Reliability Analysis of Multi-Autonomous Underwater Vehicle Cooperative Systems Based on Fuzzy Control

Abstract

Autonomous underwater vehicles (AUVs) perform a wide range of functions, since underwater circumstances are diverse and varied and resources are plentiful. However, existing prevalent theoretical computation methods for classifying underwater environments are unable to keep up with the constantly evolving and complicated undersea world. Therefore, in the design process of multi-AUV cooperative systems, there is usually some uncertainty in the parameters. This uncertainty creates some challenges in the design process and affects the system’s performance. In this study, a reliability-based multidisciplinary design optimization is presented, in which some of the problem parameters are uncertain. In this regard, it is assumed that some of the problem parameters are in the form of fuzzy numbers. We develop a multi-AUV cooperative system dependability model based on fuzzy control, leveraging the membership function of fuzzy control to thoroughly evaluate the environmental effects. We propose a method to evaluate the reliability of multi-AUV cooperative systems, which is of practical value. This solution can be used to evaluate and examine the cooperative system’s dependability in an unidentified underwater environment. The factors contributing to the reliability of multi-AUV cooperative systems are obtained. The results show that these contributing factors reflect different aspects of reliability for multi-AUV cooperative systems. It is possible to determine the reliability variation in the population of underwater vehicles operating in a complex underwater environment, which establishes a solid foundation for the further optimization of multi-AUV cooperative systems.

1. Introduction

The responsibilities of autonomous underwater vehicles (AUVs) are becoming more challenging as a result of humankind’s growing demands in relation to water. It is difficult for a single AUV to accomplish such increasingly precise and varied tasks. Thus, miniaturization, intellectualization, hybrid, and group models are being developed in relation to AUVs [1]. Currently, multi-AUV cooperative systems that are made up of several AUVs are used in real-world situations. The importance of reliability research will become clearer as the number of AUVs and the complexity of the mission increase. First, regarding the various types of multi-AUV cooperative systems, the number of AUVs differs, the system structure differs, and the complexity of the system is growing. Second, multi-AUV cooperative systems have been deployed in real marine environments where varying conditions may heighten the risk of failure. Reliability research in relation to multi-AUV cooperative systems will help them to fulfill their missions in a variety of marine environments. Third, multi-AUV cooperative systems are being developed to handle complicated tasks and achieve intensive cooperation, which has resulted in a continued increase in their assigned strength and density. The severity of underwater environments leads to an increase in system failure rates. The reliability research of multi-AUV cooperative systems has important practical significance for increasing their efficiency and ensuring the successful completion of their missions [2,3,4].

Reliability research relating to multi-AUV cooperative systems is gaining more and more attention with the increase in the potential applications of these systems. Currently, Bayesian theory and the Monte Carlo method are commonly used to calculate the reliability of systems [5,6,7]. However, these methods are not comprehensive enough to consider the environmental factors of practical applications. Mehrparvar and his partners analyzed the two methods of Fuzzy analytic hierarchy process and fuzzy VIKOR method for Auto AV shuttle, contrasting fuzzy control with other evaluation methods [8].

Gerkey and Howard of USC’s Robotics Research Lab Vaughan of its Information Sciences Laboratory [9] studied the reliability of a cooperative system composed of a multi-robot/sensor network; they presented an improved program of communication, which improved the reliability of the system. A multi-layered and distributed architecture for mission-oriented, miniature fixed-wing UAV swarms was presented by Zhihong Liu et al. [10], and each UAV achieved its goal in executing the associated decision process. Rabbath and Lé chevin of Defense Research and Development Canada [11] presented health-state estimation methods that can improve reliability based on a large number of faults and errors that have the ability to influence the reliability of multi-UAVs. In order to solve the communication challenges associated with autonomous underwater vehicle systems, Singh and Elsayed investigated the integration of diverse technologies with DWDM-FSO, using free-space optical communication to facilitate communication between autonomous underwater vehicles and ground-based systems [12,13].

