Sparse-grid Quadrature Filter Research Articles

Double‐state chi‐square test based sparse grid quadrature filtering algorithm and its application in integrated navigation

AbstractAiming to reduce the impact that the measurement uncertainty of nonlinear system brings to filtering accuracy, a novel sparse grid quadrature filter (SGQF) algorithm based on the double‐state chi‐square test is proposed. And the strap‐down inertial navigation system (SINS)/global navigation satellite system (GNSS) integrated navigation system is used to verify the algorithm. A discrete‐time system model with uncertain measurement is constructed, then the system state through the SGQF algorithm is estimated. To determine the uncertainty of the measurement, the double‐state chi‐square test is introduced. By comparing the results of the state recursor, the chi‐square test value is obtained and it is used to update the measurement noise. Further, the system state and error covariance are re‐estimated. So far a robust sparse grid filter algorithm with statistical adjustment of the measurement noise is obtained. The proposed algorithm can overcome the influence of error statistical characteristic amplification on system filtering accuracy. Taking the SINS/GNSS integrated navigation system as the application object, off‐line simulation experiments and online flight tests verify the robustness of the proposed algorithm.

Compressed Positive Quadrature Filter

A compressed positive quadrature filter (CPQF) is proposed in this technical note. Accuracy, efficiency, and numerical stability are the main concerns for quadrature-based nonlinear Gaussian filtering, but desirable performance in all these three aspects can hardly be simultaneously achieved in previously proposed nonlinear Gaussian filters, such as unscented Kalman filter, Gauss–Hermite quadrature filter (GHQF), and sparse grid quadrature filter. To improve the overall capacity of nonlinear Gaussian filtering techniques, the CPQF is proposed by numerically solving the constrained moment matching equations. By enforcing the positivity of the quadrature weights, the CPQF is numerically as stable as the GHQF. Through exploiting the underlying sparsity while solving the moment matching equations, the CPQF can also achieve the same level of accuracy as the GHQF using far fewer quadrature points. Simulations are conducted to evaluate the effectiveness of the CPQF, and the numerical results demonstrate its superior performance over conventional Gaussian filters.

Distributed State Fusion Using Sparse-Grid Quadrature Filter With Application to INS/CNS/GNSS Integration

This paper presents a new methodology of distributed state fusion by using the sparse-grid quadrature filter to deal with the fusion estimation problem for multi-sensor nonlinear systems. In this methodology, the sparse-grid quadrature filter is performed in a distributed manner to process the information at each local node; and subsequently, a distributed state fusion approach based on matrix weighting is established in the sense of mean square error, in which a flexible procedure is developed based on the sparse-grid quadrature rule to calculate the cross-covariances between any two local estimators for multi-sensor nonlinear systems. The proposed methodology of distributed fusion can obtain higher fusion estimation accuracy in a flexible way, leading to improved fusion performance for multi-sensor nonlinear systems. The simulations and experiments in INS/CNS/GNSS (inertial navigation system/celestial navigation system/global navigation satellite system) integration verifies the effectiveness of the proposed methodology.

M-estimation based sparse grid quadrature filter and stochastic stability analysis

In this study, a novel M-estimation based sparse grid quadrature filter (MSGQF) is proposed to improve the robust performance of the nonlinear system. We present a systematic formulation of the sparse grid quadrature filter (SGQF), and extend it to the discrete-time nonlinear system with abnormal measurement values. The M-estimation method is introduced in the SGQF, which uses the Huber’s cost function to update the measurement covariance. Convergence on the modified robust SGQF is established and proved. The sufficient conditions are shown to ensure stochastic stability of the MSGQF. A target tracking problem has been conducted to demonstrate the accuracy of the MSGQF. When measurement abnormal values appear, it outperforms the unscented Kalman filter (UKF), the cubature Kalman filter (CKF) and the SGQF. Theoretical analysis and simulation results prove that the MSGQF provides significant performance improvement in the robustness of the nonlinear system.

MASGQF with application to SINS alignment

To effectively improve the computation efficiency in the strapdown inertial navigation system (SINS) alignment process, the authors propose a marginalised adaptive sparse-grid quadrature filter (MASGQF). The filtering method combines the virtues of marginalisation technique and accuracy level switching structure to decrease the computational burden and improve the filtering precision simultaneously. With the adaptation criterion and the predefined tolerance value, different integration rules corresponding to the different accuracy levels nested in the proposed filtering method can be adjusted autonomously. It is proved that the MASGQF is exponentially bounded by theoretical analysis. The performance comparison of different filtering methods is demonstrated by the numerical results of the simulation and offline experiment. On the basis of the data performance of the online test, the validity of the proposed MASGQF is verified in the SINS alignment application.

