearly defined. Based on these rules and the typical encounter types defined in Section 2.1.1, this study develops the collision avoidance action constraints (Figure 3) to standardize the avoidance strategies of ships with different responsibilities in multi-ship encounter scenarios.

In overtaking situations, as defined in Rule 13 of the COLREGs, the overtaking ship assumes full responsibility for ensuring collision avoidance. It is required to alter course to port or starboard (Figure 3a) while maintaining a safe distance. The ship being overtaken is obligated maintain its course and speed.

In head-on situations, Rule 14 requires both vessels to alter course to starboard (Figure 3b). This reciprocal responsibility reduces the risk of misjudgment and enhances avoidance efficiency.

In crossing situations, Rule 15 states that if the other ship is on the starboard side, the own ship is the give-way ship and must take action to avoid collision, while the stand-on ship maintains course and speed. Typical strategies for small-angle and large-angle crossings are shown in Figure 3c,d.

2.2. Collision Risk Assessment Method

To more effectively assess collision risk and reflect the requirements of the COLREGs, this study divides the surrounding sea area into four concentric zones centered on the ship, as illustrated in Figure 4. From the outermost to the innermost, these zones are defined as the detection zone, action zone, close-quarters zone, and collision zone.

The outermost detection zone, with a radius of 10 nautical miles, initiates information exchange once a target ship enters. The implementation of this mechanism in a distributed system is discussed in detail in Section 3.1. As noted in the guide to the collision-avoidance rules [36], “the assessment of collision risk does not apply to targets that are too distant.” Therefore, in accordance with relevant COLREGs provisions and the principles of good seamanship, a radius of 6 nautical miles is set as the boundary of the action zone, serving as one of the triggering conditions for collision risk identification. The innermost collision zone represents the area where a physical collision occurs and is strictly prohibited from being entered by any vessel. A collision is typically defined as the condition in which the distance between the centers of two ships is less than or equal to half the sum of their lengths [35]. To ensure safe navigation, ships must maintain a specific safe distance of approach (SDA) when approaching one another. In this study, the SDA is used as the boundary of the light red close-quarters zone in the model. The determination of SDA is typically influenced by various factors, including the ship size, traffic density, and the surrounding navigational environment. Given the study’s focus on open waters with low traffic density, a circular ship domain model is adopted to simplify the modeling process and reduce computational complexity. This method has been widely applied and has proven to be effective and well-suited for such environments [37,38,39,40]. In the context of two-ship collision-avoidance studies, an SDA of at least 2 nautical miles is generally recommended to ensure sufficient maneuvering space. However, in complex scenarios involving multiple ships, this standard is often difficult to achieve. To address this issue, Wu et al. [41] proposed a range of acceptable safe distances, ranging from 1.0 to 2.0 ship lengths. A statistical analysis by Zhang et al. [39] suggested that the SDA be set to approximately 0.8 nautical miles. Taking into account the open-water multi-ship characteristics and general safety requirements, this study sets the collision zone radius ${\mathrm{D}}_{\mathrm{collision}}$ at 0.2 nautical miles and delineates the SDA as 1.0 nautical miles. The officer on watch (OOW) may adjust this parameter based on real-time conditions and experience.

Within the maritime domain partitioning, this study introduces the concepts of Potential Collision Risk (PCR) and Collision Risk (CR). Specifically, if a target ship enters the detection range of the own ship, and under the assumption that both vessels maintain their current courses and speeds, the DCPA is less than the predefined SDA, and the TCPA > 0, the target is identified as posing a PCR. Such targets are not immediately threatening but indicate early risk and require continuous monitoring. Furthermore, if the target ship enters the own ship’s action zone, it is considered to present a CR, triggering the collision-avoidance decision module. The system must then promptly generate and execute avoidance commands to prevent the target from entering more dangerous zones, such as the close-quarters or collision region.

Based on this risk-assessment mechanism, the target ships within the detection area are classified into three categories: Safe Ships, Potential Risk Ships, and Collision Risk Ships. Detailed definitions and parameter-based classification criteria are provided in

Table 4. Ship risk classification and detailed explanation.

Analysis of Collision Risk	Limitations	Explain
Safe Ship	${\mathrm{DCPA}}_{\mathrm{mn}}\left(\mathrm{t}\right) \ge \mathrm{SDA}$ or ${\mathrm{TCPA}}_{\mathrm{mn}}\left(\mathrm{t}\right) \le 0$	Not posing any threat or having already passed safely.
Potential Collision Risk Ship	${\mathrm{DCPA}}_{\mathrm{mn}}\left(\mathrm{t}\right) \le \mathrm{SDA}$, ${\mathrm{TCPA}}_{\mathrm{mn}}\left(\mathrm{t}\right) > 0$, ${6 \mathrm{nm} \le \mathrm{D}}_{\mathrm{mn}}\left(\mathrm{t}\right) \le 10 \mathrm{nm}$	Vessels with PCR present but not yet in the collision avoidance action area.
Collision Risk Ship	${\mathrm{DCPA}}_{\mathrm{mn}}\left(\mathrm{t}\right) \le \mathrm{SDA}$, ${\mathrm{TCPA}}_{\mathrm{mn}}\left(\mathrm{t}\right) > 0$, ${\mathrm{D}}_{\mathrm{mn}}\left(\mathrm{t}\right) \le 6 \mathrm{nm}$	An actual collision threat exists, and immediate collision avoidance action is required.

, serving as a theoretical basis for multi-ship collision risk assessment and cooperative decision-making.

2.3. Collision-Avoidance Stage Model

2.3.1. Quantitative Analysis of Collision-Avoidance Stage

According to the COLREGs and related studies [35,42], the process of two ships approaching one another until a collision occurs can be divided into four sequential stages: the Free Action, the Collision Risk (CR), the Close-quarters Situation (CS), and the Immediate Danger (ID). Although the CS and ID are not explicitly defined in the COLREGs, their concepts are implicitly reflected in COLREGs. For instance, the warnings to avoid close-quarters situations in Rules 8 and 19 correspond to the CS stage, while the provisions on good seamanship in Rules 2 and 10 are applicable to emergency judgment in the ID stage. Each stage is associated with distinct responsibilities and behavioral requirements for collision avoidance. Stage division ensures rule compliance and facilitates a multi-ship assessment and strategy planning. Based on the previously introduced domain partitioning and risk-classification framework, a stage-based risk identification mechanism is integrated into the system. The collision process is divided into four stages, forming the basis for a progressive decision-making process. The stage definitions are shown in the Figure 5.

