A novel improved CKF with adaptive generation of the cubature points and weights for target tracking

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.

Mixed-Degree Cubature H∞ Information Filter-Based Visual-Inertial Odometry

Visual–inertial odometry is an effective system for mobile robot navigation. This article presents an egomotion estimation method for a dual-sensor system consisting of a camera and an inertial measurement unit (IMU) based on the cubature information filter and H∞ filter. The intensity of the image was used as the measurement directly. The measurements from the two sensors were fused with a hybrid information filter in a tightly coupled way. The hybrid filter used the third-degree spherical-radial cubature rule in the time-update phase and the fifth-degree spherical simplex-radial cubature rule in the measurement-update phase for numerical stability. The robust H∞ filter was combined into the measurement-update phase of the cubature information filter framework for robustness toward non-Gaussian noises in the intensity measurements. The algorithm was evaluated on a common public dataset and compared to other visual navigation systems in terms of absolute and relative accuracy.

Simplex Cubature Kalman-Consensus Filter for Distributed Space Target Tracking

A simplex cubature Kalman-consensus filter, which is suitable for distributed space target tracking using multiple radars, is proposed to improve the target tracking accuracy. The detailed orbital dynamics model and radar measurement model are given as the system filtering models. The intractable nonlinear Gaussian weighted integral in the filter is decomposed into the spherical integral and radial integral, which are calculated using the spherical simplex rule and the second-order Gauss-Laguerre quadrature rule, respectively. In this way, a new simplex cubature rule is derived. By means of the statistical linear regression method, the posterior mean, covariance, and the transmitted messages in the extended Kalman-consensus filter are approximated using the deduced simplex cubature rule, which results in the proposed simplex cubature Kalman-consensus filter. No data fusion center exists in the filter, and each radar only needs to exchange the information with its neighbors to reach a consensus estimate. The simulation results show that the proposed filter can achieve more accurate results compared with the cubature Kalman-consensus filter.

A New Sparse Gauss-Hermite Cubature Rule Based on Relative-Weight-Ratios for Bearing-Ranging Target Tracking

A new sparse Gauss-Hermite cubature rule is designed to avoid dimension explosion caused by the traditional full tensor-product based Gauss-Hermite cubature rule. Although Smolyak’s quadrature rule can successfully generate sparse cubature points for high dimensional integral, it has a potential drawback that some cubature points generated by Smolyak’s rule have negative weights, which may result in instability for the computation. A relative-weight-ratio criterion based sparse Gauss-Hermite rule is presented in this paper, in which cubature points are kept symmetric in the input space and corresponding weights are guaranteed to be positive. The generation of the new sparse cubature points set is simple and meaningful for practice. The difference between our new sparse Gauss-Hermite cubature rule and other cubature rules is analysed. Simulation results show that, compared with Kalman filter with those types of full tensor-product based Gauss-Hermite rules, our new sparse Gauss-Hermite cubature rule based Kalman filter can lead to a substantially reduced number of cubature points, more stable computation capability, and maintaining the accuracy of integration at the same time.

Bivariate generalized Bernstein operators and their application to Fredholm integral equations

We introduce and study the sequence of bivariate Generalized Bernstein operators {Bm,s}m,s, m, s ? N, Bm,s=I?(I?Bm)s, Bi m = Bm(Bi?1 m), where Bm is the bivariate Bernstein operator. These operators generalize the ones introduced and studied independently in the univariate case by Mastroianni and Occorsio [Rend. Accad. Sci. Fis. Mat. Napoli 44 (4) (1977), 151- 169] and by Micchelli [J. Approx. Theory 8 (1973), 1-18] (see also Felbecker [Manuscripta Math. 29 (1979), 229-246]). As well as in the one-dimesional case, for m fixed the sequence {Bm,s(f)}s can be successfully employed in order to approximate ?very smooth? functions f by reusing the same data points f (i/m,j/m), i=0,1,...,m, j=0,1,...,m, since the rate of convergence improves as s increases. A stable and convergent cubature rule on the square [0,1]2, based on the polynomials Bm,s(f) is constructed. Moreover, a Nystrom method based on the above mentioned cubature rule is proposed for the numerical solution of two-dimensional Fredholm integral equations on [0, 1]2. The method is numerically stable, convergent and the involved linear systems are well conditioned. Some algorithm details are given in order to compute the entries of the linear systems with a reduced time complexity. Moreover the procedure can be significantly simplified in the case of equations having centrosymmetric kernels. Finally, some numerical examples are provided in order to illustrate the accuracy of the cubature formula and the computational efficiency of the Nystrom method.