Extended Probabilistic Risk Assessment of Autonomous Underwater Vehicle Docking Scenarios Considering Battery Consumption

Abstract

Autonomous underwater vehicles (AUVs) play a crucial role in marine environments, such as in inspecting marine structures and monitoring the condition of subsea pipelines. After completing their mission, AUVs dock with recovery systems at designated locations. However, underwater docking carries a significant risk of failure due to unpredictable maritime conditions. Considering the limitations in communication during the mission, docking failure can lead to the loss of collected data and failure of the entire AUV mission. In this study, a hypothetical AUV docking scenario was defined based on expert knowledge and without actual operational data. A Markov chain-based probabilistic model was employed to quantitatively assess the risk of the system during the mission. Environmental factors were excluded from the evaluation, and the simulation results were classified into five categories: success, timeout, internal component failure, exceeding a predefined sequence repetition limit, and spending the electrical energy under the battery SOC threshold. By analyzing the failure points of each category, strategies to improve the scenario success rate were discussed. This study quantitatively identified the interactions between constraints and risk factors that should be considered when establishing AUV docking plans through a virtual scenario-based failure analysis, thereby providing an evaluation framework that can be utilized in actual design.

1. Introduction

Unmanned systems, including autonomous vehicles, drones, and underwater robots are rapidly developing in various fields. They are gaining recognition for their industrial value in performing and assisting missions in environments that are difficult or dangerous for humans to access. Autonomous underwater vehicles (AUVs), in particular, are utilized in marine structure inspections, submarine pipeline monitoring, marine resource exploration, and ocean monitoring, and they are recognized as crucial systems in both the military and civilian maritime sectors [1,2].

Recently developed AUVs can dock with a recovery system after a standalone mission for data transfer, maintenance, and battery charging [3]. Docking failure can therefore result in the failure of the entire mission and the loss of collected data. In underwater environments, the likelihood of docking failure is particularly high due to communication delays and sensor errors. Therefore, assessing the risk of complex docking procedures through a hypothetical scenario is required to minimize the likelihood of docking failure in real-world operations. Recent research in this area has sought to improve docking reliability by developing system-level strategies and analytical frameworks. For example, an efficient docking system for vector-propelled robotic fish has been proposed, demonstrating that integrating hydrodynamic modeling and mechanical docking design can significantly improve the stability and energy efficiency of docking operations in dynamic underwater environments [4]. Furthermore, a path analysis-based docking management process (DMP) for torpedo-type AUVs was developed, in which docking path feasibility, docking success probability, and guidance control were hierarchically combined to quantitatively assess docking reliability [5]. These studies demonstrate the growing importance of probabilistic approaches in AUV docking assessment and emphasize the need to integrate probabilistic assessment methods when analyzing docking risks in realistic operational environments.

Probabilistic risk assessment (PRA) techniques are commonly used to proactively analyze system risk. PRA quantifies the probabilities of failure and event occurrence for system component, thereby allowing for an assessment of system reliability. By modeling complex operational procedures as a flow of events and defining transition probabilities between events, PRA can quantitatively assess the stability and risk of the overall scenario [6]. These characteristics are well-suited for modeling AUV docking procedures, where multiple devices and sensors operate sequentially. A Markov chain model for AUV operational procedures divided into 11 states was developed, combining expert knowledge and operational history to estimate transition probabilities at each stage [7]. However, this approach did not consider realistic operational constraints, such as the remaining battery life, time constraints, and sequence repetition. Furthermore, its specificity to a particular AUV case (Autosub3) makes it difficult to apply it to moving docking platforms or the early stages of a new design. A dynamic PRA technique reflecting environmental conditions and state transitions in AUV operations was explored, but it proved inefficient in environments with limited data and high computational complexity. A real-time risk model for AMSs (autonomous marine systems) was proposed, with its transition probabilities updated through Bayesian methods [8]; however, with its application is constrained in data-scarce environments, as it presupposes the availability of sufficient operational data.

Previous studies have focused on analyses in data-rich environments or fixed-point docking procedures, resulting in a lack of pre-risk assessment procedures applicable to data-deficient environments and moving docking platforms. To address these limitations, this study combines scenarios generated based on expert knowledge with a Markov chain model featuring a fixed transition probability structure to ensure computational efficiency and incorporate realistic internal constraints, such as the remaining battery capacity, time limits, and sequence repetition limits. This allows for a quantitative assessment of key internal failure factors, excluding environmental factors, and the classification of scenario outcomes into five categories: success, timeout, excessive repetition, battery drain, and equipment and sensor failure.

This study focuses on a complex system that utilizes a recovery device mounted on an underwater platform to perform unmanned docking of an AUV that has completed its mission. The docking process is accomplished through an interaction between the moving underwater platform and the AUV, with multiple mechanical components and sensors, including a case, transfer module, and recovery module, operating sequentially. While this structure offers greater flexibility than existing systems that assume a fixed recovery point, it also carries a higher risk of failure due to sensor or equipment performance degradation and communication uncertainties. Therefore, the assessment procedure in this study is essential for increasing the success rate in the real application of the docking system.

Furthermore, as evidenced in relevant case studies, the scope of AUV applications continues to expand in large-scale marine projects, such as observations of Arctic sea ice and mobile inspection of subsea pipelines. In these environments, the successful completion of the docking process after mission termination is essential to ensure the stability of AUV mission cycles. The approach in this study can be utilized as a practical tool to rapidly quantify the risk of docking scenarios, even in the early design stages where data are scarce, thereby preemptively reducing the likelihood of failure. Furthermore, by optimizing scenarios that reflect operational constraints, it can be utilized to improve the docking success rate of AUVs in diverse marine operational environments.