• Stage I: Free Action Stage

The target ship has entered the detection zone and is identified as posing a PCR, but has not yet entered the action zone, and no CR is present. The own ship retains full maneuvering freedom and may independently choose avoidance strategies without being constrained by the COLREGs.

• Stage II: Collision Risk Stage

Once the target ship enters the action zone and satisfies the criteria for CR, it is recognized as posing a real collision threat. At this stage, the system initiates avoidance actions in accordance with the COLREGs, and the ships are required to assume their respective responsibilities.

• Stage III: Close-quarters Situation Stage

At this stage, even a full avoidance maneuver by a single ship is no longer sufficient to ensure safe passage beyond the SDA. Although a collision has not yet occurred, the situation is approaching a critical threshold and requires coordinated maneuvers by both ships. The onset of this stage, referred to as the First Time of CS, is defined as the moment when the own ship executes its maximum allowable course alteration, and the resulting DCPA(t₁) equals the SDA.

• Stage IV: Immediate Danger Stage

This is the most hazardous stage, where even the maximum avoidance maneuver by one ship cannot prevent entry into the collision zone, making a physical collision unavoidable. In such cases, emergency responses beyond the scope of COLREGs are required. The First Time of ID is defined as the moment when the own ship has executed its maximum allowable avoidance maneuver, yet the DCPA(t₂) equals the radius of the collision zone.

2.3.2. Collision-Avoidance Actions at Different Stages

Based on Rules 16 and 17 of the COLREGs, as well as related literature [5,35], the collision-avoidance behavior constraints for the give-way and stand-on vessels under the four-stage model are defined as follows:

• Give-way ship actions in head-on, overtaking, and crossing situations.

Stage I: Before CR is established. The give-way ship retains full maneuvering freedom and may take any feasible action.

Stage II: Upon identification of CR, Rule 16 mandates that the give-way ship shall take early and sustained action to avoid collision, ensuring safe passing beyond the SDA and preventing escalation to a CS. Detailed responsibilities are provided in Section 2.2.

Stage III: As the situation advances toward ID, Rule 2 requires the give-way ship to execute collision-avoidance maneuvers, even if they deviate from standard rule-based responses, in order to prevent a collision.

Stage IV: Collision avoidance becomes the highest priority. The give-way ship must perform any necessary emergency maneuver, regardless of regulatory constraints.

2.

Stand-on ship actions.

Stage I: The stand-on ship, like the give-way ship retains full maneuvering flexibility before a CR is identified.

Stage II: Once a CR is present, Rule 17 instructs the stand-on ship to maintain course and speed to preserve predictability and clarify the roles of involved parties.

Stage III: To prevent the escalation to ID, Rule 17(b) allows the stand-on ship to take evasive action, provided it does not introduce new risks. At this point, the ship is authorized to maneuver.

Stage IV: If the give-way ship fails to avoid the collision independently, the stand-on ship assumes direct responsibility and must act immediately to prevent physical contact.

Table 5. Classification of collision avoidance stages and detailed explanation.

Stage Classification	Explain	Action Limitations
Stage I	Only PCR exists	Give-way ship	Free to take any available action.
		Stand-on ship
Stage II	CR exists, ${\mathrm{DCPA}}_{\mathrm{after} \mathrm{maneuver}}\left(\mathrm{t}\right) > \mathrm{SDA}$ *	Give-way ship	Collision-avoidance actions guided by the rules in Section 2.1.2.
		Stand-on ship
Stage III	CR exists, ${\mathrm{D}}_{\mathrm{collision}}$ $\le {\mathrm{DCPA}}_{\mathrm{after} \mathrm{maneuver}}\left(\mathrm{t}\right) \le \mathrm{SDA}$	Give-way ship	In an emergency, the vessel should take the most effective action under Rule 2, even if it temporarily deviates from other rules.
		Stand-on ship
Stage IV	CR exists, ${\mathrm{DCPA}}_{\mathrm{after} \mathrm{maneuver}}\left(\mathrm{t}\right) \le {\mathrm{D}}_{\mathrm{collision}}$	Give-way ship
		Stand-on ship

* ${\mathrm{DCPA}}_{\mathrm{after} \mathrm{maneuver}}\left(\mathrm{t}\right)$ refers to the DCPA at time t, after the give-way ship has executed a full avoidance maneuver. In this study, the maximum turning angle is set to 90°.

summarizes the judgment criteria and collision avoidance action guidelines for each stage.

3. Ship Collision-Avoidance Decision-Making Model Based on Distributed Stochastic Search Algorithm

3.1. Distributed Ship Collision-Avoidance Framework

The distributed collision-avoidance framework proposed in this study demonstrates strong scalability, enabling effective deployment in scenarios involving varying numbers of vessels and communication networks with arbitrary topologies. As demonstrated in Figure 6a, each ship possesses a detection range, defined as the detection zone outlined in Section 2.2, which facilitates the identification of target ships within each that range. Ships can only exchange information only when within mutual detection range. Furthermore, Figure 6b presents a communication network consisting of nine ships. The network’s more complex topology highlights the scalability and adaptability of the proposed method in multi-ship coordination scenarios.

As shown in Figure 7, the distributed ship collision-avoidance framework consists of two phases: the control phase and the search phase [29]. During the control phase, if no conflicting ships are detected, the ship proceeds to the next planned position. However, if other ships posing a collision risk are present within the detection range, the ship transitions to the search phase. In this search phase, the involved ships exchange key navigational information, including updates on collision-avoidance cost evaluations and intended heading adjustment. They then collaboratively execute a distributed collision-avoidance algorithm to generate collision-free navigation strategies and update their respective route points accordingly. These two phases alternate continuously throughout the voyage until all ships safely reach their destinations. A complete alternation of the two phases is defined as a “time step,” for which the duration can be dynamically adjusted according to factors such as the encounter type, relative position, and collision risk urgency. In emergency situations, a shorter time step can be adopted to enhance system responsiveness and enable rapid collision avoidance. For intelligent ships equipped with high-performance communication and autonomous decision-making systems, reducing the time step can further improve coordination efficiency. Conversely, traffic management systems or centralized scheduling scenarios may favor longer time steps to maintain decision stability and foresight. In order to achieve equilibrium among general collision-avoidance responsiveness, communication frequency, and system stability, the temporal resolution in this study is uniformly set to 3 min. Accordingly, each ship reassesses its course adjustments every three minutes and executes them at the next time step.

