Risk Analysis of Autonomous Underwater Vehicle Operation in a Polar Environment Based on Fuzzy Fault Tree Analysis

Abstract

Autonomous underwater vehicles have long been used in marine explorations, and their application in recent polar expeditions is particularly noteworthy. However, the complexity and extreme conditions of the polar environment pose risks to the stable operation of autonomous underwater vehicles. This study adopted the methodology of fuzzy fault tree analysis to deeply analyze the operational risks of autonomous underwater vehicles in polar environments. While traditional fault tree analysis maps the causal relationships and probabilities between basic and intermediate events, fuzzy fault tree analysis models the uncertainty of data and determines the failure probability by integrating expert opinions. This study revealed that polar environment-induced failures play a more substantial role in autonomous underwater vehicle loss in polar regions than inherent system failures. The study identified ‘recovery failure’ and ‘poor communication’ as the major risk factors facing autonomous underwater vehicles in polar environments, exhibiting the highest failure probabilities. Specifically, among various polar environmental factors, ‘large ice concentration’, ‘ice thickness’, and ‘roughness of ice underside’ under ‘bad’ conditions were found to have a significant impact on the autonomous underwater vehicle’s failure probability. The fuzzy fault tree analysis method in this study successfully filled the gap created by the absence of historical data by effectively incorporating expert opinions, enabling a quantitative presentation of the impact of polar environments, which has been previously difficult to convey in qualitative terms.

1. Introduction

Autonomous underwater vehicles (AUVs) are unmanned underwater devices designed to operate autonomously without the need for cables or continuous input from an operator [1]. Their self-directed functionality and versatility make them indispensable in numerous applications, solidifying their importance in the field of marine technology [2,3,4]. AUVs are extensively deployed in a myriad of missions, not only encompassing oceanographic data collection, seafloor mapping, and marine biology research but also environmental monitoring, underwater infrastructure inspection, and military applications [5,6,7,8,9].

The exploration of polar regions using AUVs has gained importance in recent years for multiple reasons. These include understanding the impacts of global warming; monitoring changes in the ocean’s physical, chemical, and biological properties; contributing to a deeper understanding of marine ecosystems; facilitating the planning and implementation of safer polar routes; and providing indispensable support for polar exploration and scientific missions [5]. However, despite their potential, these areas remain largely unexplored due to the inherent challenges that arise when conducting research in such harsh and extreme environments [9,10,11].

Operating AUVs in polar regions poses challenges that can result in their loss, impacting the success of polar expeditions [10,11,12]. Understanding these challenges is crucial for improving the reliability of AUVs and ensuring the success of polar expeditions [5,12]. One of the major technical challenges faced by AUVs in polar regions is the risk of collision or entrapment due to ice. The thickness, concentration, and roughness of the ice’s underside can contribute to the risks faced by AUVs, either individually or collectively. For example, navigating through regions with thick ice layers can lead to collisions or the AUV getting trapped, as thicker ice increases the risk of accidents and mechanical damages. Similarly, areas with a high ice concentration, whether large or small ice particles, pose a significant threat by affecting the AUV’s navigation sensors, making it difficult for the AUV to maneuver safely. Additionally, the roughness of the ice’s underside presents an irregular and jagged terrain, which significantly increases the risk of accidents during navigation.

Another significant challenge is inaccurate navigation, which is influenced by several factors like underwater currents, ice concentration, and the roughness of the ice’s underside. These factors can rapidly alter the AUV’s trajectory and make pre-input navigation data inaccurate. Poor communication is another critical issue, and it can be affected by ice thickness, ice concentration, and the roughness of the ice’s underside, leading to miscommunication or complete communication loss, which can be fatal for any AUV operation. Recovery failure is also a significant concern and can be influenced by various factors such as ice and weather conditions. Specifically, irregular or thick ice layers, large or small ice concentrations, and adverse weather conditions can hinder the recovery process, potentially causing damage to the AUV or resulting in its complete loss. Moreover, the integrity of the AUV can be threatened by buoyancy control failure, which can be influenced by rapid changes in salinity and underwater currents, as well as battery failure, which is exacerbated by low temperatures, significantly affecting the battery’s lifespan and performance.

In such a complex and high-risk environment, comprehensively understanding these risks becomes vital [12]. Therefore, reliability analysis, which involves assessing the probability of a system performing its intended function without failure under given conditions, emerges as a promising solution [13,14]. However, applying traditional reliability analysis techniques to polar exploration AUVs is challenging due to the high degree of uncertainty and complexity involved [4,13].

The Bayesian Belief Network (BBN), a potent probabilistic graphical model, adeptly captures and delineates the complex interdependencies among numerous variables [3]. This capability provides a more comprehensive understanding of the system’s reliability, particularly in situations where data are scarce or uncertain, such as AUV operations in polar environments, making the BBN approach especially advantageous [3,4,12]. It enables the integration of not just objective, statistical data but also subjective, expert knowledge into the model, thus offering a holistic understanding of the various factors influencing AUV operations [2,15].

A number of previous studies have endeavored to understand the influence of environmental factors on AUV operations using the BBN approach. For instance, Griffiths and Brito [12], as well as their colleagues, have laid substantial groundwork for structured, quantitative risk assessments of AUV deployment for under-ice missions. They utilized the BBN approach to estimate risk under different sea ice conditions, leveraging expert judgment elicitation as a reference for the BBN. Brito and Griffiths [2] expanded the Bayesian approach in a subsequent study to estimate the risk of AUV loss during missions, taking into account factors like ice concentration, thickness, and environmental constraints that might lead to losses. Elsewhere, Thieme and Utne [15] and Hegde et al. [4] proposed BBN risk models to calculate the probability of monitoring success or mission abortion, considering a wide range of influential factors, from technical and human aspects to organizational and environmental aspects. Yang et al. [16] applied a dynamic Bayesian approach to capture the influence of environmental changes during AUV missions. Even though their primary focus was not environmental factors, they incorporated real-time data from the AUV’s operating environment to determine dynamic risk values, highlighting the versatility of the Bayesian approach.

Other efforts involving the consideration of uncertainty in AUV risk analysis, such as the use of fuzzy set theory, have been undertaken [17,18,19]. Loh et al. analyzed the impact of operator experience on AUV loss using a fuzzy system dynamics risk analysis (FuSDRA) approach [17,18]. The FuSDRA approach yields more robust results by accounting for a broad spectrum of risk factors. However, they noted that risk factors can vary depending on the purpose and characteristics of different AUVs. It also requires modeling using multiple software, which reduces its popularity and scalability.

However, previous studies, despite employing the BBN approach, have broadly approached polar conditions and overlooked the nuanced differences within the polar environment itself. Most importantly, they lack quantitative analyses of factors such as the thickness of the ice, the presence of large icebergs, and the roughness of the ice’s underside, all of which are crucial elements that increase the risk of AUV loss. While it is beneficial to compare polar and non-polar environments, understanding the internal variations within the polar environment itself and their impact on the risks is also crucial.