This paper is structured as follows: Section 2 examines related research cases. Section 3 briefly describes the structure and characteristics of the target system, and Section 4 presents a scenario-based PRA procedure. Section 5 details the battery power consumption model and scenario evaluation model integrated into the PRA. Section 6 analyzes the simulation results, and, finally, Section 7 discusses the conclusions and future research.

2. Related Research

2.1. Probabilistic Risk Assessment

Probabilistic risk assessment (PRA) is a widely used technique in various industries for quantitatively assessing the reliability and safety of complex systems [9]. PRA calculates the overall risk based on the failure probabilities of sub-components and the causal relationships between events. Representative approaches include event trees (ETs), fault trees (FTs), Markov chains, Bayesian networks, and simulation-based approaches [10,11].

Previous studies have applied PRA to various fields, including the nuclear power industry [12], aviation [13], railways [14], and marine plants [15]. Traditional PRA methods have limitations, such as the inability to reflect time dependencies, state transitions, and changes in operating conditions. Therefore, dynamic PRA (DPRA) techniques have recently been proposed to address these limitations [10]. For example, a linking procedure that incorporates time elements and state changes into existing static PRA models has been proposed [9]. A technique for measuring the importance of risk in a DPRA environment using Markov processes and Monte Carlo simulations has also been presented [11]. Similarly, a risk quantification and visualization method has been proposed for aircraft loss-of-control scenarios [13], and dynamic risk in a railway braking system has been calculated by examining component deterioration and changes in failure probability [14].

With many use cases, PRA has become an essential tool for the risk analysis of complex systems across industries, and the scope and techniques of analysis are evolving from static analysis to dynamic analysis incorporating real-time data and state transitions.

2.2. Probabilistic Risk Assessment for Marine Systems

In the maritime field, PRA has been applied in risk analysis of subsea infrastructure, offshore plants, and unmanned surface vehicles (USVs). A redundancy monitoring system was developed for sensor communication modules in a marine observation network, and a reliability analysis was performed based on failure scenarios [16]. Multi-sensor data fusion techniques were applied to analyze the detection performance of subsea pipelines and to quantify the probability of detection failure [17]. A system for autonomously inspecting and classifying marine organisms on the surface of marine structures was proposed, and the impact of sensor data quality degradation on classification performance was analyzed [18]. A framework for the real-time condition monitoring, failure prediction, and maintenance optimization of unmanned surface vehicles (USVs) based on prognostics and health management (PHM) technology was proposed, thereby improving mission reliability [19]. A systems engineering-based risk analysis (STPA) was applied to complex marine systems to identify potential risks during the design phase and to propose process improvement measures [20].

These related cases primarily assume fixed-point-based marine system operation, and the scope of analysis focuses on the reliability of sensors, communications, and mechanical components. Therefore, the applicability of risk analysis in the early design stages, where moving docking platforms or actual measurement data are lacking, is limited, and there is a lack of PRA cases that integrate complex operating procedures and multiple constraints.

2.3. Probabilistic Risk Assessment for AUV

PRA research on AUV operations has primarily focused on the entire mission process. A Markov chain model dividing the operational procedures of the Autosub3 AUV into 11 states was developed, combining expert judgment with operational history to estimate the transition probabilities at each stage [7]. However, it did not consider internal operational constraints, such as the remaining battery life, time constraints, and sequence repetition. Dynamic PRA was applied to reflect environmental conditions and state transitions, but its high computational complexity made it ineffective in environments with limited data or in the early design stages [21]. A real-time risk model for AMSs was proposed, with transition probabilities adjusted through Bayesian updates [8], but this approach relied on sufficient real-world data.

Recent research has shown that AUV risk analysis research is shifting toward dynamic risk analysis, intelligent (machine learning-assisted) assessment, the processing of sparse historical data, and multi-vehicle cooperative missions, emphasizing the need for online decision support rather than offline prediction [22]. Furthermore, a phased-FTA framework has been developed to evaluate AUV performance across mission phases, where the expected instantaneous performance (EIP) is derived from mission-dependent fault trees and integrated via rank-sum weighting and universal generating functions (UGFs) [23]. For underwater docking, a probability-based docking assessment index has been proposed to convert position and orientation information within the docking assessment area into probabilistic values to estimate the probability of docking success in real time. Field tests have shown that a 90% confidence interval closely matches the actual docking success rate of 90%; however, the current approach does not yet incorporate sensor errors [24].

Regarding AUV docking, a modular Kalman filter-based docking navigation technique using USBL, DVL, a compass, and gyro sensors was proposed, and the docking success probability was estimated in real time based on sensor error [25]. However, its assumption of fixed-point docking makes it difficult to directly apply it to moving platforms. A framework for incorporating risk factors into multi-AUV trajectory planning was presented, but the risk of the docking procedure itself was not quantified [26].

Table 1. Related research on PRA for AUV operations.

Reference	Target System	Applied Methodology
[7]	Autosub3 AUV	Markov chain model, expert judgment, operational history
[8]	AMS (including AUV)	Real-time risk model, Bayesian update
[11]	General AUV operation	Dynamic PRA (DPRA), environmental conditions, state transition incorporation
[22]	General AUV operation	Dynamic PRA, ML-assisted risk analysis, sparse-data handling, multi-AUV collaboration
[23]	AUV phased missions	Phased fault tree analysis, rank-sum weighting UGF for performability
[24]	Underwater docking	Probability-based docking assessment index
[25]	Fixed-point AUV docking	USBL, DVL, compass, gyro-based modular Kalman filter
[26]	Multi-AUV path planning	Path planning with integrated risk factors (PRA)

presents studies where PRA was performed on AUVs.