The complicated nature of underwater environments has an impact on underwater vehicles and group communication, and the working environment has a significant impact on the dependability of a multi-AUV cooperative system. The AUV’s regular operations are impacted by the water’s turbulence, scattering, and refraction, which also make team communication more difficult; as a result, the dependability of the system is impacted. The study of idealized environmental conditions was the focus of some earlier publications, but thorough analyses and explanations of the factors that affect reliability are lacking [14,15]. Traditional techniques of dependability analysis rely on the classification of underwater environments, which presents a challenge for effective evaluation when dealing with uncharted waters. The parameters of underwater environments usually have some uncertainty. This uncertainty affects the performance of cooperative systems and creates some challenges for the design of reliability evaluations for cooperative systems. Therefore, a reliability-based multidisciplinary design optimization is presented in this study, in which some of the problem parameters are uncertain. The concept of fuzzy control is introduced into the evaluation model [16]. We found that the fuzzy control system is an effective way to deal with complex systems. It is necessary to introduce the fuzzy control system, according to the membership function, to analyze underwater environments that are difficult to quantify. Furthermore, the fuzzy rules act to judge and dispose the underwater environment, which is particularly applicable because of the uncertain nature of the underwater environment. It can improve the adaptability of the reliability model to complex environments. Therefore, an improved evaluation model for the reliability verification of multi-AUV cooperative systems is proposed in this study, which is capable of evaluating the cooperative system’s dependability in uncharted waters and uncharted underwater environments.

This section will introduce the article’s organizational structure. The influence of the undersea environment on AUVs and group communication is examined in Section 2, and its effects were comprehensively quantified using the fuzzy control’s membership function. Then, a robust model for multi-AUV cooperative systems, grounded in fuzzy control, is proposed. Real-world applications are employed as the primary reference element in Section 3, and some helpful conclusions are derived from the analysis of several indices impacting the system’s dependability. The article is concluded in Section 4.

2. Reliability Analysis of Multi-AUV Cooperative Systems

2.1. Reliability Analysis of Multi-AUV Systems

In multi-AUV systems, each AUV independently executes its designated task through the underwater communication network and collaboratively forms an integrated cooperative system. The interference caused by the underwater environment on AUV group communication is a critical factor that influences the overall reliability of the system. This study utilizes fuzzy control theory’s membership function to comprehensively analyze the aforementioned interference effects, thereby evaluating the performance stability of the system under complex underwater environments. Reliability is the ability of a system to perform specified functions within prescribed conditions and a prescribed time. The probability measure of reliability is known as the degree of reliability and is usually expressed as R. Because R is a function of time, it is defined as R = R(t), which is known as the reliability function; its value range is as follows:

$$0\le R\left(t\right)\le 1$$

(1)

Corresponding to reliability is unreliability, which is also known as the error rate; it is represented by h, and h(t) is the error function, as follows:

$$R(t)+h(t)=1$$

(2)

The operational underwater vehicle reliability of a single AUV can generally be expressed as follows:

$$R=\exp \left[-{\displaystyle {\int }_0^{t}h(t)dt}\right]$$

(3)

where h(t) is the error rate at time t. This can be obtained through relevant experiments and described by statistical methods, such as the first-order second-moment and Monte Carlo methods, among others.

As for multi-AUV cooperative systems, we estimate that a cooperative system requires several different actions and several communications, where each action and communication is likely to make mistakes and the errors are independent of each other. Then, the reliability of the AUV device in an ideal environment is expressed as R_n^S_n. In the following equations, S_n and q_mn are reliability parameters of underwater environments and communication systems, respectively.

$$R={R_1}^{S_n}\cdot {R_2}^{S_n}\cdot {R_3}^{S_n}\cdot \dots \cdot {R_N}^{S_n}={\displaystyle \prod _{n=1}^N{R_n}^{S_n}}$$

(4)

$$R_N=1-h_N={\displaystyle \prod _{m=1}^M(1-q_{mn)}}$$

(5)

Therefore, the reliability of multi-AUV cooperative systems is calculated as follows:

$$R={{\displaystyle \prod _{n=1}^N\left[{\displaystyle \prod _{m=1}^M(1-q_{mn})}\right]}}^{S_n}$$

(6)

2.2. Influence of Complex Environments on AUVs

The underwater environment has a great influence on multi-AUV cooperative systems. There are many factors that affect the reliability of cooperative systems; this study mainly analyzes the influence of complex environments on AUVs and cooperative communication.