In this study, we aim to elucidate how various aspects of the polar environment, such as ice thickness, iceberg presence, the roughness of the ice’s underside, and under-ice current patterns, affect the risk of AUV loss during missions. This research is significant not only for advancing our understanding of AUV operations in extreme conditions but also for devising practical strategies for risk reduction. The degree of relevance and applied significance of this work lies in its potential to enhance the safety and reliability of AUV operations in polar environments, thereby contributing to more effective and efficient scientific and commercial applications.

2. Methodology

The field of AUV operation in polar environments is complex and fraught with risks, as delineated in prior studies using various methodologies such as Bayesian Belief Networks (BBN) and Fuzzy System Dynamics Risk Analysis (FuSDRA) [2,4,12,15,16,17,18,19].

Table 1. Comparison of different approaches to AUV reliability analysis.

Analysis Method	Advantages	Disadvantages	Applicability in Polar Environments	Ref.
Bayesian Belief Network (BBN)	Captures complex interdependencies; includes objective and subjective data	Data-intensive; may overlook specific polar conditions	High	[3,4,12,15]
Fuzzy System Dynamics Risk Analysis (FuSDRA)	Accounts for a broad spectrum of risks; adaptable for different AUV characteristics	Requires multiple software; less popular	Moderate	[17,18]
Fault Tree Analysis (FTA)	Systematic; easy to understand	Struggles with uncertainties and complex interactions	Low	[19]
Fuzzy Fault Tree Analysis (FFTA)	Incorporates uncertainties; allows for expert judgments	Might be computationally intensive	High	[20,21]

below summarizes the different approaches used in related research works and their respective strengths and weaknesses, including the method proposed in this study.

Traditional Fault Tree Analysis (FTA) [19,22], with its top-down approach beginning with the identification of a Top Event (TE) that signifies the most adverse outcome, serves as a useful but limited tool for AUV operations in polar environments. The inherent limitations of FTA stem from its dependence on accurate failure probabilities for Basic Events (BEs) [23]. This is particularly challenging when data are scarce or when dealing with the complexities of interactions between environmental factors and AUV operations.

In response to these limitations, our study introduces FFTA [20,21]. Unlike traditional FTA, FFTA incorporates fuzzy logic, allowing for the incorporation of expert judgments to calculate failure probabilities where empirical data are lacking. This integration leads to a more nuanced, comprehensive, and practically applicable method for assessing AUV reliability in polar conditions. The specific advantage of FFTA lies in its capacity to model uncertainties related to polar-specific factors such as ice thickness and ice concentration. In summary, FFTA is an effective complement to traditional FTA, and it is particularly advantageous in data-limited or highly specific contexts like polar AUV operations. The operational environment of the AUVs considered for the risk analysis presented in this study is illustrated in Figure 1.

2.1. Construction of Fault Tree Diagram

In our analysis, the first step is identifying the TE, which is a standard procedure in the conventional FTA process [19]. For this study, we have defined the TE as ‘AUV loss’, and subsequently constructed a detailed fault tree to map out the underpinning structure of the FTA [22]. The entire FTA was then divided into two major parts. The first part includes events that contribute to system failure and are invariant across different environments, referred to as the ‘inherent system failure’ part. For this part, we were able to use historical failure rate data as these events are independent of the environment. The second part includes events that have high uncertainty due to their susceptibility to environmental factors, referred to as the ‘polar environment-induced failure’ part. By dividing the FTA into these two parts, we were able to more accurately account for the different types of failures that can occur and appropriately utilize historical failure data where applicable while also acknowledging the uncertainty and variability introduced by environmental factors. Figure 2 illustrates the algorithm principles of this study.

Consequently, we could compute the failure probability (FP) or P(T) using tailored equations corresponding to the specific logic gates (i.e., AND, OR, or XOR gates) interlinking the events. The calculation of P(T) for the AND, OR, and XOR gates can be carried out according to Equation (1) [24,25].

$$\mathrm{P}\left(\mathrm{T}\right)=\left\{\begin{array}{c}{\displaystyle \prod _{\mathrm{i}=1}^{\mathrm{n}}}\mathrm{P}\left({\mathrm{X}}_{\mathrm{i}}\right),\mathrm{for}\mathrm{AND}\mathrm{gate}\\ \\ 1-{\displaystyle \prod _{\mathrm{i}=1}^{\mathrm{n}}}\left(1-\mathrm{P}\left({\mathrm{X}}_{\mathrm{i}}\right)\right),\mathrm{for}\mathrm{OR}\mathrm{gate}\\ \\ \mathrm{P}\left({\mathrm{X}}_1\right)+\mathrm{P}\left({\mathrm{X}}_2\right)-2\mathrm{P}\left({\mathrm{X}}_1\right)\mathrm{P}\left({\mathrm{X}}_2\right),\mathrm{for}\mathrm{XOR}\mathrm{gate}\end{array}\right.$$

(1)

2.2. Calculation of Inherent System Failure Probability Using Historical Data

This section provides a brief overview of the approach employed to compute the FPs of BEs associated with inherent system failures. To facilitate this calculation, we utilized historical failure rate data amassed from a thorough review of the existing literature [25]. By making the assumption that the system adheres to an exponential distribution, we were able to calculate the FP for each BE using Equation (2), which was derived from failure rates extracted from the literature [25].

$$\mathrm{F}\left(\mathrm{t}\right)=1-{\mathrm{e}}^{-\mathsf{\lambda }\mathrm{t}}$$

(2)

2.3. Calculation of Polar Environment-Induced Failure Probability Using Fuzzy Set Theory

Expert opinions are particularly valuable in scenarios with high environmental uncertainties and challenges in collecting historical data [26,27]. These opinions are processed using fuzzy set theory to assess BEs using linguistic terms, thereby translating them into quantifiable fuzzy membership values essential for subsequent FTA [28]. This process, which is divided into three key steps—fuzzification, aggregation, and defuzzification—translates the expert assessments into quantifiable fuzzy membership values that form the basis for the FPs used in subsequent FTA [28].

Fuzzification involves converting the expert assessments into a range of possible values (referred to as ‘fuzzy membership values’). Aggregation refers to the step where these various expert assessments are combined into a single fuzzy number. Lastly, defuzzification is the process of converting the aggregated fuzzy number into a crisp, definitive value. This approach enables a more precise estimation of FPs associated with environmental factors affecting AUV operations.

2.3.1. Fuzzification

The fuzzification process involves appraising the FP of BEs using linguistic terms categorized into five levels: very high, high, medium, low, and very low. This categorization efficiently encapsulates the experts’ judgments while maintaining a manageable level of complexity for subsequent analysis [28]. The qualitative assessments are then transformed into quantifiable fuzzy membership values using the triangular fuzzy numbers (TFNs) methodology proposed by Cheng and Mon [29]. This technique has been widely deployed in recent FFTA studies [30] due to its efficacy in dealing with the inherent uncertainty and variability in FP estimations.