Overall, existing AUV studies have focused on either full-scale operational or fixed-point docking procedures, and there are no PRA cases that encompass moving docking platforms, data-poor environments, or an integrated analysis of internal constraints. To overcome these limitations, this study proposes a docking procedure risk analysis framework that combines expert knowledge-based scenarios with a Markov chain model with a fixed transition probability structure and incorporates realistic internal constraints such as battery SOC.

3. AUV—Moving Platform Docking System

The subject of this study is the docking process with a mounted AUV recovery system. The recovery process begins when the AUV completes its assigned mission and receives the recovery command. After receiving the command, the AUV moves to a pre-designated docking location and autonomously controls its speed and direction to maintain its relative position with respect to the platform. Upon completion, the platform docking system sequentially extends the case and docking device along the fixed horizontal launch tube. The drogue is extracted via a winch within the AUV, and the open docking module detects the drogue and closes to secure it, as shown in Figure 1. After mechanically engaging with the platform, the AUV shuts down its propulsion and performs position control. The fixing module, a type of locking device, prevents the docking module from opening, while the recovery module secures the rear end of the AUV to prevent mechanical collisions and instability due to shaking. The process leading up to the mechanical engagement between the platform and the AUV is illustrated in Figure 2.

Table 2. Normalized specifications of the docking system components.

Component	Length (Normalized)	Component	Mass (Normalized)
Overall Length of the Platform	1	Overall Mass of the Platform Docking System	1
Horizontal Launch Tube	0.5294	Horizontal Launch Tube	0.2706
Case	0.4706	Case	0.2764
Docking Device	0.4706	Transfer Module	0.0341
		Return Module	0.0791
		Docking Module	0.1147

presents the relative specifications of the components based on the platform length. All listed values represent relative measurements, with length normalized to the total platform length and mass referenced to the overall weight of the docking system. In the case of weight, the values represent only the mass of the devices themselves, excluding the weight of the cables and sensors.

After the engagement is complete, the platform docking system follows the reverse sequence of the deployment process. The transfer module slowly moves the docking equipment into the case, while the case structure safely protects the AUV during this process. Finally, the case is retracted into the platform along the horizontal launch tube, and all devices are returned to their original positions, completing the mechanical recovery process. Figure 3 shows the docking system and the AUV secured within the platform after the docking process. The horizontal launch tube and case enclose the docking device and the AUV, but they are not included in the figure to clearly illustrate the completed docking process.

This AUV recovery process is generally more challenging than the launching process due to the uncertainty of the external marine environment during recovery, the potential of sensor signal anomalies, and the need for precise engagement under relative motion.

Any synchronization failure or malfunction of a component directly affects the success rate of the overall recovery scenario. Additionally, after recovery, the AUV transfers the data acquired during the mission to the platform and wirelessly recharges batteries using the platform’s power sources. Ensuring the stability of the docking process is therefore essential for subsequent missions. In this context, this study quantitatively evaluates the reliability and safety of docking scenarios through a scenario-based risk analysis that incorporates these complex constraints.

4. PRA Based on the Docking Scenario

In this section, a docking scenario between an AUV and a moving platform is constructed using an event tree format. Transitions between sequences(events) occur based on clearly defined transition probabilities, and as transitions at any given time are unaffected by previous states, the transition matrix is modeled using a Markov chain model. In real-world operations, if communication-related anomalies occur, then the system must be floated regardless of whether the issue is resolved, as self-response is impossible. Furthermore, as this is an initial design system without actual operational data, environmental factors affecting the system are excluded. Therefore, the transition probabilities between sequences are assumed to be the failure probabilities of the physical elements that make up the system.

4.1. Markov Chain

A Markov chain is a probabilistic model based on the Markov property, which states that the state transition of a system depends only on the current state and is independent of history [27]. It is defined as a discrete state space $S=\{s_1, s_2, \dots , s_n\}$ and transition probability $P_{ij}=P(X_{t+1}=s_j|X_t=s_i)$, and the entire transition structure is expressed through the transition matrix P of Equation (1):

$$\mathbf{P}=\left(\begin{array}{cccc}{P}_{11}& P_{12}& \dots & P_{1n}\\ P_{21}& P_{22}& \dots & P_{21}\\ ⋮& ⋮& \ddots & ⋮\\ P_{n1}& P_{n2}& \dots & P_{nn}\end{array}\right), \sum _{i,j=1}^n{P}_{ij}=1$$

(1)

Markov chains offer the following advantages: First, they can clearly describe system behavior using only state definitions and transition probabilities, thereby simplifying complex operational procedures step by step [28,29]. Second, they can quantitatively compare and evaluate various operational situations by adjusting transition probabilities based on scenario-specific conditions [30]. Third, they can be combined with event trees (ETs) or fault trees (FTs) to enable dynamic reliability assessment [31].

Table 3. Research cases utilizing Markov chains.

Reference	Field	Application
[27]	Nuclear Power	Application of state transition model for uncertainty analysis in level 1 PRA
[28,32]	Land Use	Derivation of transition probability matrix for land use change prediction and long-term pattern analysis
[29]	Geotechnical Engineering	Probabilistic site characterization using Markov chain Monte Carlo (MCMC)
[30]	Complex System Reliability	Automatic generation of event and fault trees using Markov approach
[33]	Meteorological Hazards	Prediction of SPI drought class transitions

summarizes examples of Markov chains applied to various domains.

This study adopted a Markov chain model for the following reasons:

• The simplicity and efficiency of calculations using a transition probability matrix enable a rapid risk analysis, even in data-poor environments.
• The AUV docking procedure, characterized by step-by-step state definitions and clear transition conditions, lends itself well to state-based modeling.
• It structurally reflects various constraints, such as the remaining battery capacity, number of iterations, and timeouts. Furthermore, it can be expanded to a dynamic model once real-world data becomes available.