3.2. Cost and Improvement Function Design

After completing the exchange of navigational information with the target ship, the own ship evaluates the total collision avoidance cost for all feasible heading alternatives. In accordance with Rule 8 of the COLREGs, which emphasizes the use of course alteration for effective avoidance, and considering the ship’s maneuverability constraints [28,30], candidate headings crs are selected from a discrete set ranging from −45° (port) to +45° (starboard) in 5° increments. In addition, if the heading angle that directly connects the ship’s current position to its destination, denoted as (${\mathrm{crs}}_{\mathrm{dest}}$), lies within this range, it is also included in the candidate set for collision avoidance decision-making.

$$\mathrm{crs} \in \left\{-{45}^{°},-{40}^{°},\dots ,-5^{°},0^{°},+5^{°},\dots ,+{40}^{°},+{45}^{°},{\mathrm{crs}}_{\mathrm{dest}}\right\},$$

(1)

where,

$${\mathrm{crs}}_{\mathrm{dest}}\equiv \left\{\begin{array}{cc}{\mathrm{C}}_{\mathrm{dest}}-{\mathrm{C}}_{\mathrm{m}},& \mathrm{if} \left|{\mathrm{C}}_{\mathrm{dest}}-{\mathrm{C}}_{\mathrm{m}}\right| < {45}^{°}\\ \mathrm{empty},& \mathrm{otherwise}\end{array}\right.,$$

(2)

${\mathrm{C}}_{\mathrm{dest}}$ denotes the destination heading, which is defined as the direction of the line connecting the ship and its destination.

In previous studies [24,25,28], collision-avoidance costs are typically calculated by simply aggregating the weighted sum of collision risk and navigational efficiency across multiple ships. However, such a linear combination fails to capture the varying levels of influence exerted by different target ships in complex multi-ship encounter scenarios. To more faithfully incorporate the guiding principles of COLREGs, this study proposes a novel cost function that explicitly incorporates navigational safety, rule compliance, avoidance efficiency, and the dynamically weighted vessel influence based on the urgency of collision risk. This function is designed to improve the safety, compliance, and rationality of cooperative collision avoidance among multiple ships.

3.2.1. Navigational Safety Cost

In ship navigation, DCPA and TCPA quantify collision risk in the spatial and temporal dimensions, respectively. As the most intuitive indicators to evaluate the success of collision avoidance, these metrics have been widely adopted in both nautical practice and research on autonomous ship collision avoidance [26,35]. In conjunction with the safety assessment method described in Section 2, this study establishes membership functions for DCPA and TCPA and adopts a fuzzy comprehensive evaluation approach to calculate safety costs [33,43]. At time t, the spatial and temporal cost memberships between ships S_m and S_n are defined as follows:

$${\mathsf{\mu }}_{{\mathrm{DCPA}}_{\mathrm{mn}}}=\left\{\begin{array}{cc}1& {\mathrm{DCPA}}_{\mathrm{mn}} \le {\mathrm{D}}_{\mathrm{collision}}\\ {\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}\sin \left[{\displaystyle \frac{\mathsf{\pi }}{\mathrm{SDA} - {\mathrm{D}}_{\mathrm{collision}}}}\left({\mathrm{DCPA}}_{\mathrm{mn}} - {\displaystyle \frac{{\mathrm{D}}_{\mathrm{collision}} + \mathrm{SDA}}{2}}\right)\right]& {\mathrm{D}}_{\mathrm{collision}} < {\mathrm{DCPA}}_{\mathrm{mn}} \le \mathrm{SDA}\\ 0& \mathrm{otherwise}\end{array}\right.$$

(3)

$${\mathsf{\mu }}_{{\mathrm{TCPA}}_{\mathrm{mn}}}=\left\{\begin{array}{cc}1& \mathrm{TCPA} \le {\mathrm{T}}_1\\ {\displaystyle \frac{{\mathrm{T}}_2 - \mathrm{TCPA}}{{\mathrm{T}}_2 - {\mathrm{T}}_1}}& {\mathrm{T}}_1 < \mathrm{TCPA} \le {\mathrm{T}}_2\\ 0& \mathrm{TCPA} > {\mathrm{T}}_2\end{array}\right. ,$$

(4)

where ${\mathrm{D}}_{\mathrm{collision}}$ and SDA denote the collision distance and the safe distance of approach, respectively, as defined in Section 2.2. Target ships with a DCPA exceeding the SDA are considered safe ships and are excluded from the distributed collision-avoidance optimization queue of the own ship. To reflect the COLREGs principle of taking early action for collision avoidance, ${\mathrm{T}}_1$ represents the latest moment for a ship to initiate avoidance maneuvers. It is defined as the time required for a ship to travel from SDA to the closest point of encounter. ${\mathrm{T}}_2$ denotes the moment when the ship begins to monitor the target ship, corresponding to the TCPA when the two ships are at the action distance ${\mathrm{D}}_{\mathrm{action}}$. The calculations of ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$ are given as follows:

$${\mathrm{T}}_1={\displaystyle \frac{\sqrt{{\mathrm{SDA}}^2 - {{\mathrm{DCPA}}_{\mathrm{mn}}}^2}}{\mathrm{R}{\mathrm{V}}_{\mathrm{mn}}}},$$

(5)

$${\mathrm{T}}_2={\displaystyle \frac{{\mathrm{D}}_{\mathrm{action}}}{{\mathrm{RV}}_{\mathrm{mn}}}},$$

(6)

Based on the above definitions, the safety cost between ships S_m and S_n at time t is defined as follows:

$${\mathrm{COST}}_{{\mathrm{safe}}_{\mathrm{mn}}}\left(\mathrm{crs}\right) = {\displaystyle \frac{1}{2}}({\mathsf{\mu }}_{{\mathrm{DCPA}}_{\mathrm{mn}}}+ {\mathsf{\mu }}_{{\mathrm{TCPA}}_{\mathrm{mn}}}),$$

(7)

3.2.2. COLREGs Compliance Cost

Collision-avoidance responsibilities vary across various encounter situations and stages. Compliance with the COLREGs should be assessed based on the specific role assigned to each ship. According to the COLREGs quantification model described in Section 2, a ship’s responsibility during the collision-avoidance process varies depending on the collision situation and the risk level. To account for this, we introduce a COLREGs compliance cost matrix ${\mathrm{COST}}_{{\mathrm{rule}}_{\mathrm{mn}}}$, which is defined in detail as shown in

Table 6. The cost of COLREGs under different situations and responsibilities.

Encounter Situation	Explain		Rule Violation Cost Matrix *
Head-on	Give-way ship		[1 1 0]
Overtaking	Give-way ship		[0 1 0]
	Stand-on ship		[1 0 1]
Crossing	Small angle crossing	Give-way ship	[1 1 0]
		Stand-on ship	[1 0 1]
	Large angle crossing	Give-way ship	[0 1 0]
		Stand-on ship	[1 0 1]

* Matrix values represent port turn, straight course, and starboard turn, respectively. A cost value of 0 indicates that the crs is within the permitted range of the rules. A cost value of 1 means that the crs is outside the permitted range of the rules and will be subject to punishment.