TFNs are characterized by three parameters, namely a₁, a₂, and a₃, that range between 0 and 1. These parameters denote the lower bound, modal value, and upper bound of the fuzzy number, respectively [20]. Therefore, for any fuzzy number a = (${\mathrm{a}}_1$, ${\mathrm{a}}_2$, ${\mathrm{a}}_3$), the membership function, ${\mu }_{\mathrm{A}}\left(\mathrm{x}\right)$, is defined as:

$${\mu }_{\mathrm{A}}\left(\mathrm{x}\right)=\left\{\begin{array}{c}\frac{\mathrm{x}-{\mathrm{a}}_2}{{{\mathrm{a}}_2-\mathrm{a}}_1},{\mathrm{a}}_1\le \mathrm{x}\le {\mathrm{a}}_2\\ \frac{{\mathrm{a}}_3-\mathrm{x}}{{{\mathrm{a}}_3-\mathrm{a}}_2},{\mathrm{a}}_2\le \mathrm{x}\le {\mathrm{a}}_3\\ 0,\mathrm{otherwise}\end{array}\right.$$

(3)

where a₁, a₂, a₃ range between 0 and 1.

Table 2. Possibility of occurrence in form of linguistic terms to triangular fuzzy number.

Possibility of Occurrence in Form of Linguistic Terms	Symbol	Possibility of Occurrence in Form of Triangular Fuzzy Number
Very high	VH	(0.7, 0.9, 1)
High	H	(0.5, 0.7, 0.9)
Medium	M	(0.3, 0.5, 0.7)
Low	L	(0.1, 0.3, 0.5)
Very Low	VL	(0, 0.1, 0.3)

offers a representative mapping of linguistic terms to TFNs [29]. For example, a linguistic term such as “low” could equate to TFNs with parameters (0.1, 0.3, 0.5). Figure 3 visually represents a typical fuzzy membership function for TFNs, facilitating the comprehension of the distribution and shape of the fuzzy sets and aiding the interpretation of the expert evaluations [20].

The evaluations of the experts, denoted as ${\mathrm{E}}_{\mathrm{i}}(\mathrm{i}=1,2,\dots ,\mathrm{m})$, are represented as $\widetilde{{\mathrm{A}}_{\mathrm{i}}}$ for the BEs. The membership functions ${\mu }_{\mathrm{A}}\left(\mathrm{x}\right)$ are used to represent the subjective estimation of a rating for a given evaluation. The evaluated results $\widetilde{{\mathrm{A}}_{\mathrm{i}}}(\mathrm{i}=1,2,\dots ,\mathrm{m}$) need to be converted into the representative value $\tilde{\mathrm{A}}$. Here, $\tilde{\mathrm{A}}=f(\widetilde{{\mathrm{A}}_1},\widetilde{{\mathrm{A}}_2},\dots ,\widetilde{{\mathrm{A}}_m})$ represents the collective opinion of all the experts. The culmination of this process is the transformation of qualitative expert assessments into quantifiable fuzzy membership values, which subsequently serve as the foundation for the FPs used in FTA [28].

2.3.2. Aggregation

Aggregation, a pivotal stage in the FFTA, serves to harmonize the various inputs from multiple experts into a collective judgment. As experts differ in their experiences and understandings, their evaluations of BEs also tend to differ [26,27]. The aggregation phase is designed to reconcile these variances through following a sequence of steps [31]:

1.

Calculating the degree of similarity

The first step is to quantify the similarity between the opinions of different experts [31]. This is achieved using TFNs to represent the degree of agreement between each pair of experts’ assessments of the BEs. The degree of similarity between the opinion $\widetilde{{\mathrm{A}}_{\mathrm{i}}}$ and $\widetilde{{\mathrm{A}}_{\mathrm{j}}}$ of experts ${\mathrm{E}}_{\mathrm{i}}$ and ${\mathrm{E}}_{\mathrm{j}}$ can be calculated as follows:

$$\mathrm{S}(\widetilde{{\mathrm{A}}_{\mathrm{i}}},\widetilde{{\mathrm{A}}_{\mathrm{j}}})=1-\frac{1}{3}\sum _{\mathrm{h}=1}^3\left|{\mathrm{a}}_{\mathrm{i}\mathrm{h}}-{\mathrm{a}}_{\mathrm{j}\mathrm{h}}\right|$$

(4)

where $\widetilde{{\mathrm{A}}_{\mathrm{i}}}$= (${\mathrm{a}}_{\mathrm{i}1}$, ${\mathrm{a}}_{\mathrm{i}2}$, ${\mathrm{a}}_{\mathrm{i}3}$) and $\widetilde{{\mathrm{A}}_{\mathrm{j}}}$ = (${\mathrm{a}}_{\mathrm{j}1}$, ${\mathrm{a}}_{\mathrm{j}2}$, ${\mathrm{a}}_{\mathrm{j}3}$) for the TFNs. The greater value of $\mathrm{S}(\widetilde{{\mathrm{A}}_{\mathrm{i}}},\widetilde{{\mathrm{A}}_{\mathrm{j}}})$ is the best similarity between two experts with respect to the fuzzy numbers of $\widetilde{{\mathrm{A}}_{\mathrm{i}}}$ and $\widetilde{{\mathrm{A}}_{\mathrm{j}}}$. The similarity index, $\mathrm{S}(\widetilde{{\mathrm{A}}_{\mathrm{i}}},\widetilde{{\mathrm{A}}_{\mathrm{j}}})$, is closest to 1 when the fuzzy values from two experts are identical, indicating a high degree of consensus [32].

2.

Computing the average agreement (AA) degree

The average agreement (AA) degree for each expert is calculated by averaging the degrees of similarity between the viewpoints of an expert and those of all other experts [31]. For each expert ${\mathrm{E}}_{\mathrm{i}}(\mathrm{i}=1,2,\dots ,\mathrm{m})$, the average agreement degree $\mathrm{A}\mathrm{A}\left({\mathrm{E}}_{\mathrm{i}}\right)$ is obtained as follows:

$$\mathrm{A}\mathrm{A}\left({\mathrm{E}}_{\mathrm{i}}\right)=\frac{1}{\mathrm{m}-1}\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^{\mathrm{m}}\mathrm{S}\left(\widetilde{{\mathrm{A}}_{\mathrm{i}}},\widetilde{{\mathrm{A}}_{\mathrm{j}}}\right)$$

(5)

3.

Computing the relative agreement (RA) degree

The relative agreement (RA) degree for each expert is calculated. The RA is obtained by comparing the AA of a particular expert with the AAs of all other experts [31]. The relative agreement degree $\mathrm{R}\mathrm{A}\left({\mathrm{E}}_{\mathrm{i}}\right)$ for each expert ${\mathrm{E}}_{\mathrm{i}}(\mathrm{i}=1,2,\dots ,\mathrm{m})$ is calculated as follows:

$$\mathrm{R}\mathrm{A}\left({\mathrm{E}}_{\mathrm{i}}\right)=\frac{\mathrm{A}\mathrm{A}\left({\mathrm{E}}_{\mathrm{i}}\right)}{\sum _{\mathrm{i}=1}^{\mathrm{m}}\mathrm{A}\mathrm{A}\left({\mathrm{E}}_{\mathrm{i}}\right)},\mathrm{i}=1,2,\dots ,\mathrm{m}$$

(6)

4.