4.2. AUV Docking Scenario

The docking scenario of this system consists of 13 normal sequences, and detailed descriptions of each event can be found in

Table 4. Normal sequences of the docking scenario.

Step	Description
D1	AUV Autonomous Mission
D2	Front Recovery Start
D3	Move to Front Recovery Position
D4	Deploy Case/Transfer Module
D5	Release Recovery Module
D6	Deploy Docking Module
D7	Drogue Winch Payout, AUV Attitude Control
D8	Fold Docking Module
D9	AUV Position Fixing, Drogue Hull In
D10	AUV Seated in Docking Module, Fold Recovery Module
D11	Stop AUV Thrust, Fold Transfer Module/Case
D12	Wireless Charging
D13	Visible Light Communication

. Each sequence is expressed by combining the first letter of “docking” and a number.

Additionally, to enhance system stability, a total of six response sequences for different anomalies were designed. For emergency stop situations, two sequences were designed: one before and one after the AUV and moving docking platform engagement. This is because, if an emergency stop occurs while the AUV is not engaged with the docking platform, then emergency surfacing must be carried out to safely recover the AUV. The names and details of the defined response sequences and emergency stop sequences can be found in

Table 5. Fault and emergency stop sequences of the docking scenario.

Step	Description
DF1	Fold Transfer Module/Case
DF2	Fold Recovery Module
DF3	Fold Docking Module
DF4	Drogue Winch Hull In
DF5	Deploy Docking Module
DF6	Deploy Recovery Module
Break1	Forced Termination and AUV Surfacing
Break2	Forced Termination

. The response sequences are expressed by combining the first letters of the term “docking fault” with a number.

The key physical elements and failure probabilities associated with the docking scenario are summarized in

Table 6. Failure probabilities of main physical components and sensors.

Probability Variable	Detail Components	Role	Value (%)
$P_c$	$D_{case}$	Docking system Operation	0.5
$P_t$	$D_{transfer}+S_{transfer}$		1.5
$P_r$	$D_{recovery}+S_{recovery}$		1.5
$P_m$	$S_{magnetic}$	Drogue Detection via Magnetic Field	2
$P_d$	$D_{dock}+S_{dock}$	Docking Module	1.5
$P_f$	$D_{fixing}$	Fixing Module	0.5
$P_w$	$D_{winch}$	Drogue Winch	1
$P_l$	$S_{drogLED}$	Drogue Illumination	0.5
$P_p$	$D_{propulsion}$	AUV Thruster	2.5
$P_{ps}$	$D_{p\_sending}$	Wireless Charging Power Transmission (platform) and Reception (AUV)	1
$P_{pr}$	$D_{p\_receiving}$		1

. Here, “D” represents the failure probability of the device itself, and “S” represents the failure probability of the sensor. Due to the lack of actual operational data, environmental factors were excluded. Furthermore, as it is assumed that system operation will have to be halted without a separate countermeasure if an anomaly occurs in the communication unit, the scenario transition is assumed to be based solely on the failure probabilities of the physical elements. For elements that require simultaneous consideration of the failure probabilities of both the device and the sensor, the failure probability is defined as a simple sum, and the probability value is assigned different values depending on the number of devices and sensors and the importance judgment based on expert knowledge.

While the defined failure probabilities are based on assumptions, different failure probabilities were assigned according to the elements directly involved in the docking process and the number of sequences in which they operate, to construct a more conservative scenario. The AUV thruster was assigned the highest failure probability of 2.5%, as it operates in nearly all sequences. Furthermore, elements directly involved in the mechanical engagement between the AUV and the moving platform were assigned a 1.5% failure probability, while elements operating in specific or single sequences were assigned a 1% failure probability. Finally, auxiliary elements that constitute the system but do not directly affect success were assigned the lowest failure probability of 0.5%.

Meanwhile, regarding the system specifications presented in Table 2, changes in element length and mass may affect failure probability during actual operation, influenced by environmental factors such as currents and obstacles. However, the currently assumed probabilities concern the intrinsic reliability of the devices and sensors themselves, and thus the impact of specification changes is disregarded.

Based on the above information, the docking scenario for the target system is represented in an event tree format, as shown in Figure 4. Green represents the normal sequence, yellow represents the response sequence, and red represents the emergency shutdown sequence. The indication that the normal sequence repeats means that, even if no abnormality has occurred, the system will retry the normal sequence to account for a temporary error in the event of failure to transition to the next sequence.

When multiple probability factors interact in a complex manner, the probability of at least one event occurring can be calculated using Equation (2). However, when the individual probabilities are sufficiently low, the rare event approximation can be applied, in which the overall probability is expressed as a simple summation, as shown in Equation (3). In Figure 4, the transition from DF4 to Break1 is influenced by the largest number of failure probability factors. The failure probability for this transition, calculated using Equation (2), is 5.88%, whereas the value obtained using the approximation in Equation (3) is 6.00%. The difference between the two results is negligible. Therefore, in this study, a transition matrix was constructed by simply summing the failure probabilities of each physical element.

$$P\left(X_1\cup X_2\cup \cdots \cup X_n\right)=1-\prod _{i=1}^n(1-P(X_i\left)\right)$$

(2)

$$P\left(X_1\cup X_2\cup \cdots \cup X_n\right)\approx \sum _{i=1}^{n}P\left(X_i\right)$$

(3)

In summary, the docking scenario of the system is modeled using a 22 × 22 transition matrix, and the sequences are classified as shown in Equation (4):

$$\begin{array}{c}{S}_{normal}=\left\{\mathrm{1,2},\dots ,13\right\},\\ S_{success}=\left\{14\right\},\\ S_{fault}=\left\{15,\dots ,20\right\},\\ S_{break}=\left\{\mathrm{21,22}\right\}\end{array}$$

(4)

4.3. Extended PRA Framework for AUV Docking Scenario

A previous study conducted a probabilistic risk assessment (PRA) of the same moving platform in the context of a launching scenario [34]. In that work, scenarios were modeled step-by-step based on the physical components of the platform and AUV, and transition probabilities were derived from expert judgment to analyze changes in success probability due to physical failures and visit restrictions. In this analysis, the scenario success rates and failure frequencies were determined, and each failure factor was compared and evaluated.