Composite weighted average consensus filtering for space object tracking

In this paper, a composite weighted average consensus filtering (CWACF) algorithm is proposed for space object tracking by combining two distributed heterogeneous nonlinear filters. In light of the sensors' different sensing accuracy and computational capability, extended Kalman filter (EKF) and sparse-grid quadrature filter (SGQF) are compositely adopted on different sensors as local filters. Then, estimates from neighbours are fused based on the weighted average consensus framework to attain better estimation performance. Moreover, it is proved that the estimation error is exponentially bounded in mean square. The performances of the proposed algorithm, the distributed extended Kalman filtering (DEKF) and the distributed sparse-grid quadrature filter (DSGQF) are compared in a space object tracking problem.

Relations Between Sparse-Grid Quadrature Rule and Spherical-Radial Cubature Rule in Nonlinear Gaussian Estimation

The recently emerging cubature Kalman filter and the sparse-grid quadrature filter approximate the numerical integrations for the mean and covariance in Gaussian filters using the spherical-radial cubature rule and the sparse-grid quadrature rule, respectively. This technical note reveals that 1) spherical rules can be obtained by the projection of sparse-grid quadrature rules; 2) the third- and some of the fifth-degree spherical-radial cubature rules can be directly constructed from sparse-grid quadrature rules.

Delayed Genz–Keister Sequences-Based Sparse-Grid Quadrature Nonlinear Filter With Application to Target Tracking

An improved quadrature nonlinear filter named delayed Genz-Keister sequences-based sparse-grid quadrature filter (DGKSGQF) is developed for the target tracking problems. The filter changes the non-nested Gaussian quadrature points of the quadrature filters to the nested Genz-Keister points for selecting the unvariate points, which are the basis point sets extended to form a multidimensional grid using the sparse-grid theory. As a result, the points used for lower accuracy levels DGKSGQF can be reused for any higher accuracy level. Thus, it can further reduce the number of total points used for the conventional Gauss-Hermite SGQF without sacrificing performance. The proposed filter is applied to the reentry ballistic target tracking problem. The simulation results show that the DGKSGQF achieves higher accuracy than the EKF and the UKF. In addition, it can more flexibly control the performance in terms of the number of points and accuracy level.

Vision-Based Spacecraft Relative Navigation Using Sparse-Grid Quadrature Filter

In this paper, vision-based relative navigation of two spacecraft is addressed using the sparse-grid quadrature filter. The relative navigation provides the estimates of the relative orbit and relative attitude as well as the gyro biases. It is a challenging problem because of its high nonlinearity and dimensionality. The extended Kalman filter (EKF) and the unscented Kalman filter (UKF) have been used in the past to solve this problem. However, these filters are not accurate enough in the presence of large initial uncertainties or high nonlinearities. Moreover, although other filters, such as the Gauss-Hermite quadrature filter and the particle filter, can be more accurate than the EKF and UKF, they are hard to use in this high-dimensional estimation problem since a large number of quadrature points or particles are required and therefore the computation complexity is prohibitive. It is shown in this paper that the new sparse-grid quadrature filter can achieve much higher estimation accuracy than EKF, UKF, and the cubature Kalman filter without excessive computation load.

Sparse-grid quadrature nonlinear filtering

In this paper, a novel nonlinear filter named Sparse-grid Quadrature Filter (SGQF) is proposed. The filter utilizes weighted sparse-grid quadrature points to approximate the multi-dimensional integrals in the nonlinear Bayesian estimation algorithm. The locations and weights of the univariate quadrature points with a range of accuracy levels are determined by the moment matching method. Then the univariate quadrature point sets are extended to form a multi-dimensional grid using the sparse-grid theory. Compared with the conventional point-based methods, the estimation accuracy level of the SGQF can be flexibly controlled and the number of sparse-grid quadrature points for the SGQF is a polynomial of the dimension of the system, which alleviates the curse of dimensionality for high dimensional problems. The Unscented Kalman Filter (UKF) is proven to be a subset of the SGQF at the level-2 accuracy. The performance of this filter is demonstrated by an orbit estimation problem. The simulation results show that the SGQF achieves higher accuracy than the Extended Kalman Filter (EKF), the UKF, and the Cubature Kalman Filter (CKF). In addition, the SGQF is computationally much more efficient than the multi-dimensional Gauss–Hermite Quadrature Filter (GHQF) with the same performance.