Expert elicitation

Subsequently, the importance assigned to each expert’s evaluation, known as the weighting factors, are determined through the elicitation process. This process involves rating each expert based on factors such as their experience, knowledge, and consistency in evaluation using

Table 3. Score rating according to the experts.

Job Position	Educational Degree	Experience (Years in Service)	Weighting Score (WS)
Professor/Chief Engineer/Director	PhD	>20	5
Assistant professor/Manager	Master’s	15~20	4
Engineer, Supervisor	Bachelor’s	10~15	3
Foreman, Technician	Higher National Diploma	5~10	2
Operator	Secondary school	<5	1

[32,33]. For each expert ${\mathrm{E}}_{\mathrm{i}}(\mathrm{i}=1,2,\dots ,\mathrm{m})$, the weighting factors $\mathrm{W}\mathrm{F}\left({\mathrm{E}}_{\mathrm{i}}\right)$ are computed as follows:

$$\mathrm{W}\mathrm{F}\left({\mathrm{E}}_{\mathrm{i}}\right)=\frac{\mathrm{W}\mathrm{S}\left({\mathrm{E}}_{\mathrm{i}}\right)}{\sum _{\mathrm{i}=1}^{\mathrm{m}}\mathrm{W}\mathrm{S}\left({\mathrm{E}}_{\mathrm{i}}\right)}$$

(7)

5.

Aggregated weight calculation

An important step is the calculation of the aggregated weight $w_i$ using the relaxation factor β, which reflects the importance of one expert over another [34,35], as follows:

$${\mathrm{w}}_{\mathrm{i}}=\beta ·\mathrm{W}\mathrm{F}\left({\mathrm{E}}_{\mathrm{i}}\right)+(1-\beta )·\mathrm{R}\mathrm{A}\left({\mathrm{E}}_{\mathrm{i}}\right)$$

(8)

In this study, the value for β was chosen as 0.5, considering the heterogeneous nature of the expert panel [36]. When β = 0, the weight of an expert is not considered, indicating that a group of homogeneous experts should be employed [34]. Conversely, if β = 1, the importance given to $\mathrm{R}\mathrm{A}\mathrm{D}\left({\mathrm{E}}_{\mathrm{i}}\right)$ is not considered, suggesting a high level of consensus among the different expert opinions [35]. It is crucial for the decision maker to obtain a proper value of β.

At the end, the aggregated result of the expert’s judgement can be calculated as shown below:

$$\tilde{\mathrm{A}}={\sum }_{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{w}}_{\mathrm{j}}{\mathrm{A}}_{\mathrm{i}\mathrm{j}},\mathrm{i}=1,2,\dots ,\mathrm{m}$$

(9)

where $\mathrm{m}$ is the number of experts, $\mathrm{n}$ is the number of BEs, ${\mathrm{A}}_{\mathrm{i}\mathrm{j}}$ is the linguistic variable for the BEs given by expert $\mathrm{j}$.

2.3.3. Defuzzification

Defuzzification is a crucial step in a fuzzy logic system that transforms a fuzzy output set into a precise, actionable value. For this study, the Center of Area (COA) defuzzification method, a widely-accepted technique in fuzzy decision making, was used [37,38]. The COA method, also known as the Center of Gravity method, calculates the center of the fuzzy set area to derive a crisp, non-fuzzy output. This method finds the balance point of the membership function graph, representing the most representative point of the fuzzy output set. The COA method was used due to its robust ability to reflect the entire scope of a fuzzy set rather than focusing on maximum values or outliers. It incorporates the whole output set, providing a comprehensive and weighted representation of the defuzzified output. This is particularly important in scenarios marked by ambiguity and uncertainty, aligning seamlessly with our objective of robust decision-making in fuzzy contexts [37,38]. The precise output produced by the COA defuzzification method is essential for converting the crisp failure possibility (CFP) of BEs into FPs, forming a solid foundation for subsequent steps in the FFTA process. The defuzzified output X* can be calculated as follows:

$${\mathrm{X}}^{\mathrm{*}}=\frac{\int {\mu }_{\mathrm{A}}\left(\mathrm{x}\right)\mathrm{x}\mathrm{d}\mathrm{x}}{\int {\mu }_{\mathrm{A}}\left(\mathrm{x}\right)\mathrm{d}\mathrm{x}}=\frac{{\displaystyle \underset{{\mathrm{a}}_1}{\overset{{\mathrm{a}}_2}{\int }}}\frac{\mathrm{x}-{\mathrm{a}}_2}{{{\mathrm{a}}_2-\mathrm{a}}_1}\mathrm{x}\mathrm{d}\mathrm{x}+{\displaystyle \underset{{\mathrm{a}}_2}{\overset{{\mathrm{a}}_3}{\int }}}\frac{{\mathrm{a}}_3-\mathrm{x}}{{{\mathrm{a}}_3-\mathrm{a}}_2}\mathrm{x}\mathrm{d}\mathrm{x}}{{\displaystyle \underset{{\mathrm{a}}_1}{\overset{{\mathrm{a}}_2}{\int }}}\frac{\mathrm{x}-{\mathrm{a}}_2}{{{\mathrm{a}}_2-\mathrm{a}}_1}\mathrm{d}\mathrm{x}+{\displaystyle \underset{{\mathrm{a}}_2}{\overset{{\mathrm{a}}_3}{\int }}}\frac{{\mathrm{a}}_3-\mathrm{x}}{{{\mathrm{a}}_3-\mathrm{a}}_2}\mathrm{d}\mathrm{x}}=\frac{1}{3}({\mathrm{a}}_1+{\mathrm{a}}_2+{\mathrm{a}}_3)$$

(10)

2.3.4. Converting Crisp Failure Possibility of Basic Events into Failure Probability

The process of converting the CFP into a FP is a crucial step in reconciling the differences between probability and fuzzy theory. The CFP, derived from the defuzzification of fuzzy numbers, represents the failure possibility as a ‘crisp’ or clear number. However, this value does not inherently align with the probabilistic nature of FP, necessitating a conversion process. In this stage, a transformation function is employed to define the relationship between the CFP and the FP, playing a pivotal role in bridging the gap between fuzzy theory and probability and providing a FP estimate that takes uncertainty into account. A commonly used transformation function is the logarithmic function proposed by Onisawa [23]. This function can be represented as follows:

$$\mathrm{F}\mathrm{P}=\left\{\begin{array}{c}\frac{1}{{10}^{\mathrm{K}}},\mathrm{CFP}\ne 0\\ 0,\mathrm{CFP}=0\end{array},\mathrm{K}={\left[(\frac{1}{\mathrm{C}\mathrm{F}\mathrm{P}}-1)\right]}^{\frac{1}{3}}\right.\times 2.301$$

(11)

This function ensures that when CFP values lie between 0 and 1, the FP also falls within the same range, respecting the basic tenet of probability that FP should range from 0 (indicating no possibility of failure) to 1 (indicating absolute certainty of failure). Through this transformation process, the CFP, which encompasses uncertainty, is converted into a probabilistic FP, enabling a comprehensive assessment of the overall possibility of system error occurrence while maximizing the utilization of uncertainty information provided by fuzzy theory and enabling the application of probabilistic analysis methods. This value provides a quantifiable measure of risk, considering the inherent uncertainties involved in AUV operations in polar environments.