However, previous studies considered only a limited set of physical elements when defining anomaly probabilities, and they did not sufficiently reflect the time-related constraints or power consumption factors that are critical in actual operations. Recognizing these limitations, this study proposes an extended PRA framework that expands the analysis scope from the launching process to the docking process and incorporates realistic operational constraints. The docking process is inherently more challenging than the launching process, involving complex factors such as uncertainty in the marine environment, sensor signal attenuation, and the requirement for precise engagement under relative motion. Accordingly, the analysis scope is expanded to include the recovery procedure, which consists of sequential steps: moving the AUV to the docking location, engagement and fixation between the platform and the AUV, data transmission, wireless charging, and recovery completion. The outcome of each step directly affects the stability of the entire mission cycle and readiness for subsequent missions.

Furthermore, unlike previous research that focused narrowly on physical failures, this study extends the PRA framework by integrating a dedicated power consumption module and assigning priorities among constraints, such as the remaining battery capacity, timeouts, physical failures, and visit count restrictions. This extension captures realistic mission conditions and allows for a quantitative assessment of success rates and failure frequencies under these constraints. Based on the results, it also enables the proposal of risk mitigation strategies and performance enhancement measures for the recovery process.

Both studies assumed values for failure probabilities centered on physical components, excluding communication issues and environmental factors from the analysis. However, even though anomaly probability is assumption-based, the results remain meaningful in that they systematically identify risk factors before actual operations and enable proactive improvement. Therefore, this study overcomes the limitations of previous PRA research by establishing a recovery-centered target scenario and proposing an extended PRA framework that integrates realistic constraints and energy consumption factors directly related to mission success. Section 5 presents the detailed structure of the newly introduced power consumption module and evaluation logic and explains how these are integrated into the recovery scenario risk assessment.

5. Extended PRA Considering Battery Capacity

This study aims to improve the practical applicability of risk assessment by adding a power consumption module to the Markov chain-based risk assessment process. Furthermore, logic is added to determine the primary causes of scenario failure in the following order: scenario time limits, excessive repetition, battery capacity limitations, and physical element failures.

5.1. Battery Module

Figure 5 illustrates a schematic of the battery power system. Each battery is connected through a fuse box for safety, which prevents overvoltage and malfunction. Power is then supplied to a system that controls the assigned components, and, finally, the terminal delivers it to the terminal components to operate the entire system.

The power consumption module integrated into the Markov chain-based transition model presented in Section 4 is shown in Figure 6. However, as this study does not address the AUV mission execution process, it is assumed that the battery capacity of the AUV is sufficient until docking is complete. Therefore, only the battery capacity of the underwater moving platform—the docking target—is included in the evaluation.

A total of four batteries are included in the target system, and the specifications of each can be found in

Table 7. Battery specifications of the system.

Battery	Nominal Voltage [V] (Min~Max)	Maximum Discharge Current	DOD (Depth of Discharge)	Capacity [Wh]	Cycle Life	SOH (State of Health)
Battery 1	51.2 (44.8~58.4)	200	70%	7168	2500	60%
Battery 2	96 (84.0~109.5)	200	70%	13,440	2500	60%
Battery 3	25.6 (22.4~29.2)	400	70%	7168	2500	60%
Battery 4	192 (168.0~219.0)	25	70%	3430	2000	60%

. Hereafter, the capacity of each battery is denoted as $C_n$($n=1, 2, 3, 4)$.

Additionally, the supply elements allocated to each battery are shown in

Table 8. Components powered by each battery.

Battery	Power Supply Elements	Power Consumption [W]
Battery 1 (48 V)	Main Thruster	0.022
	DVL, IMU	0.040
	USBL	0.022
	Camera, Light	0.286
	Buoyancy Control Device	0.214
	Side Thruster (2 EA) Rudder (4 EA)	0.022
	Docking Module	0.2
Battery 2 (96 V)	Main Thruster Driver	6.111
Battery 3 (24 V)	Side Thruster Driver (2 EA) Rudder Driver (4 EA)	2.453
Battery 4 (196 V)	AUV Wireless Charging	2.13

. At this time, the power consumption of each element reflects the system specifications given in Table 2. Increases in length and mass may lead to additional power consumption due to the increased load.