Currently, it is generally believed that the accurate hydrodynamic model and advanced control methods are two key factors that can be used to improve the performance of AUVs. Firstly, the dynamic and hydrodynamic models of AUVs are nonlinear in complex underwater environments; among them, the quadratic damping effect significantly reduces the performance of conventional controllers. Secondly, the dynamic and hydrodynamic characteristics of AUVs in practical applications are uncertain. There are many uncertain factors in the current situation, such as changes in model parameters caused by the aging or damage of parts; disturbances caused by the action of wind, waves, and currents in the external environment; and so on [17]. Therefore, the synthetic evaluation method based on fuzzy control is used to describe the impact of environmental factors on AUVs, which is more realistic.

We assume that the environmental factors affecting the AUV constitute the following factor set:

$$A=\left\{a_1\right.,a_2,\dots ,\left.a_i\right\}$$

(7)

Among them, the larger i is, the more comprehensive the consideration of environmental factors is.

The influence degree of the environment on the AUV is expressed according to the following influence degree set:

$$B=\left\{b_1\right.,b_2,\dots ,\left.b_j\right\}$$

(8)

In the formula, the J-value is proportional to the influence of the environment on the vehicle, which can be expressed according to the corresponding parameter value, as given below:

$$c_k=k^2(k=1,2\dots ,j)$$

(9)

where c_k is the parameter value of the B_k level in the influence degree set.

The fuzzy relationship between set A and set B can be determined using statistical methods to quantify the influence of environmental factors on the AUV as follows:

$$r_{nk}={\mu }_{B_k}(a_k)$$

(10)

Then, the fuzzy relation matrix between set A and set B can be obtained as follows:

$$R_m={\left[r_{nk}\right]}_{i\times i}$$

(11)

In the working process of AUVs, various environmental factors have different influences. The effect of each environmental factor on the influence degree set constitutes a fuzzy subset of set A as follows:

$$W={\displaystyle \sum _{n=1}^N{\omega }_n/a_n}$$

(12)

where ω_n represents the measure of the effect of environmental factors on the influence degree. The concept of weight is as follows:

$${\displaystyle \sum _{n=1}^N{\omega }_n=1}$$

(13)

In this way, the impact of each environmental factor on the AUV is evaluated as follows:

$$D=W\cdot R_m=(d_1,d_2,\dots ,d_i)$$

(14)

where

$$d_i={\displaystyle \sum _{n=1}^N{\omega }_n}\cdot r_{nk}(k=1,2,\dots ,j)$$

(15)

After processing set B, the influence degree of various influencing factors on the AUV can be obtained, as given below:

$$y={\displaystyle \sum _{k=1}^i{d}_k^m}\cdot c_k/{\displaystyle \sum _{k=1}^i{d}_k^m}$$

(16)

Among them, k is mainly used for the purpose of distinguishing between the parameters of the influence level, depending on the specific situation—in general, k = 2. The comprehensive impact of the environment on the AUV is expressed as a fuzzy subset, and the following can be obtained:

$${\mu }_1(y)=(y-1)/({\alpha }^2-1)$$

(17)

The membership function μ₁(y) quantifies the comprehensive impact of the environmental factors on the AUV.

2.3. Influence of Complex Environments on Group Communication

Multi-AUV cooperative systems require continuous underwater wireless optical communication (UWOC) to communicate instruction transmission, position confirmation, and other related information. However, the underwater environment is exceedingly complicated due to the existence of stochastic elements such as turbulence, pollutants, colloids, and so forth. There are three main factors that relate to the optical characteristics of the water environment—pure water, dissolved matter, and suspensions. A diagrammatic sketch of an UWOC environment is shown in Figure 1. There are two important optical properties of various components in water—the absorption and scattering of lasers. Lasers suffer from extreme absorption and Mie scattering in water or seawater. Blue–green wavelengths, as a distinct transmission window of the seawater for underwater optical wireless communication, also suffer from Mie scattering due to the water molecules, salt ions, phytoplankton, and other transparent biological organisms [18,19].

Absorption and scattering characteristics will cause attenuation characteristics of laser transmission in the water. The attenuation coefficient is the sum of the water absorption coefficient and the scattering coefficient [20,21].

