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• A GNSS/INS integrated navigation system with the SW-KF, SHKF , and VBAKF is implemented in this paper. • The differences and connections of three adaptive filters for measurement noise are discussed from multiple perspectives. • Mathematical derivation proves the three adaptive filters schemes are equivalent. • The VBAKF always estimates positive definite matrices and demonstrates the best performance in measurement noise estimation. Adaptive filters are widely used to deal with the problem of inaccurate measurement noise in nonlinear systems, such as state estimation of GNSS/INS integrated systems. In the complex environment, GNSS signals are frequently interrupted, and the prior observation noise is no longer suitable for the observations with time-varying measurement conditions. This uncertainty will interfere with the confidence of each sensor’s observations and reduce the filtering accuracy. The existing adaptive filters have their limitations in practical applications, and their performance in the GNSS/INS integrated system needs to be evaluated in depth. Given this, a GNSS/INS integrated system with the variational Bayesian-based Kalman filter, Sage-Husa filter, and sliding window-based filter is implemented in this paper. The differences and connections of different adaptive filters for measurement noise are discussed from multiple perspectives through mathematical derivation, and the equivalence of the three filtering schemes is proved. The performances of filtering under different parameters are compared and analyzed using simulation and field experiments. The results show that the sliding window-based filter is adaptable to the abrupt environment. Still, it cannot guarantee the estimation accuracy of the noise covariance matrix in the stationary observation environment. Both the Sage-Husa filter and variational Bayesian-based filter are suitable for the environment where the observation noise is stationary; the difference is that the covariance matrix of Sage-Husa filter estimation cannot be guaranteed to be positive definite, and the effect in a practical application depends on the setting of the minimum value and cannot achieve adaptive estimation. The estimated noise covariance matrix of the variational Bayesian-based filter is always positive definite, which has the best estimation effect in the application.
Li et al. (Sun,) studied this question.