Based on these allocations, the battery power allocation vector ${\mathit{b}}_{\mathit{n}}(n=1, 2, 3, 4)$ is defined, and the overall battery power allocation matrix B is expressed as Equation (5). Elements not powered by a specific battery are assigned a value of 0, indicating the absence of a supply relationship.

$$\mathbf{B}=\left[\begin{array}{c}{\mathit{b}}_1\\ {\mathit{b}}_2\\ {\mathit{b}}_3\\ {\mathit{b}}_4\end{array}\right], \mathbf{B}\in {\mathbb{R}}^{4\times 10}$$

(5)

Based on the scenario in the event tree format and transition matrix, a usage matrix $\mathit{U}\in {\left\{\mathrm{0,1}\right\}}^{22\times 10}$ can be defined to represent the devices operating in each sequence. ${\mathit{U}}_{i,j}$, which denotes the $j$-th element of Table 8 in sequence $i$, is set to 1 when on and 0 when off. Accordingly, the unit power consumption $R_{i,n}$ of battery $n$ in sequence $i$ is given by Equation (6):

$$R_{i,n}=\sum _m{\mathit{U}}_{i,m}{\mathbf{B}}_{n,m}$$

(6)

When the dwell time in the $k$-th visited sequence $X_k$ is ${\tau }_{X_k}$, the decrease in the SOC of battery $n$ in the sequence can be defined as $\Delta SO{C}_n^{\left(k\right)}$:

$$\Delta SO{C}_n^{\left(k\right)}={\displaystyle \frac{100}{C_n}}{R}_{X_k,n}{\tau }_{X_k}$$

(7)

Let $t$ denote the elapsed simulation time until termination. The accumulated SOC of battery $n$ up to time $t$ can then be expressed as follows:

$$SO{C}_n^{\left(t\right)}=100-\sum _{k=0}^t\Delta SO{C}_n^{\left(k\right)}$$

(8)

5.2. Scenario Assessment Logic

Excluding the influence of environmental variables, the major failure factors that can occur in the designed AUV–underwater moving docking system can be categorized as follows: exceeding the allocated docking completion time, excessive repetition of internal scenario processes, exceeding battery capacity limits, and physical component failure. These failure factors directly impact not only the success probability of the mission procedure but also the stability of subsequent operational cycles. Therefore, this study developed a hierarchical judgment logic that reflects the order of occurrence and causal relationships among these failure factors.

First, the scenario execution time serves as the absolute boundary for docking success in actual operations and acts as the standard for evaluating other constraints; thus, it is set to the highest constraint. The predefined mission completion target time for the scenario is 1200 s, and the dwell time per sequence is considered the most influential factor. Therefore, the accumulated scenario execution time is used to determine whether the target time has been exceeded. Let $T$ denote the accumulated execution time until the termination of the simulation. The indicator for the timeout condition is expressed as follows:

$$E_T=\left\{\begin{array}{c}1, T>T_{max}\\ 0, otherwise\end{array}\right.$$

(9)

Second, exceeding the repeat visit limit directly increases the execution time and the probability of docking failure due to unnecessary procedure repetition. To assess this, the number of visits per sequence is counted during the simulation. However, the fault sequence is designed to respond flexibly to anomalies. According to the assumptions of this study, even if a transition to the response sequence occurs, there is a very high probability of re-transitioning to the normal sequence. Therefore, the repetition limit is applied only to the normal sequence. Furthermore, transitioning to the break sequence leads directly to the termination of the simulation, so it is excluded from the evaluation. Let $N_i$ denote the cumulative number of visits to the normal sequence $i$ until termination, and let $L_i$ be the allowable visit limit of sequence $i$. The indicator for the visit limit condition is expressed as follows:

$$E_V=\left\{\begin{array}{c}1, \left|\mathcal{L}\right|\ge 1 \\ 0, otherwise\end{array}\right.,\mathcal{L}=\left\{i\in \left\{1,\dots ,13\right\} | N_i=L_i\right\}$$

(10)

Third, battery SOC depletion is a consequential indicator of an increased execution time. This is determined based on the accumulated power consumption per battery, as described in the previous section. Let ${SOC}_n$ denote the state of charge of battery $n$ at the termination of the simulation, and let ${\theta }_{SOC}$ be the SOC threshold. The indicator for the visit limit condition is expressed as follows:

$$E_B=\left\{\begin{array}{c}1, \underset{n}{\mathit{min}}SO{C}_n\le {\theta }_{SOC} \\ 0, otherwise\end{array}\right.$$

(11)

Therefore, the decision process first evaluates whether the execution time has been exceeded, then assesses whether the repeat visit limit has been exceeded and finally verifies whether the SOC limit has been exceeded to determine the step-by-step causal relationship.

Probabilistic failure refers to the possibility of failure due to uncertainty in physical components, as defined in Table 6, and it does not have a direct causal relationship with time. Furthermore, as a threshold-based boundary setting is not required, probabilistic failure is placed as the lowest-level decision factor, separate from cumulative constraints such as scenario time limits, visit limits, and SOC. Let $X_{last}$ denote the last visited sequence at the termination of the simulation. The indicator for the emergency-transition condition is expressed as follows:

$$E_P=\left\{\begin{array}{c}1, X_{last}\in \left\{\mathrm{21,22}\right\} \\ 0, otherwise \end{array}\right.$$

(12)

In summary, the decision logic sets whether the mission execution time has been exceeded as the highest-level branch condition. It then sequentially verifies whether the repeat visit limit has been exceeded and whether the battery has been depleted. Probabilistic failures are additionally considered to evaluate the scenario failure factors. If this is expressed in the form of an event tree, then it is as shown in Figure 7, where each failure factor is hierarchically classified according to the order of occurrence and causal relationship, so that the cause of docking failure can be clearly distinguished and improvement directions can be derived according to priority. Utilizing the previously defined indicators, the scenario assessment logic is expressed in Equation (13):

$$\mathrm{O}\mathrm{u}\mathrm{t}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{e}=\left\{\begin{array}{c}Timeout fail { E}_T=1 \\ Visit limit fail E_T=0, E_V=1 \\ Battery fail { E}_T=0, E_V=0,E_B=1 \\ Probabilistic fail { E}_T=0, E_V=0,E_B=0, E_P=1 \\ Success E_T=E_V=E_B=E_P=0 and X_{last}=14\end{array}\right.$$

(13)

An extended PRA framework that can identify the success and failure factors of a scenario by considering a Markov model based on fixed transition probabilities and battery power consumption is shown in Figure 8.