. The value of ${\mathrm{COST}}_{{\mathrm{rule}}_{\mathrm{mn}}}$ is set based on the action norms of the giving-way ship and the stand-on ship under different encounter situations as described in Section 2.1.

3.2.3. Navigational Efficiency Cost

During navigation, a ship should maintain a course closely aligned with the planned route. However, once the ship alters its heading, this may result in a significant deviation from the destination, unnecessary detours, or difficulty in returning to the intended heading. To ensure a reasonable collision-avoidance trajectory, this study introduces a destination efficiency cost ${\mathrm{COST}}_{\mathrm{dest}}$, defined by the angle between the intended heading and the direction toward the destination. A smaller angle indicates a lower deviation and thus a lower cost. The S_m’s ${\mathrm{COST}}_{\mathrm{dest}}$ is calculated as follows:

$${\mathrm{COST}}_{{\mathrm{dest}}_{\mathrm{m}}}\left(\mathrm{crs}\right) = {\displaystyle \frac{\left|{\mathrm{C}}_{\mathrm{dest}} - {\mathrm{C}}_{\mathrm{m}} \left(\mathrm{crs}\right)\right|}{{180}^{°}}} ,$$

(8)

3.2.4. Total Cost and Improvement Function

This study constructs a multi-factor total cost model based on three key indicators: collision risk, COLREGs compliance, and deviation from the destination. Under different collision-avoidance options of S_m, the total collision-avoidance cost between S_m and S_n is given by Equation (9). The model employs a weighted linear combination of three indicators with differentiated importance in navigational decision making. This weighting strategy is designed to dynamically balance risk avoidance, regulatory compliance, and route efficiency, thereby increasing the effectiveness of cooperative collision-avoidance decisions.

$${\mathrm{COST}}_{\mathrm{mn}}\left(\mathrm{crs}\right) = {\mathrm{COST}}_{{\mathrm{safe}}_{\mathrm{mn}}}\left(\mathrm{crs}\right) + {\mathrm{k}}_1\left({\mathrm{k}}_2 · {\mathrm{COST}}_{{\mathrm{rule}}_{\mathrm{mn}}}\left(\mathrm{crs}\right) + {\mathrm{k}}_3 · {\mathrm{COST}}_{{\mathrm{dest}}_{\mathrm{m}}}\left(\mathrm{crs}\right)\right)$$

(9)

Navigational safety is always prioritized; only when safety is ensured are COLREGs compliance and route efficiency considered. Furthermore, COLREGs compliance is weighted more heavily than the navigational economy to reflect the system’s prioritized response to safety and rule compliance in multi-ship collision-avoidance scenarios. The weights ${\mathrm{k}}_2$ and ${\mathrm{k}}_3$ are set to 0.7 and 0.3, respectively, while the value of ${\mathrm{k}}_1$ depends on the collision-avoidance stage. In Stages 1 and 2, where collision risk remains low, ${\mathrm{k}}_1$ is set to 1 to retain the influence of COLREGs compliance and navigational efficiency. In Stages 3 and 4, where the collision risk increases significantly, priority should be given to taking any effective action to avoid collision; therefore, ${\mathrm{k}}_1$ is set to 0 [37,44].

The above model effectively evaluates the collision avoidance cost between the own ship and a specific target ship. However, in multi-ship encounters, a simple linear aggregation of costs between each ship pair may overlook spatial distribution, allowing distant vessels to exert the same influence as nearby high-risk ones. To address this, a distance-based weighting indicator ${\mathrm{Dis}}_{\mathrm{mn}}\left(\mathrm{t}\right)$ is introduced, which assigns differentiated weights to each ship pair cost. This approach emphasizes the impact of nearby vessels while reducing that of distant vessels, thereby increasing the model’s sensitivity to local risks and ensuring greater stability and robustness in dynamic multi-ship environments. ${\mathrm{Dis}}_{\mathrm{mn}}\left(\mathrm{t}\right)$ is defined as follows:

$${\mathrm{Dis}}_{\mathrm{mn}}\left(\mathrm{t}\right) = {\mathrm{e}}^{\mathrm{SDA} - {\mathrm{D}}_{\mathrm{mn}}\left(\mathrm{t}\right)},$$

(10)

In a multi-ship encounter scenario, the total collision-avoidance decision cost of the own ship S_m relative to all its j neighboring ships is given by the following:

$${\mathrm{COST}}_{\mathrm{m}}(\mathrm{crs}) =\sum _{\mathrm{j}}{\displaystyle \frac{{\mathrm{Dis}}_{\mathrm{mn}}\left(\mathrm{t}\right)}{\sum _{\mathrm{n}\in \mathrm{j}}{\mathrm{Dis}}_{\mathrm{mn}}\left(\mathrm{t}\right)}}·{\mathrm{COST}}_{\mathrm{mn}}(\mathrm{crs}),$$

(11)

During the search procedure shown in Figure 7, a ship tentatively selects a heading as its intended course crs and applies the cost defined in Equation (11). However, it may further reduce this cost by adjusting its intended heading. Equation (12) calculates the maximum possible cost reduction, representing the improvement itself. This means that ship will always seek the avoidance route that yields the greatest cost reduction.

$${\mathrm{improvement}}_{\mathrm{m}} = \max \left\{{\mathrm{COST}}_{\mathrm{m}}\left(0\right) - {\mathrm{COST}}_{\mathrm{m}}\left(\mathrm{crs}\right)\right\},$$

(12)

where ${\mathrm{COST}}_{\mathrm{m}}\left(0\right)$ represents the cost of the ship continuing its current course at the present time.

3.3. Ship Collision-Avoidance Algorithm Based on Distributed Stochastic Search

3.3.1. Distributed Stochastic Search

The global workflow of the DSA-based collision avoidance decision-making algorithm (DSSA) is shown in the Figure 8. In each decision cycle, a ship sets its current heading as its intention_self and exchanges this with nearby ships. It then computes the total cost of intention_self and the maximum possible cost improvement (improvement_self) and generates a new candidate heading (new_intention_self). The candidate does not immediately replace the current intention. When improvement_self equals zero, the ship is considered satisfied. If all ships are satisfied with their intentions, the system converges to a “quiescent” state.

If a ship’s improvement_self is greater than zero, it indicates a potentially better course. The ship then accepts new_intention_self with probability p, otherwise it maintains its original intention with probability 1 − p. In this study, the probability p is set to 0.5 [28,29]. This update is performed in parallel on all ships and iterates until the system reaches quiescence or a predefined iteration limit. Compared to DLSA and DTSA, DSSA offers improved communication efficiency and concurrency. Its stochastic mechanism also enhances the ability to escape local optima, making it effective for dynamic, multi-ship collision avoidance with reduced communication overhead.