2.4. Ranking of Minimal Cut Sets

After calculating the FP in the FTA, we assessed the significance of each Minimal Cut Set (MCS) using the Fussell–Vesely Importance Measures (FV-I) [39]. This method determines the importance of each BE within the MCS, considering not only the probability of the BEs occurring but also the change in system FP should the event be prevented. The FV-I can be computed using the following equation:

$$\mathrm{FV}-\mathrm{I}=\frac{Q_i\left(t\right)}{Q_{TE}\left(t\right)}$$

(12)

where $Q_i\left(t\right)$ is the FP of MSC $i$, and $Q_{TE}\left(t\right)$ is the FP of TE. Ranking the BEs according to the FV-I enables us to identify which events exert the most substantial impact on the system’s overall reliability, aiding in prioritizing BEs for further investigation or preventative measures.

3. Results of Fuzzy Fault Tree Analysis

Section 3 presents the derivation of the FTA results for our study, as outlined in the methodology detailed in Section 2. We began by identifying ‘AUV loss’ as the TE, as discussed in Section 2. The potential failures leading to this TE were classified into two primary categories: (1) inherent system failures and (2) polar environment-induced failures triggered by environmental factors. The identification and categorization of these failures were based on the methodology outlined in Section 2. The subsequent fault tree connects these two types of failures to the TE via OR gates, illustrating that the occurrence of either failure type could lead to the TE. The complete fault tree that illustrates these relationships is provided in Figure 4, and

Table 4. Abbreviations for fault tree diagram.

ID	Definition	ID	Definition
TE	AUV loss
I	Inherent system failure	P	Polar environment-induced failure
I1	Power system failure	P1	Collision or trap due to ice
I2	Propulsion and maneuvering failure	P2	Inaccurate navigation
I3	Navigation failure	P3	Poor communication
I4	Abnormal condition detection failure	P4	Recovery failure
I11	Power management failure	P5	Buoyant control failure
I21	Motor failure	P6	Battery failure
I22	Direction control failure	P41	Ice condition
I221	Shaftlesstruster failure	P42	Weather condition
I23	Buoyancy control failure	B1-1	Ice thickness
I31	Navigation information acquiring failure	B2-1	Ice concentration
I32	Base camp networking failure	B2-11	Large ice concentration
I33	Navigation calculation failure	B2-12	Small ice concentration
I41	Leakage detection failure	B3-1	Roughness of ice underside
I42	Abnormal temperature detection failure	B4-1	Underwater current
I411	Alarm sensor failure	B1-2	Ice thickness
I412	Case/o-ring damage	B2-2	Ice concentration
A1	Low quality battery	B2-21	Large ice concentration
A2	Connecter or cable failure	B2-22	Small ice concentration
A3	Batter failure due to aging	B3-2	Roughness of ice underside
A4	Motor malfunction	B1-3	Ice thickness
A5	Bearing failure	B2-3	Ice concentration
A6	Short circuit	B2-31	Large ice concentration
A7	Thruster bearing failure	B2-32	Small ice concentration
A8	Lack of lubrication for thruster	B5	Surface wind
A9	Actuator failure	B6	Surface wave
A10	Thruster failure	B7	Invisibility
A11	iUSBL failure	B8	Rapid salinity change
A12	USBL with ATM failure	B4-2	Underwater current
A13	DVL/ADCP failure	B9	Low temperature
A14	IMU failure
A15	Depth meter failure
A16	iUSBL failure
A17	USBL with ATM failure
A18	PC hardware failure
A19	Software error
A20	Internal networking interface failure
A21	Leakage alarm failure
A22	Pressure alarm failure
A23	Poor coating thickness
A24	O-ring aging
A25	Temperature alarm failure
A26	Abnormal temperature due to overcurrent

provides abbreviations for fault tree diagram. In the case of inherent system failures, the probabilities are founded on historically acquired failure rates, which inherently incorporate a time dimension as they are expressed in reciprocal units of time. This allows for a dynamic representation of the system’s reliability over time. On the other hand, the failure probabilities associated with polar environment-induced failures are not time-dependent; rather, they are situation-dependent and calculated via fuzzy logic based on the prevailing environmental conditions at the time of the mission. Unlike the inherent system failures, these probabilities are fundamentally different in that they do not require a time element for their calculation and interpretation. In this context, we have identified intermediary events attributed to AUV failure modes that could potentially act as precursors to the top event. Importantly, these intermediary events are assumed to be independent, aligning with the foundational assumptions of traditional FTA. Further complicating this framework, we categorized the causal events that contribute to these intermediary failure modes as extreme environmental factors. While some environmental factors are considered in multiple causal events, each factor’s unique impact on different intermediary events is separately assessed.

3.1. Inherent System Failures and Probability Analysis

The primary objective of this study was to understand the impact of factors associated with polar environments on AUV loss; therefore, a general AUV model was considered. This section addresses failures intrinsic to conventional AUVs rather than those specific to a particular model. The failures are grouped into four categories corresponding to the key components of a typical AUV: power system failure (I1), propulsion and maneuvering failure (I2), navigation failure (I3), and abnormal condition detection failure (I4) [7,40]. These categories not only represent the primary components of a typical AUV but also form the basis of system failures. The failure rate data for the reliability analysis were obtained from previous studies and are presented in

Table 5. Failure rates for basic events of system failure.

	Failure Rate (λ × 10−6/h)	Failure Probability (for 24 h)
A1	0.24	5.76 × 10−6
A2	5	1.20 × 10−4
A3	1	2.40 × 10−5
A4	3.33	7.99 × 10−5
A5	1	2.40 × 10−5
A6	10	2.40× 10−4
A7	1	2.40 × 10−5
A8	1	2.40 × 10−5
A9	1	2.40 × 10−5
A10	33.3	7.99 × 10−4
A11	2	4.80 × 10−5
A12	2	4.80 × 10-5
A13	0.1	2.40 × 10−6
A14	2	4.80 × 10−5
A15	2	4.80 × 10−5
A16	2	4.80 × 10−5
A17	2	4.80 × 10-5
A18	1	2.40 × 10−5
A19	1	2.40 × 10−5
A20	3.62	8.70 × 10−5
A21	13.59	3.26 × 10−4
A22	1	2.40 × 10−5
A23	1	2.40 × 10−5
A24	10	2.40 × 10−4
A25	1	2.40 × 10−5
A26	6.5	1.56 × 10−4

. These data were then utilized to calculate the FP of the system, represented as P(I) in Equation (2) [7,40].