Monte Carlo simulation employs four typical algorithms for modeling photon migration [19,22]. In this study, the transmission model of UWOC is established and simulated using the Monte Carlo method. The laser source is assumed to be a collection of photons, and the final results are obtained by tracking the transmission process of a large number of photons. Assuming that the light source is located at the origin of the spatial rectangular coordinate system, the positive z axis is in the zenith direction. The laser intensity of the source obeys the Gaussian distribution, and the initial scattering angle θ is determined by the source [23] as follows:

$${\theta }_0=-r_0/f_l$$

(18)

Among them, r₀ is the sampling radius:

$$r_0=d_0{\left[-\ln (1-\xi )\right]}^{1/2}$$

(19)

$$f_l=-d_0/{\vartheta }_{div/2}$$

(20)

where d₀ is the laser beam width, ϑ_div/2 is the divergence half angle, and the azimuth angle φ = 2πξ. The free step length between the twice-scattered photons is determined according to the probability of scattering or absorption. According to Bill’s Law, the attenuation coefficient underwater is determined by the absorption coefficient and scattering coefficient of the water.

The bubbles in the water can cause photons to scatter; moreover, it is mainly Mie scattering, for which its spatial scattering angle adopts the function of Henyey–Greenstein to approximate its scattering phase function as follows:

$$p(\cos \theta )={\displaystyle \frac{1-g^2}{2{(1-2g\cos \theta +g^2)}^{3/2}}}$$

(21)

where θ is the scattering angle and satisfies the following relationship:

$$\theta =\left\{\begin{array}{c}\arccos (\frac{1}{2g}[1+g^2-{(\frac{1-g^2}{1-g=2g\xi })}^2])\\ \arccos (2\xi -1)\end{array}\right.\begin{array}{c},g\ne 0\\ ,g=0\end{array}$$

(22)

where g is an asymmetric factor with a threshold value of [−1, 1], and ξ is a random number with a threshold value of [0, 1].

The weight statistical forecast method believes that photons are likely to be transmitted with a certain probability at each collision point. The cosine value of the forward direction of the photon can be confirmed according to the scattering angle and azimuth angle of the next scattering as follows:

$$\left\{\begin{array}{l}{{\mu }_x}^{\prime }={\displaystyle \frac{\sin \theta }{{(1-{\mu }_z^2)}^{1/2}}}({\mu }_x{\mu }_z\cos \phi -{\mu }_y\sin \phi )+{\mu }_x\cos \theta \\ {{\mu }_y}^{\prime }={\displaystyle \frac{\sin \theta }{{(1-{\mu }_z^2)}^{1/2}}}({\mu }_y{\mu }_z\cos \phi +{\mu }_x\sin \phi )+{\mu }_y\cos \theta \\ {{\mu }_z}^{\prime }=-\sin \theta \cos \phi {(1-{\mu }_z^2)}^{1/2}+{\mu }_z\cos \theta \end{array}\right.$$

(23)

For the influence of turbulence on UWOC, Hassan Making Oubei and other researchers have proposed the Generalized Gamma Distribution (GGD) as a new description function of underwater optical channels [16]. The researchers showed that the GGD is closer to the experimental measurement data than the Weber distribution, gamma distribution, and exponential distribution. Additionally, the GGD can represent the turbulence of different intensities; its probability density function is as follows:

$$f(I;h,j,k)={\displaystyle \frac{c{I}^{hk-1}}{j^{hk}}}\times \frac{exp(-{(\frac{I}{j})}^k)}{\Gamma \left(h\right)},I>0;h,j,k>0$$

(24)

where j is the scale parameter, h and k are shape parameters, and Γ(h) presents the gamma function. Different GGDs can be obtained with different h and j values; for example, the Weber distribution can be obtained when h = 1 and the exponential distribution can be obtained when h = j = 1. The measure of turbulence light fluctuation intensity in relation to underwater turbulence is the scintillation index, which is expressed as σ_I²:

$${\sigma }_I^2={\displaystyle \frac{\Gamma \left(h\right)\Gamma (h+\frac{2}{k})}{{\Gamma }^2(h+\frac{1}{k})}}-1$$

(25)

The GGD model was used to simulate underwater turbulence of different intensities; the results are shown in Figure 2. Finally, different scintillation coefficients are obtained, which are shown in

Table 1. Generalized Gamma Distribution channel parameters.