3.3.2. Distributed Ship Collision-Avoidance Decision-Making Under COLREGs

The overall framework of the proposed COLREGs-compliant distributed stochastic search algorithm for ship collision avoidance (CC-DSSA) is illustrated in Figure 9. It consists of three main modules: (1) navigation data processing and COLREGs-based situation assessment, (2) comprehensive collision-avoidance cost evaluation that integrates safety, rule compliance, navigational efficiency, and risk urgency, (3) a distributed stochastic search algorithm for determining the optimal avoidance strategy. The operational procedure is structured as the following steps.

Step 1: Each ship acquires real-time navigational-state data of itself and surrounding ships through radar, AIS, or other equivalent navigational sensor modules. The communication structure and protocol among ships are described in Section 3.1, with exchanged information including the position, speed, heading, and relative bearing.

Step 2: At each time step, each ship computes dynamic parameters—including distance, TCPA, and DCPA—with respect to nearby target ships, based on current state information and the method described in Appendix A Subsequently, following the COLREGs quantitative model outlined in Table 3, Table 4 and Table 5 in Section 2, the ship evaluates the collision risk level for each encounter, determines the encounter type, and identifies the current collision-avoidance stage. Based on this, and in accordance with the relevant provisions of the COLREGs, the ship determines its corresponding collision avoidance responsibility.

Step 3: If none of the nearby ships are identified as posing a CR, as defined in Table 4, the ship maintains its current course and speed without taking any avoidance action. Otherwise, if any nearby ship is identified as posing a CR, the own ship will trigger the search procedure within CC-DSSA. The control and search framework of CC-DSSA is illustrated in Figure 7.

Step 4: Ships in the search phase will evaluate their collision-avoidance cost relative to its neighboring ships, considering three aspects: navigational safety, COLREGs compliance, and navigational efficiency. The interdependencies among these cost components and the formulation of the total cost function are detailed in Section 3.2.4. Each ship then selects the course option with the highest improvement value as its current optimal intention.

Step 5: CC-DSSA then activates the distributed stochastic search coordination mechanism described in Section 3.3.1, enabling ships to iteratively update and refine their navigation intentions through probabilistic local interactions. This process continues until the system reaches a quiescent state, in which all ships are considered “satisfied” with their current collision avoidance strategies—i.e., no further improvements can be achieved through local adjustments. Once convergence is achieved, each ship executes its selected navigation intention and advances to the next decision-making time step.

Step 6: Steps 1 through 5 are performed iteratively until all ships have passed through without collisions and have safely reached their respective destinations.

4. Simulation Experiment and Result Analysis

4.1. Simulation Scenario Setup

Five simulation scenarios were designed to evaluate the proposed CC-DSSA. Scenarios 1 through 3 represent typical two-ship encounters and are designed to validate the algorithm’s effectiveness and compliance with the COLREGs in these situations. Scenario 4 involves a three-ship encounter, assessing the algorithm’s adaptability in a multi-ship environment. These four scenarios were simulated based on real AIS data from typical open-sea encounters in the offshore waters near the Port of Tianjin. Scenario 5 is designed as an extreme case to comprehensively assess the algorithm’s performance in dense and complex multi-ship encounter environments. It involves five vessels converging toward a central point and encompasses various types of encounter situations. These scenarios offer representative and practical validations for autonomous collision avoidance in real-world maritime traffic.

Table 7. The initial state parameters of the ship in the experimental scenarios.

Scenario	Ship ID	Initial Position	Destination	Course (°)	Speed (Knots)	Relationship with Collision Risk
1	Ship 1	(2.2, −11.5)	(−1, 10)	351.5	20	Ship 2
	Ship 2	(1, −5.37)	(−0.2, 6)	354	10	Ship 1
2	Ship 1	(10, 10)	(−10, 10)	225	12.5	Ship 2
	Ship 2	(−10, −10)	(10, 10)	45	13	Ship 1
3	Ship 1	(−10, 0)	(10, 0)	90	12	Ship 2
	Ship 2	(0, −10)	(0, 10)	0	12	Ship 1
4	Ship 1	(8, 8)	(−8, −8)	225	10.5	Ship 2, Ship 3
	Ship 2	(8, −8)	(−8, 8)	315	10	Ship 1, Ship 2
	Ship 3	(−8, 0)	(8, 0)	90	8	Ship1, Ship 2
5	Ship 1	(−10, 10)	(10, −10)	135	12	Ship 2, Ship 3, Ship4
	Ship 2	(10, 10)	(−10, −10)	225	12	Ship1, Ship 3, Ship 4, Ship5
	Ship 3	(−10, −10)	(10, 10)	45	12	Ship 1, Ship 2, Ship 4, Ship 5
	Ship 4	(10, −10)	(−10, 10)	315	12	Ship1, Ship 2, Ship 3
	Ship 5	(−7, −7)	(8, 8)	45	7	Ship 2, Ship 3

summarizes the initial positions, headings, and speeds of the ships in each scenario. Figure 10 illustrates the spatial distribution and encounter configurations of the vessels. The arrow indicates the direction of the ship’s movement. The stars indicate the planned destinations. As shown, the initial DCPA values for several ship pairs falls significantly below the predefined safety threshold without any collision avoidance measures, with some cases showing a clear risk of direct collision.

4.2. Experimental Results of Two-Ship Encounter Situations

4.2.1. Simulation Scenario 1

Table 8. The initial DCPA between ships in Scenario 1.

Ship ID	Ship 1	Ship 2
Ship 1	—	0.011 n miles
Ship 2	0.011 n miles	—

shows the calculated DCPA values for Scenario 1, if both ships keep their initial courses and speeds. Figure 11 illustrates the simulation process of collision avoidance. Figure 11a–d depict the two vessels’ trajectories at key time steps. Figure 11e,f show the variations in the inter-ship distance and heading over time, respectively. In the initial phase, although the DCPA between the two ships was already significantly below the safety threshold, the ships remained sufficiently apart and outside the action range. As they continued to converge and entered the designated action zone, the collision avoidance algorithm was activated. Ship 1, identified as the give-way ship, executed a 5° starboard turn as its avoidance maneuver. Following this course change, Ship 1 maintained a steady heading following the avoidance action. By minute 39, the two ships had safely passed the closest point of approach, successfully completed the overtaking maneuver, and eliminated the collision risk. Ship 1 then resumed its course, adjusting its heading to 344.8°, and proceeded steadily toward its intended destination. Ship 2, as the stand-on ship, consistently maintained its original course and speed throughout the encounter without executing any evasive maneuvers. The collision-avoidance strategy generated by the algorithm effectively increased the DCPA from an initial value of 0.011 nautical miles to 1.019 nautical miles, thus surpassing the predefined SDA threshold and thereby ensuring safe navigation. In addition, all actions undertaken during the overtaking process strictly adhered to the relevant provisions of the COLREGs.

