In the nonlinear state estimation, the generation method of cubature points and weights of the classical cubature Kalman filter (CKF) limits its estimation accuracy. Inspired by that, in this paper, a novel improved CKF with adaptive generation of the cubature points and weights is proposed. Firstly, to improve the accuracy of classical CKF while considering the calculation efficiency, we introduce a new high-degree cubature rule combining third-order spherical rule and sixth-degree radial rule. Next, in the new cubature rule, a novel method that can generate adaptively cubature points and weights based on the distance between the points and center point in the sense of the inner product is designed. We use the cosine similarity to quantify the distance. Then, based on that, a novel high-degree CKF is derived that use much fewer points than other high-degree CKF. In the proposed filter, based on the actual dynamic filtering process, the simultaneously adaptive generation of cubature points and weight can make the filter reasonably distribute the cubature points and allocate the corresponding weights, which can obviously improve the approximate accuracy of one-step state and measurement prediction. Finally, the superior performance of the proposed filter is demonstrated in a benchmark target tracking model.

Summary This article investigates the state estimation problem of the nonlinear system with the large process prior uncertainty but high‐accuracy measurement information, the situation is prone to occur in the inertial navigation system (INS)/global navigation satellite system (GNSS) tightly coupled navigation system. Furthermore, the unknown heavy‐tailed measurement noises induced by the inaccurate prior navigation information and random measurement outliers are also considered. Given existing methods such as progressive cubature Kalman filter (PCKF) cannot effectively solve the above issues, a novel robust PCKF with variable step size (RVSS‐PCKF) is proposed. First, a new Gaussian‐uniform‐mixing inverse Gamma (GUMIG) distribution is presented to model the variable step size and measurement noise. Next, the GUMIG distribution is expressed as a hierarchical Gaussian presentation and then RVSS‐PCKF is derived with the help of the variational Bayesian (VB) inference. In the filter, the state, variable step size and noise statistic are jointly estimated by the fixed‐point iterations. Finally, the simulations and real data of the tightly coupled navigation illustrate the superiority of the filter in terms of accuracy and steady‐state performance.

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