6. Simulation Result Analysis

6.1. Simulation Results

This section presents the evaluation results of the docking scenario utilizing the framework proposed in this study. For successful docking, the relationship between the simulation time and the SOC of each battery is shown in Figure 9. The target completion time of the scenario was set to 20 min, and the dwell time in both the normal and response sequences was set to 90 s. For failed docking, an SOC graph of each battery over the elapsed time of the scenario is shown in Figure 10. Figure 10a shows the simulation result where the scenario execution time was exceeded, which is the top-priority evaluation criterion. Figure 10b shows the simulation result where the maximum number of repeat visits to any normal sequence was set to 1, which should lead to successful docking immediately without any abnormalities. As can be seen, an excessive number of visits occurred in the sequence in the early part of the scenario. Figure 10c shows the simulation result where the lower limit of the battery SOC was exceeded. It should be noted that this is the result of arbitrarily adjusting the lower limit to 50% in order to obtain an intuitive understanding, as the number of simulations of failure caused by exceeding the battery capacity was very low when the lower limit of all batteries in the system was set to 30%, which is the specification standard. Finally, Figure 10d shows the simulation result where all three constraints were satisfied but reached the break sequence due to the influence of the previously defined physical element failure probability.

Figure 9 and Figure 10 show examples of the arbitrary constraint threshold adjustments made to visualize the simulation results. To more realistically simulate and evaluate the designed docking system, it is necessary to apply various constraints based on constraint priority and analyze the results.

6.1.1. Success Probability According to Sequence Dwell Time

First, to evaluate the system docking scenario time constraints, the dwell times for each sequence were assigned in 10-s increments, from 30 to 120 s. Based on existing research, the number of visits was set to three (no more than two allowed) to ensure stable system operation, and the lower SOC limit for each battery was set to 30% based on specifications. The anomaly probability of each physical element in the system is shown in Table 6. To verify the results, the probability of occurrence of each result was calculated for 10 trials of 100 simulations each (resulting in a total of 1000 simulations), and the mean value was used. The impact of varying the dwell times for each sequence on the system success probability is shown in Figure 11.

As the dwell time increased, the total time required increased, leading to a higher probability of timeout failure. This trend was particularly pronounced at 80 s or more. Starting at 100 s, the increased time required reversed the probability of success and timeout, making simulation success impossible. Under the 70 s condition, only 1 timeout occurred out of 1000 simulations, but this is significant given that it was the first occurrence. This suggests that, under ideal operating conditions, a dwell time of 60 s or less would prevent timeouts.

Meanwhile, failures due to physical component malfunctions appeared to have little trade-off with the sequence dwell time. However, the decrease observed in the 100 s range is believed to be due to timeouts occurring before failures.

6.1.2. Success Probability According to Visit Limit

Based on the above results, the simulation used a dwell time of 60 s per sequence, considering the most conservative conditions. The visit limit for normal sequences was then increased from 2 to 5 times, with the lower battery SOC limit and physical component failure probability maintained as constant. The simulation results are shown in Figure 12.

When the visit limit was set to 2, the success probability was relatively low because each normal sequence had to be visited only once. Setting the visit limit to 3 significantly reduced the failure rate. When the visit limit was set to 4, the number of times the visits exceeded the limit was only 1 in 1000. When the visit limit was set to 5, no failures caused by exceeding the limit were observed. Therefore, to ensure the stability of the scenario, the number of iterations must be set to at least 3. Considering the trade-off between the additional time consumed by the iterations and the stable success probability, the visit limit should be set between 3 and 5.

Physical component failures still occur independently of the primary cause of failure, with the only difference being the timing of occurrence.

In this study, the visit limit was set to 3 visits based on the belief that this would ensure a sufficient success rate (74.9%).

6.1.3. Success Probability According to Battery SOC Threshold

The SOC lower limit was set in 10% increments from 30% to 70%, and the results are shown in Figure 13. Significant changes were observed under the 60% condition, while under the 70% condition, the failure probability due to battery overcapacity reached 75.6%, with no success cases. According to Table 6 and Table 7, Batteries 3 and 4 each supply power only during specific sequences, while Battery 1 has numerous supply elements but low individual power consumption. Therefore, Battery 2 consumes the most power, and the capacity-related issues are solely confined to Battery 2, as can be directly confirmed in Figure 9. Therefore, the battery overcapacity observed in the simulation results is likely caused by Battery 2. Based on the results, both the 30% and 40% conditions are considered safe. Analyzing the failure time and overcapacity under the 50% condition will broaden the range of possible configurations, suggesting potential cost savings from a battery design perspective. Considering the expansion of the power supply target and stability in the future, the lower SOC limit for this verification was set to 30%.

The failure trends due to physical component failures are consistent with the previous analysis. In other words, the only factor that influences failure is the timing of failures due to other elements. Therefore, to improve system stability, it is essential to consider the sensitivity to physical element failures.

6.1.4. Success Probability According to Component Failures

In this study, 11 potential failure factors were defined, and the failure probability of these factors did not show a significant correlation with that of the other factors. The physical factors considered to be crucial in this scenario were the AUV propulsion system, the AUV winch, and the platform docking module. The failure probabilities of winches $P_p, P_w$, and $P_d$, corresponding to each of these factors, were set as ranges, and the success rate of the scenario was evaluated based on an increasing probability. Furthermore, the failure probability of $P_l$ (drogue LED illumination for detection), which was expected to be relatively low in importance, was additionally selected and subjected to the same analysis. This allowed the effect of component importance on the scenario success rate to be determined intuitively. The results are presented in Figure 14.