H	j	k	σI2
1.620	0.765	1.440	0.311
1.622	1.020	0.851	0.880
1.408	0.630	0.840	1.025

. According to the findings, when there is a lot of turbulence in the water, the scintillation index is higher than 1. It is commonly accepted that there is moderate turbulence when the scintillation index is between 0.75 and 1, and weak turbulence is observed when the scintillation index is less than 0.75. We can classify underwater turbulence based on the results above, and we can use this to model how complicated underwater environments affect signals.

During the transmission of photons, the weights are attenuated by collisions with various impurities in water, and the photons carry the weights for the next scattering. We can regard the propagation process of UWOC as a Markov process, whereby each evolution of the attenuation and propagation process is not affected by the previous evolution. When photons are transmitted underwater, their interaction process also satisfies the Markov property [24], which can be used to directly superimpose the attenuation results. Many studies have demonstrated that the Markov-based statistical sampling method is effective for developing risk, performance, and other evaluation models [25,26,27]. In this study, the attenuation of wireless optical transmission underwater is simulated. At the same time, in order to better adapt to real-world communication environments, there are a small number of relatively agglomerated small molecular impurities in the transmission path, and the evolution of the attenuation and propagation process is not affected by the previous evolution. The wavelength of the blue–green laser is 532 nm. In order to observe the influence of the underwater environment on optical communication, we set the initial weight as 100, assuming that the photons will be scattered 70 times during the underwater transmission process. Meanwhile, the asymmetry factor g is 0.924 and σ_I² is 0.86 [19]. Through 100 rounds of simulation tests, according to the remaining weight value of each laser at the end of transmission, we randomly selected ten of these process simulations; the attenuation diagram of the laser weight during each simulation is shown in Figure 3.

In Figure 3, an obvious descending process can be seen in the weight attenuation curve of each simulation, especially after quadratic fitting, between the 30th and the 40th scattering. These rapid and strong energy decrements are a result of the significant impacts that underwater turbulence has on the laser. This also shows that the quality of information interactions between the multi-AUVs is closely related to the water environment.

In multi-AUV cooperative systems, the water quality of the working environment varies with the working area; unpredictable environments in water can cause unpredictable interference with group communications. Therefore, the influence of underwater environments on communication is a fuzzy concept. The interference degree of communication is graded as a variable x; in the actual case, the degree of interference is the variable in [0, +∞]. We can consider that the membership function of the interference degree of the laser signal is as follows:

$${\mu }_2(x)=\left\{\begin{array}{cc} & ,x<1\\ 0.5+0.5{\sin }_1^0\left[(x-1)/4\right]\pi ,& 1\le x\le 3\\ & ,x>3\end{array}\right.$$

(26)

2.4. Reliability Calculation Model Based on Environmental Factors

The complexity of underwater environments will affect every communication of the multi-AUV cooperative system and the quality of communication will be impaired. It directly affects the execution and judgment of the multi-AUV cooperative system. Hence, the harsh environment leads to an increase in the number of group communications and the error rate of the AUV will also increase. The action of multi-AUV cooperative systems may need to be repeated many times.

The influence of AUVs and group communications on sn and qmn is exponential. At the same time, the influence degree of complex environments on AUVs and group communications is different, and this difference can be expressed by the weight of machine influence ω₁ and the weight of communication influence ω₂. The complex environment of the tasks managed by the cooperative system influences the value of ω₁, while ω₂ is primarily determined according to the complexity of the system itself. By adjusting the ratio of ω₁ to ω₂, we can obtain evaluation results in scenarios where the task is complex, the number of cooperative machines is substantial, or both factors are intricate. Therefore, the sum of ω₁ and ω₂ is 1, assuming that s_n and q_mn are statistical data in an ideal setting (e.g., a stable underwater environment and a stable communication system). Considering the fuzzy influence of the working environment, the sum of parameters s_n and q_mn in Equation (2) can be obtained as follows:

$$s_n^{\prime }=s_n\left\{{\omega }_1\exp \left[{\mu }_1(y)\right]+{\omega }_2\exp \left[{\mu }_2(x)\right]\right\}$$