4.2.2. Simulation Scenario 2

Table 9. The initial DCPA between ships in Scenario 2.

Ship ID	Ship 1	Ship 2
Ship 1	—	0 n miles
Ship 2	0 n miles	—

lists the initial DCPA between the two ships in Scenario 2. Figure 12 depicts the ship’s trajectories and the time evolution of inter-ship distance and individual headings. As illustrated in Figure 12a–d, the two ships consistently maintained a safe distance throughout the encounter until they safely passed each other. The model precisely identified the head-on situation and determined that both vessels were give-way ships. Ships 1 and 2 executed starboard turns in accordance with Rule 14 of the COLREGs, fully satisfying their prescribed maneuvering obligations. In the early phase of the encounter, the two ships were sufficiently far apart, which corresponded to Stage I of the collision-avoidance process—both vessels were free to navigate. At minute 47, they entered the action range (Stage II), triggering the distributed collision avoidance algorithm. At this point, Ship 1 and 2 executed starboard turns of 10° and 15°, respectively, adjusting their headings to 235° and 60°. Throughout the encounter, the two ships maintained a safe separation, with a minimum distance of 1.221 nautical miles. By minute 69, the collision risk had been eliminated, and both vessels resumed their original routes by turning to 223.1° and 42.3°, respectively, proceeding steadily toward their intended destinations.

4.2.3. Simulation Scenario 3

Scenario 3 presents a typical crossing situation.

Table 10. The initial DCPA between ships in Scenario 3.

Ship ID	Ship 1	Ship 2
Ship 1	—	0 n miles
Ship 2	0 n miles	—

lists the initial DCPA values between the two ships. Figure 13a–d illustrate the trajectories at critical moments during the collision avoidance process, whereas Figure 13e,f depict the time series of the inter-ship distance and heading changes, respectively. The simulation maintained a DCPA of 1.241 nautical miles, confirming effective avoidance. At minute 30, the collision avoidance algorithm was activated. Ship 1, acting as the give-way ship in accordance with the COLREGs, executed an avoidance maneuver by turning starboard to adjust its heading to 115°. By minute 48, the collision risk had been successfully resolved, and Ship 1 resumed its course by turning port to 105°, ultimately stabilizing at a heading of 82.4° toward its intended destination. During this period, Ship 2, as the stand-on ship, maintained its original course and speed without taking any action. The entire avoidance process strictly adhered to the relevant provisions of the COLREGs.

The three representative two-ship encounter experiments demonstrate the CC-DSSA’s strengths in accurately identifying encounter situations, adhering to COLREGs, and generating safe and rational avoidance strategies. Achieving safe and compliant decisions in two-ship cases forms the foundation for extending the method to multi-ship coordination. These validation results affirm the algorithm’s reliability and support its application in more complex encounter scenarios.

4.3. Experimental Results of Multi-Ship Encounter Situations

4.3.1. Simulation Scenario 4

Table 11. The initial DCPA between ships in Scenario 4.

Ship ID	Ship 1	Ship 2	Ship 3
Ship 1	—	0.488 n miles	0.336 n miles
Ship 2	0.488 n miles	—	0.558 n miles
Ship 3	0.336 n miles	0.558 n miles	—

presents the initial DCPA values between ships in Scenario 4, all of which are substantially below the predefined safe distance, indicating a high risk of collision. Figure 14 depicts the simulation results of collision-avoidance decisions in this scenario. As illustrated in Figure 14a–d, the overall avoidance process can be divided into three distinct phases: active maneuvering, course recovery, and steady navigation toward the destination. Throughout this process, all ships performed avoidance actions strictly in accordance with COLREGs.

As depicted in Figure 14e, the distance between ships initially decreases and subsequently increases, with all final DCPA values surpassing the predefined SDA. The minimum DCPA recorded is 1.053 nautical miles, demonstrating that the algorithm ensures encounter safety while minimizing unnecessary maneuvers, thereby enhancing navigational efficiency. Figure 14f illustrates that at minute 51, the CC-DSSA generates a coordinated maneuvering scheme: Ship 1 turns starboard by 5°, Ship 2 by 15°, and Ship 3 by 35°. Subsequently, Ship 3 reaches its CPA with the other two vessels first and, after the collision risk is resolved, adjusts its heading to 120° before stabilizing at 82.7° toward its planned destination. Meanwhile, Ships 1 and 2 remain stable for a period, and at minute 66, they begin recovering to 223.8° and 311.8°, respectively, continuing toward their destinations. Many multi-ship avoidance studies adopt a single-ship focus, overlooking interactions between ships. This may result in risk transfer, uncoordinated maneuvers, and potential violations of COLREGs. In contrast, the CC-DSSA, operating within a decentralized structure, generates globally coordinated avoidance strategies that effectively balance safety, efficiency, and COLREGs compliance.

4.3.2. Simulation Scenario 5

To further evaluate the adaptability of the CC-DSSA under highly complex encounter scenarios, Scenario 5 simulates an extreme case where five vessels approach a central area from various directions. The scenario encompasses crossing, head-on, and overtaking encounters, involving multiple collision risks and COLREGs constraints.

Table 12. The initial DCPA between ships in Scenario 5.

Ship ID	Ship 1	Ship 2	Ship 3	Ship 4	Ship 5
Ship 1	—	0 n miles	0 n miles	0 n miles	1.425 n miles
Ship 2	0 n miles	—	0 n miles	0 n miles	0 n miles
Ship 3	0 n miles	0 n miles	—	0 n miles	0 n miles
Ship 4	0 n miles	0 n miles	0 n miles	—	1.425 n miles
Ship 5	1.425 n miles	0 n miles	0 n miles	1.425 n miles	—

summarizes the initial DCPA values between vessels. This experiment aims to assess whether the proposed method can generate globally coordinated collision-avoidance strategies that ensure safety, efficiency, and compliance with COLREGs and good seamanship in complex multi-ship situations. Figure 15 depicts the collision-avoidance process in Scenario 5. As illustrated in Figure 15a–e, all vessels successfully mitigated collision risks, safely passed the encounter area, and proceeded steadily toward their respective destinations. Following the execution of avoidance maneuvers, the DCPA between all ships exceeded the predefined SDA threshold, with the minimum value of 1.074 nautical miles recorded between Ships 2 and 4.