As expected, an increased failure probability of the AUV thruster significantly impacted the success rate. When the failure probability reached 10%, the success rate dropped sharply to below 40%, indicating that this element is a critical component throughout the docking process. Figure 4 confirms that the AUV thruster failure probability is accounted for in most transitions. While the impact of the platform docking module and AUV winch failures was smaller than that of the AUV thruster failure, the success rate tended to decrease as the failure probability increased. This suggests that both components can accumulate at certain stages and cause performance degradation.

As expected, failure of the AUV LED had a much smaller impact on the overall success rate than that of the other components. This suggests that, if this component fails, it can be compensated for by other sensors and additional sequences. The pre-assessed criticality based on the event tree was largely consistent with the simulation results.

6.2. Strategies to Improve Docking Success Rates

Based on the condition where the timeout occurred initially (70 s), the optimal dwell time for each sequence can be determined using the pre- and post-conditions of 60 and 80 s. If effective methods for increasing the success rate can be found under the subsequent constraints, the dwell time can be set to around 70 s. Subsequently, considering the operating speed and expected time required during actual system operation, it is possible to allocate more time to time-consuming sequences and reduce the dwell times for simpler sequences, thereby improving the overall scenario success rate.

The visit limit is a key factor in balancing the trade-off between the time reduction and success rate, with a setting of 3 to 5 visits being optimal. As no timeouts occurred under the current experimental conditions, where the sequence dwell time was set to 60 s, there is no concern about deviating from the absolute standard when setting constraints. This advantage can also be applied to constraint settings based on actual operational results. Furthermore, considering cases where normal sequence repetition is unavoidable due to environmental changes or obstacle avoidance, the visit limit for each sequence can be adjusted to increase the success rate.

The current design standard of 30% SOC provides sufficient stability. However, to reduce system design costs and weight, a battery with a lower limit of 40% of the current battery SOC can be substituted. Depending on the results of further analysis, a lower limit up to 50% SOC may be acceptable. However, given the significant impact of Battery 2′s power consumption on system stability, its capacity and lower limit should be prioritized. Furthermore, in the long term, the optimization of battery management algorithms can help secure a safety margin relative to the lower limit.

The simulation results showed that AUV thruster failure had the greatest impact on the reduced success rate, with a sharp decline in the success rate as the failure probability increased. Therefore, measures such as designing the thruster to minimize failure rates, securing spare parts, and conducting periodic maintenance are necessary. Appropriate measures to prevent malfunctions in the platform docking module and AUV winch are also expected. For components with relatively minor impacts, such as the drogue LED, it would be sufficient to develop countermeasures to mitigate performance degradation from a long-term operational perspective.

7. Conclusions

This study proposes a probabilistic risk assessment (PRA) procedure that integrates a battery power consumption module into a Markov chain-based state transition model for docking scenarios between AUVs and underwater moving platforms. The proposed procedure is designed to quantitatively assess the causes of docking failures by considering realistic operational constraints such as mission time constraints, scenario revisit limits, battery state of charge (SOC) thresholds, and probabilistic failures of physical components.

However, this study was conducted without actual operational data. The target system is in the early design stage, and the lack of actual operational data presents a limitation. Consequently, this research faced the following limitations: First, the potential failure factors and their failure probabilities were determined based on assumptions, not actual statistics or operational records. This implies that the calculated success probability and risk indicators may not fully reflect the actual operational environment. Second, regarding battery consumption, dynamic load changes, such as propulsion speed fluctuations, could not be reflected. This means that the energy consumption patterns during actual missions were modeled in a simplified manner, potentially limiting the accuracy of the power consumption predictions.

Nevertheless, the proposed framework is significant in that it allows for a quantitative assessment of system safety and success rate by applying various assumptions, even in the early stages. In particular, the proposed Markov chain-based assessment model allows for the flexible replacement of transition probabilities. Therefore, once actual operational data become available, they can be easily incorporated into the model, resulting in a more realistic and reliable assessment. Furthermore, once the actual operational sea area and operational cycle are confirmed, operational profiles for various components, including the propulsion system, can be obtained. By incorporating these into the Markov chain model, real-world factors such as marine environmental uncertainty, dynamic load conditions, and mission time constraints can be effectively reflected. In other words, the extended PRA framework proposed in this study has the potential to develop into an evaluation tool directly applicable to actual operations.

Therefore, future research should obtain operational data to calibrate the failure probabilities of physical components and incorporate environmental uncertainties into the model, thereby improving the realism of the framework.

Through the proposed framework, the following improvements are suggested to enhance the stability of the system: First, the risk of overtime can be minimized by optimizing the dwell time, considering the complexity and duration of each sequence in a real-world operational environment. Second, unnecessary failures can be prevented by adjusting the visit limit based on scenario characteristics and environmental changes. Third, conditions that would allow the battery lower limit to be relaxed to 40% should be examined, with priority given to enhancing the capacity and consumption patterns of Battery 2. However, for actual system operation, the safety margins of all batteries, not just Battery 2, must be accurately determined. Furthermore, it is essential to assess changes in battery conditions during long-term operation before establishing appropriate thresholds. Fourth, strategies should be developed for design refinement, regular maintenance, and spare parts acquisition to minimize AUV thruster failure rates; the stability of the docking module and winch should be prioritized; and supplementary measures should be prepared for the AUV LED to ensure reliability during long-term operation.

The proposed methodology offers the advantage of enabling a rapid and systematic scenario-based risk assessment, even in the early design stages or in environments with limited data. In future work, actual operational data will be utilized to incorporate environmental uncertainty into the Markov chain model and adaptively update transition probabilities, thereby expanding its applicability to a wider range of marine environments.