3.2. Fuzzy Set Theory for Environment Factors

For the environmental factors, we defined six intermediate events. These encompass collisions or traps due to ice (P1), inaccurate navigation (P2), poor communication (P3), recovery failure (P4), buoyant control failure (P5), and battery failure (P6). Failures due to these factors are potentially influenced by ten environmental variables: ice thickness, large ice concentration, small ice concentration, the roughness of the ice’s underside, underwater current, surface wind, surface wave, invisibility, salinity change, and low temperature. Notably, we divided the ice concentration into large and small based on its size. Large ice typically refers to land-derived ice such as ice shelves, icebergs, and bergy bits. For sea ice, large ice also encompasses multi-year ice and other ice formations larger than the AUVs with an egg code of 3 or higher. This classification stems from the potential for significant AUV damage upon collision with sizable ice formations. Conversely, small ice refers to sea ice formations that are smaller than the AUVs with an egg code of 3. This includes new ice, thin ice, and smaller ice floes or fragments that are not land-derived. Although small ice may seem less threatening compared to large ice formations, it can still pose significant challenges for AUV operations. For example, small ice can cause abrasions to the AUV’s surface, obstruct sensors, or lead to entrapment in narrow passages. Additionally, small ice concentrations can vary rapidly with changing environmental conditions, making it difficult to predict and navigate through.

Furthermore, some environmental variables influence multiple IEs simultaneously. For example, ice thickness, large ice concentration, small ice concentration, and the roughness of the ice’s underside were evaluated for their impacts on several IEs, relying on expert opinions. As a result, as can be seen in Figure 4, there are 18 BEs in total.

In this study, four experts were involved, including two researchers for AUV and Remotely Operated Vehicle (ROV) development, one polar environment scientist, and one AUV engineer specializing in matters related polar areas.

Table 6. Expert weighting score and factor.

	Job Position	Educational Degree	Experience	Weighting Score (WS)	Weighting Factor (WF)
Expert 1	5	5	4	14	0.2593
Expert 2	5	5	5	15	0.2778
Expert 3	5	4	5	14	0.2593
Expert 4	4	4	3	11	0.2037

details the weights and calculated weighting factors (WF) assigned to each expert.

The experts were asked to evaluate the impact of each environmental factor on AUV loss using five linguistic levels: very high, high, medium, low, and very low (Table 2). The above-mentioned 18 BEs were further classified into three categories—good, moderate, and bad—depending on the environmental conditions. The experts provided their judgments on the severity of the environmental factors and the corresponding impacts on AUV loss.

Table 7. Experts’ judgements on basic events.

	Environmental Factors		No.	Condition	Expert
					1	2	3	4
Collision or trap due to ice (P1)	Ice thickness (B1-1)		B1-1G	Thin	VL	L	M	L
			B1-1M	Moderate	L	M	M	M
			B1-1B	Thick	M	H	M	H
	Ice concentration (B2-1)	Large (B2-11)	B2-11G	0~25%	L	M	M	M
			B2-11M	25~75%	L	M	M	M
			B2-11B	75~100%	H	M	M	VH
		Small (B2-12)	B2-12G	0~25%	VL	L	M	L
			B2-12M	25~75%	L	M	M	L
			B2-12B	75~100%	M	H	H	M
	Roughness of ice underside (B3-1)		B3-1G	Low	VL	L	M	M
			B3-1M	Moderate	M	M	H	M
			B3-1B	Irregular	H	H	VH	VH
Inaccurate navigation (P2)	Underwater current (B4-1)		B4-1G	Weak	VL	M	M	L
			B4-1M	Moderate	M	H	M	M
			B4-1B	Strong	VH	VH	H	H
Poor communication (P3)	Ice thickness (B1-2)		B1-2G	Thin	H	M	M	M
			B1-2M	Moderate	H	H	M	M
			B1-2B	Thick	H	VH	M	H
	Ice concentration (B2-2)	Large (B2-21)	B2-21G	0~25%	M	M	M	L
			B2-21M	25~75%	M	H	M	M
			B2-21B	75~100%	M	VH	M	H
		Small (B2-22)	B2-22G	0~25%	L	L	M	L
			B2-22M	25~75%	L	M	M	L
			B2-22B	75~100%	L	H	H	M
	Roughness of ice underside (B3-2)		B3-2G	Low	M	L	M	M
			B3-2M	Moderate	H	M	H	H
			B3-2B	Irregular	VH	H	VH	VH
Recovery failure (P4)	Ice thickness (B1-3)		B1-3G	Thin	L	M	M	L
			B1-3M	Moderate	H	H	H	M
			B1-3B	Thick	VH	VH	H	VH
	Ice concentration (B2-3)	Large (B2-31)	B2-31G	0~25%	M	M	M	M
			B2-31M	25~75%	H	H	H	H
			B2-31B	75~100%	VH	VH	H	VH
		Small (B2-32)	B2-32G	0~25%	VL	L	M	L
			B2-32M	25~75%	L	M	H	M
			B2-32B	75~100%	M	H	VH	H
	Surface wind (B5)		B5G	Weak	L	L	L	VL
			B5M	Moderate	M	M	M	L
			B5B	Strong	M	H	H	M
	Surface wave (B6)		B6G	Calm	H	M	L	L
			B6M	Moderate	VH	H	M	M
			B6B	High	VH	VH	H	VH
	Invisibility (B7)		B7G	Negligible	L	M	L	L
			B7M	Partial	M	H	M	M
			B7B	Complete	M	VH	H	H
Buoyant control failure (P5)	Rapid salinity change (B8)		B8G	Negligible	H	L	L	L
			B8M	Moderate	VH	M	M	M
			B9B	Rapid	VH	H	H	H
	Underwater current (B4-2)		B4-2G	Weak	L	M	M	L
			B4-2M	Moderate	M	H	M	M
			B4-2B	Strong	M	VH	H	H
Battery failure (P6)	Low temperature (B9)		B9G	Good	H	M	L	VL
			B9M	Moderate	H	H	M	L
			B9B	Very low	VH	VH	VH	M

shows the results regarding the expert judgments for each of the 18 BEs under the three respective conditions. In conclusion, the experts evaluated the three conditions of the 18 BEs, meaning that they gave their opinions on a total of 54 BEs.

The fuzzy numbers given in Table 2 were utilized to aggregate the experts’ judgments. An example result for B1-1 is demonstrated in

Table 8. Aggregation for B1-1G.