At the initial moment, Ships 3 and 5 were identified as being in an overtaking situation during stage II of collision avoidance. To prevent escalation, the search mechanism was triggered, and Ship 3, as the give-way ship, executed a starboard turn. As the maneuver took effect, the DCPA between the two ships gradually increased. To improve navigational efficiency, Ship 3 resumed its heading of 45°, completing the overtaking. During this phase, Ship 1, which was concurrently approaching both Ships 3 and 5, assumed a higher avoidance responsibility and adjusted its heading to 180°. As shown in Figure 15e,f, after the minimum distance among the three vessels was reached, all ships resumed their intended courses by minute 72 and proceeded steadily toward their destinations, with inter-ship distances continuing to widen. Throughout the encounter, Ship 5, acting as the stand-on vessel, maintained her course and speed in accordance with Rule 13 of the COLREGs, avoiding any uncoordinated maneuvers. The experimental results demonstrate that the CC-DSSA exhibits strong decision stability and effective action coordination in complex multi-ship encounter scenarios. The resulting strategies ensure safety, efficiency, and COLREGs-compliant seamanship.

4.4. Result Discussion and Comparative Analysis

To comprehensively evaluate the practical applicability and overall performance of the proposed CC-DSSA in complex multi-ship, multi-environment scenarios, this section conducts a comparative analysis between Scenario 5 and the conventional DSSA [28]. Figure 16 presents the collision-avoidance results produced by the conventional DSSA. Although the DSSA exhibits a certain level of collision-avoidance capability, several critical limitations are observed. First, some ships fail to comply with COLREGs requirements during critical situations. Second, the absence of differentiated weighting for nearby and distant targets leads to suboptimal global coordination and, in some cases, introduces new collision risks. Third, some ships demonstrate unstable course behavior due to unnecessary oscillatory maneuvers, thereby undermining overall navigational stability. A detailed comparison across four dimensions highlights the advantages of CC-DSSA: safety, COLREGs compliance, efficiency, and computation.

4.4.1. Navigation Safety

Navigational safety remains the core objective in collision-avoidance-algorithm design. In dynamic multi-ship encounter environments, insufficient safe passing distances can easily result in collisions.

Table 13. The DCPA results between ships after the operation of the conventional DSSA.

Ship ID	Ship 1	Ship 2	Ship 3	Ship 4	Ship 5
Ship 1	—	1.051 n miles	1.576 n miles	2.954 n miles	2.888 n miles
Ship 2	1.051 n miles	—	1.120 n miles	1.114 n miles	1.091 n miles
Ship 3	1.576 n miles	1.120 n miles	—	1.851 n miles	0.977 n miles
Ship 4	2.954 n miles	1.114 n miles	1.851 n miles	—	0.996 n miles
Ship 5	2.888 n miles	1.091 n miles	0.977 n miles	0.996 n miles	—

and

Table 14. The DCPA results between ships after the operation of the proposed CC-DSSA.

Ship ID	Ship 1	Ship 2	Ship 3	Ship 4	Ship 5
Ship 1	—	2.027 n miles	2.660 n miles	2.723 n miles	1.107 n miles
Ship 2	2.027 n miles	—	2.396 n miles	1.074 n miles	1.330 n miles
Ship 3	2.660 n miles	2.396 n miles	—	1.225 n miles	1.203 n miles
Ship 4	2.723 n miles	1.074 n miles	1.225 n miles	—	1.707 n miles
Ship 5	1.107 n miles	1.330 n miles	1.203 n miles	1.707 n miles	—

present the DCPA values between ship pairs after applying the conventional DSSA and the proposed CC-DSSA. Compared with the results obtained without any collision avoidance measures (Table 12), both methods achieve a significant improvement in DCPA between vessels and successfully prevent all potential collisions. However, when using the conventional DSSA, two ship pairs still exhibit DCPA values below the SDA threshold, which suggests inadequate safety margins. In contrast, the CC-DSSA consistently ensures that DCPA values remain above the SDA threshold for all vessel pairs, thereby showcasing superior performance in enhancing navigational safety.

Additionally, maintaining heading stability during the collision-avoidance process is equally critical. Frequent or unreasonable course changes can not only disrupt the planned voyage but also hinder target vessels from accurately interpreting the own ship’s avoidance intentions, thereby violating the COLREGs principle of “clear and recognizable” maneuvering. As illustrated in Figure 16, under the conventional DSSA, Ship 5—despite being designated as the stand-on ship—executed an unnecessary course change at minute 3. Furthermore, Ship 3 performed an abrupt port turn at minute 57, crossing directly in front of Ship 5’s bow, which represents an extremely dangerous maneuver explicitly prohibited in practical navigation.

4.4.2. COLREGs Compliance

The CC-DSSA quantifies and models COLREGs and good seamanship, introduces and embedded into the multi-objective cost function of a distributed stochastic search algorithm. Based on this, the algorithm can effectively resolve rule conflicts among ships during multi-ship collaborative decision-making, thereby identifying globally optimal collision-avoidance strategies that adhere to COLREGs and embody the principles of good seamanship. To more intuitively assess the compliance of collision-avoidance decisions, this paper categorizes all ship pair encounter situations in the initial global Scenario 5 into 16 types (Figure 17) and further evaluates the compliance performance of the algorithm’s results.

Table 15. Analysis of the rule compliance of the conventional DSSA.

No.	Compliance	No.	Compliance	No.	Compliance	No.	Compliance
No. 1	N	No. 5	N	No. 9	N	No. 13	Y
No. 2	N	No. 6	N	No. 10	Y	No. 14	N
No. 3	N	No. 7	N	No. 11	N	No. 15	N
No. 4	N	No. 8	Y	No. 12	N	No. 16	N

and

Table 16. Analysis of the rule compliance of the CC-DSSA.

