## 1 Introduction  
Nowadays, autonomous underwater vehicles (AUVs) have been widely used by many communities for commercial, offshore, and defense applications [1]. Especially in deep-sea exploration and development, AUVs are playing significant roles [2]. This is largely due to the rapid development of underwater navigation technology. To complete predefined underwater tasks, such as surveying, underwater detection, and underwater reconnaissance, the importance of accurate position and velocity information for AUVs is increasingly being emphasized [3]. The inertial navigation system (INS) is the most widely used position- and speed-acquisition technology for underwater navigation [4]. For cost reasons, the microelectromechanical systems (MEMSs)-based inertial sensors are commonly used in AUVs. However, because of the characteristics of MEMS-based sensors, the error introduced by the INS solution increases over time. These errors can lead to a sharp drop in the positioning performance of AUVs [5]. In addition, due to the existence of various noises, such as ocean currents, salt cliffs, and surrounding ships, the marine environment is complicated and time-varying, resulting in the loss of measurement information [6, 7]. To eliminate the influence of complex measurement noise and the measurement information randomly lost, several Bayesian-theory-based filtering algorithms have been proposed and applied, such as the extended Kalman filter (EKF) [8, 9], unscented Kalman filter (UKF) [10], CubatureKalman filter (CKF) [11], and so on. The performance of the filtering algorithms is crucial in judging whether the AUV can properly work and safely return [12; 13; 14]. EKF is the most widely used algorithm for integrating navigation signals, such as position, velocity, and so on, which can restrict the system defects and improve the calculation accuracy. However, it is well-known that the measurement and process noises of the EKF are assumed to be uncorrelated, and both are assumed to satisfy the zero-mean Gaussian distribution. Due to the presence of stochastic non-Gaussian noise, the inaccurate noise assumptions may cause velocity and position drifts. Moreover, EKF struggles to deal with the impact of system uncertainty. In recent years, many improved algorithms based on EKF have been developed, such as adaptive EKF [15], fuzzy EKF [16], iterated EKF, and so on. However, for strongly nonlinear systems, the truncation error, caused by the Taylor-series expansion-based linearization process [17; 18; 19], leads to serious degradation of filter performance. To overcome the influence of truncation error, improved UKF-algorithm-based unscented transformation and UKF-based algorithms are proposed as maximum correlation entropy-based UKF and adaptive UKF, respectively [20; 21; 22]. For MEMS-grade INS, UKF is more adaptable to highly nonlinear models than EKF [23; 24]. Similar to UKF, CKF is a sigma point Gaussian approximation filter. However, unlike UKF, which is a uniform selection of 2n+1 sigma points around the expectation, the spherical cubature rule is used by the CKF to select 2n cubature points and thus estimate the mean and correlation covariance of the posterior probability density by solving for the spherical mean integral. Some CKF-based algorithms, such as embedded CKF and variational Bayesian adaptive CKF, have been proposed to achieve navigation signal integration in recent years [25; 26]. When the same matrix decomposition methods, such as square root decomposition, are used for UKF and CKF, the computational burden of CKF is lower than that of UKF. Moreover, due to the insensitivity of the CKF to increases in dimension, the stability of CKF is better than that of UKF if the system has a high dimension, e.g., the dimensional order is greater than three. However, the complex marine environment can cause a loss of measurement information, so these navigation algorithms tend to diverge when working underwater and have very limited practical application effects.