(27)

$$q_{mn}^{\prime }=q_{mn}\left\{{\omega }_1\exp \left[{\mu }_1(y)\right]+{\omega }_2\exp \left[{\mu }_2(x)\right]\right\}$$

(28)

After considering the probabilistic impact of complex environments on AUVs and group communication, on the basis of the reliability calculation model of ideal environments, we can obtain the calculation model of reliability based on environmental factors as follows:

$$R^{\prime }={{\displaystyle \prod _{n=1}^N\left[{\displaystyle \prod _{m=1}^M(1-q_{mn}^{\prime })}\right]}}^{s_n^{\prime }}$$

(29)

Figure 4 shows a comparison of reliability between the ideal case and a case in which environmental factors are introduced. We assume that underwater environments are kept at a medium level to obtain the above data. It is not difficult to make out the influence of the model on reliability calculations. As the number of group communications increases, the reliability between the ideal situation and the real situation will gradually increase. The difference is caused by the influence of environmental factors on AUVs and group communications. The model proposed by this study is closer to that of the real application scenario, and the obtained data have a higher credibility and authenticity.

3. Experimental Results

In the common industrial underwater vehicle, there are three major components—the sensor system, the core system, and the actuator system. Then, based on the calculation model in this study, we can obtain the variation in reliability for dual AUV cooperative systems with water quality, communication frequency, and different weights of machine and communication influence.

First of all, each time a dual AUV cooperative system is used, the reliability gradually decreases, the water quality gradually becomes worse, and the communication frequency increases. Additionally, as the ratio of ω₁ and ω₂ gradually decreases, the reliability of the cooperative system is gradually improved. The color changes in Figure 4 illustrate the changes in the reliability of the cooperative system when the ratio of ω₁ and ω₂ changes. Figure 5a–c depict the ratio values of 3/7, 1, and 7/3, respectively. Hence, the impact of a complex environment on the reliability of AUVs is the primary element influencing the dependability of cooperative systems. Future research should focus on how to ensure the stable and efficient operation of multi-AUV cooperative systems in complex environments.

Based on the dual AUV cooperative model, we also conducted reliability simulations for small- and medium-sized AUV cooperative systems. These simulations account for variations in the underwater environment and the number of AUVs within the cooperative system. We assumed a ratio of ω₁ and ω₂ as 1 and that the multi-AUV system requires 100 communication exchanges to complete the cooperative task, as illustrated in Figure 6a. We can also obtain the relationship between system reliability, water quality, and quantity of AUVs following 20 communications, as shown in Figure 6b.

When there are too many AUVs in cooperative systems, the reliability of the system drops dramatically. At the same time, if cooperative systems are too complex and require a lot of communication, the reliability of the system can also drop dramatically. In contrast, the variation curve of water quality to system reliability is relatively smooth, so the quantity of AUVs is the decisive factor for system reliability. Based on this model, the influence of different water environments, different quantities of AUVs, and different communication amounts on system reliability can be obtained. The average reliability of the cooperative system can be ascertained under specific environmental conditions, and the fluctuations in reliability can be comprehended. This model provides a strong basis for the further optimization of multi-AUV cooperative systems and even for their practical applications.

4. Conclusions

Unlike other reliability models, the one proposed by this study integrates environmental factors, which are grounded in the inherent fuzziness of the operational environment for multi-AUV cooperative systems. AUVs are applied in water, causing the whole system to be in a fuzzy varying state, i.e., a number of reliability indices will change with environmental factors. Despite the cooperative system’s working mode and structure remaining unchanged, the reliability evaluation index will vary due to alterations in the state of the underwater environment. Therefore, this study introduces fuzzy control and establishes a computational model to study the cooperative system in fuzzy environments; additionally, the model can evaluate the dependability of the cooperative system without having to investigate and evaluate the underwater environment. The affecting factors of systems are analyzed in the nearby natural environment, which have a certain reference value in practical applications. The multi-agent system designed based on this scheme can achieve more robust functionalities. However, it also encounters several challenges, including the need for an effective information synchronization mechanism and a cooperative working framework among multiple agents [28]. Moreover, the determination of the failure rate of the AUVs used in this study is based on certain experience and experiments; more profound research will be carried out in relation to the failure rates of AUVs in the future.