	Linguistic Opinion	Fuzzy Numbers
Expert 1	VL	0	0.1	0.3
Expert 2	L	0.1	0.3	0.5
Expert 3	M	0.3	0.5	0.7
Expert 4	L	0.1	0.3	0.5
Similarity degree	S12	0.833
	S13	0.633
	S14	0.833
	S23	0.800
	S24	1.000
	S34	0.800
Average agreement degree	AA (E1)	0.767
	AA (E2)	0.878
	AA (E3)	0.744
	AA (E4)	0.878
Relative agreement degree	RA (E1)	0.235
	RA (E2)	0.269
	RA (E3)	0.228
	RA (E4)	0.269
Aggregated weight calculation	w (E1)	0.247
	w (E2)	0.273
	w (E3)	0.244
	w (E4)	0.236
Aggregated results	0.124	0.299	0.499
Defuzzification	0.308
Failure probability	0.001

. The experts shared four differing opinions: very low, low, medium, and low. Using Equation (4), we computed S12, the degree of similarity between the opinions of experts $E_1$ and $E_2$. This process was repeated to establish the degree of similarity across all pairs of experts. Once we computed all degrees of similarity, the average agreement degree could be determined using Equation (5). Sequentially, the relative agreement degree for each expert was calculated using Equation (6). With reference to Table 6, the weighting factors $\mathrm{W}\mathrm{F}\left(E_i\right)$ were obtained using Equation (7). As a result, we could derive the aggregated TFNs using Equations (8) and (9). Subsequently, the aggregated TFNs were converted into CFP through a defuzzification process using Equation (10). The CFP was further transformed into the FP of TE utilizing Equation (11). By repeating this procedure for all BEs, we obtained aggregated TFNs, as shown in Figure 5.

After determining the FP, we utilized Equation (12) to compute the FV-I, thereby identifying the MCSs that increase the TE probability. Considering that the fault tree associated with environmental factors is linked by OR gates, each BE constitutes an MCS. Therefore, in this study, the ranking of MCS is synonymous with the ranking of BEs.

The impact of changing environmental conditions on AUV loss is illustrated through the alterations in FV-I and the consequential changes in the rankings shown in

Table 9. Importance of basic events and their rankings.

			Good		Moderate		Bad
			FV-I	Ranking	FV-I	Ranking	FV-I	Ranking
Collision or trap due to ice (P1)	Ice thickness (B1-1)		0.022	15	0.027	15	0.028	17
	Ice concentration (B2-1)	Large (B2-11)	0.079	4	0.027	15	0.034	14
		Small (B2-12)	0.022	16	0.018	17	0.029	15
	Roughness of ice underside (B3-1)		0.036	12	0.053	12	0.089	7
Inaccurate navigation (P2)	Underwater current (B4-1)		0.039	11	0.053	8	0.092	6
Poor communication (P3)	Ice thickness (B1-2)		0.157	1	0.076	5	0.051	9
	Ice concentration (B2-2)	Large (B2-21)	0.083	3	0.053	8	0.036	13
		Small (B2-22)	0.033	14	0.018	17	0.021	18
	Roughness of ice underside (B3-2)		0.078	5	0.102	3	0.121	4
Recovery failure (P4)	Ice thickness (B1-3)		0.054	8	0.107	2	0.122	1
	Ice concentration (B2-3)	Large (B2-31)	0.113	2	0.140	1	0.122	1
		Small (B2-32)	0.022	16	0.038	13	0.050	12
	Surface wind (B5)		0.013	18	0.028	14	0.029	15
	Surface wave (B6)		0.078	6	0.097	4	0.122	1
	Invisibility (B7)		0.033	13	0.053	8	0.051	9
Buoyant control failure (P5)	Rapid salinity change (B8)		0.048	10	0.066	6	0.068	8
	Underwater current (B4-2)		0.054	8	0.053	8	0.051	9
Battery failure (P6)	Low temperature (B9)		0.059	7	0.058	7		5

. Some environmental factors, when in a ‘good’ state, have a minimal impact on AUV loss. However, as these conditions worsen to ‘moderate’ or ‘bad’, they can significantly increase the probability of AUV loss. This is exemplified by the effects of ‘ice thickness’ and ‘invisibility’ on failures in AUV recovery. Conversely, there are certain environmental factors that, surprisingly, may exert less influence on AUV loss in ‘bad’ conditions than when the conditions are ‘good’. An instance of this trend is noted in the impact of ‘large ice concentration’ on events such as ‘collision or trap due to ice’ and ‘poor communication’.

4. Discussion

This section assesses the potential for AUV loss in polar regions by utilizing the fault tree shown in Figure 4. As illustrated in the fault tree, the assessment was primarily based on two key events: (1) inherent system failures, representing the intrinsic malfunctions of the system, and (2) polar environment-induced failures, representing the failures caused by the harsh polar conditions. When the FTA considers inherent system failures exclusively, the result will only reflect the AUV loss probability due to system malfunctions. Conversely, the integration of both events provides a comprehensive evaluation of AUV loss probability in polar regions.

The results of the FTA are shown in Figure 6. When only considering polar environment-induced failures, the calculated AUV loss probabilities are 0.044, 0.132, and 0.350 in good, moderate, and bad environments, respectively, showing a significant increase with worsening environmental conditions. On the other hand, when only considering inherent system failures, the AUV loss probability was calculated to be 1.79 × 10⁻³ based on a 24 h operating mission. This value is very small compared to the values calculated when only considering polar environment-induced failures, indicating that the proportion of inherent system failures in AUV loss in polar regions is negligible compared to polar environment-induced failures. Moreover, the AUV loss probability in polar regions, calculated considering both polar environment-induced failures and inherent system failures, is very similar to that calculated considering only polar environment-induced failures, as shown in Figure 6. This is because the value for inherent system failures is insignificant. The AUV loss probabilities at the 75th quartile in previous studies [5,13] are 0.17 and 0.40 for sea ice and ice shelf conditions, respectively. When it is assumed that the bad and moderate environmental conditions in this study are similar to the ice shelf and sea ice environments, the AUV loss probabilities derived in this study are similar to those presented in previous studies. This suggests that the FFTA methodology used in this study can effectively analyze the risks despite their variability due to various environmental factors.

The risk analysis provides a snapshot of AUV loss probability, reflecting specific conditions at the time of the assessment. While this model is adept at incorporating immediate environmental variations by dynamically updating the probabilities related to polar environment-induced failures, it is currently not designed to adapt to changes in operator skill and experience over time. However, our FTA model is not static; it is built to evolve. As operator teams gain more experience and skills, reducing the overall operational risk, it would be prudent to revisit the expert evaluations contributing to the FTA. This will allow for a recalibration of the risk estimates to account for enhanced operator efficiency and expertise. Therefore, at points where it is empirically evident that the team’s skill level has led to reduced risk, a comprehensive update of the FTA risk metrics should be initiated. Regarding inherent system failures, they are calculated based on empirically acquired failure rates, which inherently factor in the aging effects of the system over time. This ensures that our FTA model remains a dynamic and adaptive tool that is capable of accounting for changing environmental conditions and improvements in operational maturity, serving as a comprehensive tool for evaluating the risks associated with AUV operations in polar regions.