No.	Compliance	No.	Compliance	No.	Compliance	No.	Compliance
No. 1	N	No. 5	N	No. 9	N	No. 13	Y
No. 2	Y	No. 6	Y	No. 10	Y	No. 14	N
No. 3	Y	No. 7	Y	No. 11	Y	No. 15	N
No. 4	Y	No. 8	Y	No. 12	Y	No. 16	Y

present the rule compliance evaluations for each ship pair under the conventional DSSA and the CC-DSSA. The collision-avoidance behaviors of 11 pairs of ships in the CC-DSSA meet the collision-avoidance action requirements set in Section 2. For the remaining 5 pairs identified as non-compliant, further analysis reveals that in Cases No. 1, No. 9, No. 14, and No. 15, the stand-on ships deviated from their prescribed courses by voluntarily assuming give-way responsibilities in response to interactions with other target ships. These actions are in accordance with the discretionary provisions for stand-on ships stipulated in Rules 2 and 17 of the COLREGs. In Case No. 5, Ship 5 acted as the stand-on ship in an overtaking situation—taking evasive action could have introduced a new collision risk. After Ship 2 executed a large-angle starboard turn, the collision threat between them was effectively mitigated, demonstrating the model’s capability to handle such complex scenarios. Overall, the CC-DSSA generates globally coordinated and COLREGs-compliant avoidance strategies while accommodating context-aware responses to individual deviations, thereby reflecting both the flexibility of the COLREGs and their practical applicability.

By contrast, the results of the conventional DSSA (Table 15, Figure 16) resulted in several explicit violations of COLREGs, such as port-side avoidance by Ship 1 and Ship 4, failure to yield by Ship 2, avoid action taken by the stand-on ship (Ship 5), and overtaking across another ship’s bow by Ship 3. These behaviors indicate a lack of rule compliance, as the algorithm does not account for COLREGs constraints, thereby rendering its decisions difficult to justify and compromising both safety and practical applicability.

4.4.3. Navigation Efficiency

To evaluate the navigational efficiency of the collision-avoidance strategies, this section evaluates the route deviation distances of each ship from their originally planned routes under both methods. The specific values are provided in

Table 17. The distance of path deviation of the conventional DSSA.

	Ship 1	Ship 2	Ship 3	Ship 4	Ship 5	Total
Deviation distance	0.372 n miles	0 n miles	0.314 n miles	0.598 n miles	0.136 n miles	1.420 n miles

and

Table 18. The distance of path deviation of the CC-DSSA.

	Ship 1	Ship 2	Ship 3	Ship 4	Ship 5	Total
Deviation distance	1.024 n miles	0.306 n miles	0.312 n miles	0.021 n miles	0 n miles	1.663 n miles

. While the total deviation distance under the proposed method is slightly greater than that of the conventional DSSA, this increase is primarily attributed to the method’s more comprehensive consideration of safety and COLREGs compliance during strategy generation. Moreover, the deviation levels of all vessels remain within acceptable deviation ranges, with no excessive detours or redundant trajectories observed, ensuring that the overall efficiency of navigation remains satisfactory.

4.4.4. Computational Efficiency

In practical navigation system, collision-avoidance systems must exhibit high real-time performance. A rapid response and efficient convergence are critical to ensuring operational applicability for enhancing an algorithm’s applicability in intelligent navigation systems. The DSA-based method significantly reduces the computational time compared to compared to DLSA and DTSA [28]. Simulation results from Scenarios 1 to 5 demonstrate that, in the absence of communication latency, the conventional DSSA achieves an average computation time of 0.062 s per time step, whereas the CC-DSSA averages 0.068 s. This slight increase is attributable to the inclusion of more complex collision risk assessments and COLREGs compliance costs in its cost function. Despite the higher computational demand, the CC-DSSA maintains satisfactory real-time performance, fulfilling the practical requirements of multi-ship collision avoidance.

To further assess the computational efficiency of the distributed collision-avoidance algorithm, we analyzed the number of information exchange rounds required for each ship to achieve coordination and convergence within each time step. This metric serves as a critical indicator for evaluating the communication efficiency and convergence performance of distributed decision-making algorithms. Figure 18a,b show the statistical results of the number of information exchange rounds for the conventional DSSA and the CC-DSSA proposed in this study under Scenario 5.

The results demonstrate that both algorithms achieve global convergence within each time step by utilizing a limited number of coordination rounds. This performance is primarily attributed to the probabilistic update mechanism integrated into the distributed stochastic search framework, which enhances convergence performance. The average number of information exchange rounds per time step throughout the entire collision-avoidance process is 1.75 for the conventional DSSA, compared to 1.51 for the CC-DSSA. This difference is primarily be attributed to the CC-DSSA’s ability to resolve critical collision risks and recover viable courses within the first 50 time steps, after which coordination stabilized at a single round per step. In contrast, the conventional DSSA, which does not incorporate COLREGs constraints, generated uncoordinated avoidance actions during the early phase, resulting in new encounter conflicts and prolonged information exchange and trajectory adjustments up to approximately the 60-time step. This phenomenon has also been verified in other multi-ship scenario experiments.

Table 19. Comparison of result parameters between conventional DSSA and CC-DSSA.

	Safety		Number of Unreasonable Violations of COLREGs	Total Yaw Distance	Computational Cost (per Time Step)
	Collision Count	Occurrences of DCPA Below SDA Threshold			Average Decision-Making Time	Average Exchange Rounds Count
DSSA	0	2	13	1.420 n miles	0.062 s	1.750 rounds
CC-DSSA	0	0	0	1.663 n miles	0.068 s	1.510 rounds

presents a detailed comparison of the before-mentioned indicators. As shown in the results, the CC-DSSA—through the integration of COLREGs and good seamanship principles—achieves more globally coordinated collision-avoidance strategies while significantly reducing the likelihood of secondary conflicts. These improvements contribute to notable improvements in the overall efficiency, safety, and stability of the collision-avoidance process.

5. Conclusions

This study proposes a distributed, multi-ship, cooperative collision-avoidance framework (CC-DSSA) that adheres to the COLREGs and the principles of good seamanship. A quantitative model was developed to represent the encounter type identification, safety assessment, and phase classification under the COLREGs. Then, a distributed stochastic search algorithm was designed that integrating localized communication structure and a probabilistic perturbation to enable adaptive coordination. Additionally, a multi-factor cost function was constructed that integrates collision risk, COLREGs compliance, navigational efficiency, and distance-based weighting to enhance coordination and decision-making robustness in complex, dynamic environments. Validation across multiple simulation scenarios demonstrates that the proposed method enables safe, compliant, and efficient collision avoidance. A comparative analysis with the conventional DSSA was conducted in terms of navigational safety, COLREGs compliance, path efficiency, and computational real-time performance, in which the proposed CC-DSSA exhibited consistently strong overall performance, confirming its effectiveness and operational relevance in multi-ship autonomous navigation.

However, this study has certain limitations. The current model does not incorporate more accurate ship maneuvering dynamics and does not explicitly account for environmental disturbances such as wind, waves, currents, and adverse weather. In addition, the framework is primarily designed for open-water scenarios. Future work will focus on improving maneuverability modeling, enhancing adaptability to environmental uncertainties (including sensor noise and data transmission error), and extending the framework to complex near-shore environments to better meet practical maritime operations collision-avoidance demands.