While our FTA model and risk analysis have taken into account some essential elements that contribute to AUV loss, it is imperative to delve deeper into the polar environment-induced failures for a more comprehensive understanding. Various factors in the polar environment significantly impact the probability of AUV loss. Here, we enumerate and discuss these in detail:

• Ice Thickness: The varying thickness of ice in polar regions can obstruct AUV recovery and affect communication systems, as thicker ice layers inhibit signal penetration.
• Large Ice Concentration: A high concentration of ice can cause mechanical stress on the AUV or lead to collision and entrapment scenarios.
• Small Ice Concentration: Even a sparse amount of ice can affect AUV operations by obstructing sensors or causing navigational errors.
• Roughness of Ice’s Underside: The irregular surface of the underside of ice poses a substantial navigation risk for the AUV, increasing the likelihood of collision or entrapment.
• Rapid Salinity Change: Quick changes in water salinity can affect the AUV’s buoyancy control systems, destabilizing its navigation and potentially leading to loss.
• Low Temperature: Extremely cold temperatures can adversely affect battery life and electronic components, increasing the risk of system failure.
• Underwater Current: Unpredictable underwater currents can significantly alter the AUV’s planned trajectory, making navigation challenging.
• Surface Wind: Although our study found this to have a relatively minor impact, strong surface winds can still affect surface operations related to AUV deployment and recovery.
• Surface Wave: Surface wave conditions can also impede AUV operations at the water surface, affecting both deployment and recovery.
• Invisibility: Poor visibility conditions can hinder effective operation management and real-time decision-making, increasing operational risk.

Figure 7 represents the results of P1–P6, which constitute the polar environment-induced failure in the FTA of Figure 4. Recognizing the challenges AUVs encounter in polar environments and issues with high-failure probabilities is critical.

In bad conditions, ‘recovery failure’, with an FP of 0.161, stands out as a primary concern. Influencing factors such as ‘ice thickness’ and ‘ice concentration’ increase this challenge. Thick or irregular ice formations can hinder the AUV’s recovery process, leading to potential damage or its complete loss. ‘Poor communication’, with an FP of 0.078 in bad conditions, significantly hampers AUV operations. ‘Ice thickness’, ‘ice concentration’, and the ‘roughness of ice underside’ are key factors affecting communication. In bad conditions, thick ice layers make it challenging for signals to penetrate effectively. The risk of ‘collision or trap due to ice’ increases as conditions worsen to bad. In bad conditions, the FP is 0.061. Specifically, the ‘roughness of ice underside’ poses a substantial risk, making navigation for the AUV more challenging. ‘Inaccurate navigation’ poses significant threats under moderate to bad conditions. In bad conditions, its FP stands at 0.032. The unpredictable ‘underwater currents’ in polar terrains can severely deviate the AUV’s planned trajectory, making navigation difficult. For ‘buoyant control failure’, the concern escalates with an FP of 0.041 in bad conditions. In such environments, maintaining buoyancy can become a challenging task for the AUV. Lastly, ‘battery failure’ is another concern, particularly in ‘low temperatures’. In bad conditions, the FP for this issue is 0.034. Low temperatures adversely affect the battery’s performance and overall lifespan.

Figure 8 illustrates the degree of environmental impact on the loss of AUVs, as it represents a reorganization of the FP values from Figure 5, aggregated for each environmental factor. As shown in Figure 8, ‘roughness of ice underside’, ‘ice thickness’, and ‘large ice concentration’ are among the environmental factors that exhibit the highest FP values across all conditions (good, moderate, and bad). Conversely, ‘surface wind’ has a relatively minor impact on the AUV’s FP compared to other environmental variables. ‘Low temperature’, ‘rapid salinity change’, ‘surface wave’, ‘invisibility’, and ‘underwater current’ also significantly affect the AUV’s FP in bad conditions.

In good conditions, environmental factors like ‘low temperature’, ‘rapid salinity change’, ‘surface wave’, ‘invisibility’, and ‘underwater current’ have relatively low impacts on the AUV’s FP, indicating that they do not significantly compromise the AUV’s stability. On the other hand, in moderate conditions, environmental factors like ‘low temperature’, ‘rapid salinity change’, ‘surface wave’, ‘invisibility’, and ‘underwater current’, although higher compared to good conditions, are still relatively lower than in bad conditions. This suggests that while these factors do impact the AUV’s stability in moderate conditions, proper management and measures can minimize the risk.

The analysis conducted in this section is crucial for enhancing our comprehension of the environmental factors that escalate the FP during the operation and management of AUVs. Such analyses, (i.e., those based on the opinions of experts) facilitate a better understanding of which environmental factors increase the FP during the operation and management of AUVs. However, it is important to note that the specific values and impacts may vary based on the opinions of the experts consulted. Therefore, the selection of a broader and more appropriate range of experts is of utmost importance. Additionally, this data can be utilized in the design and improvement process of AUVs, contributing to the development of more stable underwater robots. Ultimately, this highlights the need for a comprehensive approach that considers a wide range of expert opinions and leverages this knowledge for the ongoing development and enhancement of AUVs.

5. Conclusions

This study conducted a risk analysis utilizing the fuzzy fault tree analysis methodology to clarify the loss risk of autonomous underwater vehicles during polar explorations. The basic events of the fault tree were categorized into inherent system failures based on historical failure rate data, and polar environment-induced failures were found to have vast uncertainties due to environmental factors. Inherent system failures were calculated using traditional methods, whereas polar environment-induced failures were evaluated based on expert opinions using fuzzy set theory. The research presented in this study detailed the influence of nine environmental factors, and experts assessed the importance of each environmental factor in linguistic terms. The assessment of the experts were then converted into failure probabilities through fuzzification, aggregation, and defuzzification processes. The results of the fuzzy fault tree analysis revealed that the autonomous underwater vehicle loss probability significantly increases with the worsening of environmental conditions. Specifically, polar environment-induced failures play a more substantial role in autonomous underwater vehicle loss in polar regions compared to inherent system failures. The calculated autonomous underwater vehicle loss probabilities are considerably similar to those presented in previous studies, validating the effectiveness of the fuzzy fault tree analysis methodology in analyzing risks due to environmental factors. Furthermore, the analysis identified ‘recovery failure’, ‘inaccurate navigation’, ‘roughness of ice underside’, and ‘poor communication’ as the most significant threats to autonomous underwater vehicle operation in polar environments. Environmental factors like ‘ice thickness’, ‘large ice concentration’, and ‘roughness of ice underside’ exhibited the highest failure probabilities across all conditions. Conversely, factors like ‘surface wind’, ‘low temperature’, ‘rapid salinity change’, ‘surface wave’, ‘invisibility’, and ‘underwater current’ had relatively minor impacts on autonomous underwater vehicle’s failure probability in ‘good’ and ‘moderate’ conditions and, while they showed increased effects in bad conditions, their impacts were still less pronounced compared to factors such as ‘ice thickness’, ‘large ice concentration’, and ‘roughness of ice underside’. This study’s results underscore the importance of considering environmental factors in the design and operation of autonomous underwater vehicles in polar regions, as well as the necessity of implementing proper management and measures to minimize the risks associated with these factors.