Fifth-degree Cubature Kalman Filter Research Articles

Autonomous Navigation Based on the Earth-Shadow Observation near the Sun–Earth L2 Point

This paper is devoted to a novel autonomous navigation method for spacecraft around the Sun–Earth L2 point. In contrast to the previous navigation methods, which rely on ground-based or inter-spacecraft measurements, the proposed method determines the orbit based on Earth-shadow measurements. First, the navigation framework using the Earth-shadow measurement is proposed. Second, the geometric analysis is used to derive the mathematical model of the Earth-shadow measurements. Then, the fifth-degree Cubature Kalman filter (CKF) is designed to estimate the states of the spacecraft. Numerical simulations are implemented to validate the performance of the proposed navigation method. Finally, the simulation results show that the navigation system is observable and that the proposed method could be potentially useful for an autonomous navigation mission near the Sun–Earth L2 point in the future.

Multilayer Low-Cost Sensor Local-Global Filtering Fusion Integrated Navigation of Small UAV

Aimed at improving the nonlinear integrated navigation solution performance of multiple low-cost sensors fusion, this paper presents a multilayer loosely-coupled, local-global, and step-optimized MF5DCKF (Multisensor Federated fifth-degree Cubature Kalman filter) state estimation algorithm for the small unmanned aerial vehicle (UAV). This method establishes a multilayer nonlinear integrated navigation model composed of the nonlinear attitude and heading reference system (AHRS) error model, strapdown inertial navigation system/global positioning system (SINS/GPS) error model, and strapdown inertial navigation system/barometer (SINS/BARO) error model to enhance the robustness and richness of the navigation module. Further, based on the above navigation models, a loosely-coupled error state fusion frame is designed to obtain the local convergent state vector. Simultaneously, a three-layer fifth-degree Cubature Kalman filter is proposed to improve the local state estimation accuracy. Subsequently, to optimize the estimated local state, this paper presents a novel distributed MF5DCKF scheme fusing the local state vector to calculate the global optimal state parameters in a step-optimized process. The experimental flight test results show that the proposed algorithm achieves a higher state solution accuracy and a better convergent performance compared with some conventional multisensor fusion algorithms. The new algorithm framework can provide applicability and reliability for the small UAV during the flight.

MATLAB-based general approach for square-root extended-unscented and fifth-degree cubature Kalman filtering methods

A stable square-root approach has been recently proposed for the unscented Kalman filter (UKF) and fifth-degree cubature Kalman filter (5D-CKF) as well as for the mixed-type methods consisting of the extended Kalman filter (EKF) time update and the UKF/5D-CKF measurement update steps. The mixed-type estimators provide a good balance in trading between estimation accuracy and computational demand because of the EKF moment differential equations involved. The key benefit is a consolidation of reliable state mean and error covariance propagation by using delicate discretization error control while solving the EKF moment differential equations and an accurate measurement update according to the advanced UKF and/or 5D-CKF filtering strategies. Meanwhile the drawback of the previously proposed estimators is an utilization of sophisticated numerical integration scheme with the built-in discretization error control that is, in fact, a complicated and computationally costly tool. In contrast, we design here the mixed-type methods that keep the same estimation quality but reduce a computational time significantly. The novel estimators elegantly utilize any MATLAB-based numerical integration scheme developed for solving ordinary differential equations (ODEs) with the required accuracy tolerance pre-defined by users. In summary, a simplicity of the suggested estimators, their numerical robustness with respect to roundoff due to the square-root form utilized as well as their estimation accuracy due to the MATLAB ODEs solvers with discretization error control involved are the attractive features of the novel estimators. The numerical experiments are provided for illustrating a performance of the suggested methods in comparison with the existing ones.

Square‐rooting approaches to accurate mixed‐type continuous‐discrete extended and fifth‐degree cubature Kalman filters

The fifth-degree cubature Kalman filter (5D-CKF) has been recently developed for both the discrete- and continuous-time non-linear stochastic systems. In the published works, it has been mentioned that numerically stable square-root 5D-CKF implementations are not feasible to derive, although they are easily obtained for the third-degree CKF. The key problem of the underlying mathematical derivation is negative weight coefficients appeared in the fifth-degree cubature rule, which prevents the required square-root factorisation of the filters’ equations. The authors resolve this essential problem existed for the 5D-CKF methodology by utilising hyperbolic transformations instead of the usual ones traditionally used for square-rooting in the engineering literature. The authors’ solution is given within both the Cholesky and singular value decomposition (SVD) and is based on matrix calculus with the hyperbolic QR and SVD transformations involved. The theoretical results are illustrated in a case of the continuous-discrete mixed-type estimator ACD-EKF-5DCKF and can be applied to any other 5D-CKF strategy. Numerical experiments are also provided.

Estimation for state-of-charge of lithium-ion battery based on an adaptive high-degree cubature Kalman filter

Accurate estimation for state-of-charge of the battery is very important for energy storage systems in electric vehicles and smart grids. To improve the accuracy and reliability of state-of-charge estimation, accurate model equations and a set of robust algorithm are necessary. Different from the commonly used method, this paper adopts a polynomial based on Gaussian function to build up the open circuit voltage function, and proposes an adaptive fifth-degree cubature Kalman filter algorithm to estimate the battery state-of-charge. Two typical driving cycles, including the dynamic stress test and the Worldwide harmonized Light Vehicles Test Cycle are applied to evaluate the performance of the proposed estimator. The results indicate that compared with the unscented Kalman filter and the adaptive cubature Kalman filter, the adaptive fifth-degree cubature Kalman filter can achieve higher state-of-charge estimation accuracy and better overcome the impact of large measurement error and initial error.

Application of improved fifth-degree cubature Kalman filter in the nonlinear initial alignment of strapdown inertial navigation system.

This paper addresses the state estimation of the nonlinear initial alignment of the strapdown inertial navigation system (SINS), which mainly focuses on the initial alignment on the swaying base and under the in-motion condition with the measurement uncertainties. In order to achieve a higher alignment precision, stronger numerical stability, and lower computational cost for the initial alignment of SINS on the swaying base, a new discrete large azimuth misalignment error model of SINS is established, and an improved fifth-degree cubature Kalman filter (5th-CKF) algorithm is proposed, which combines the 5th-CKF and a simplified dimensionality reduction filtering algorithm. The 5th-CKF is introduced to solve the nonlinear filtering problem, a simplified dimensionality reduction algorithm is derived to reduce the large calculation values of 5th-CKF. Furthermore, under the Bayesian framework, a novel filtering approach named the fifth-degree variational Bayesian (VB) adaptive cubature Kalman filter is deduced for the in-motion alignment with a large azimuth misalignment angle and unknown and time-varying measurement noise statistics, which combines the iterative VB approach and 5th-CKF. The 5th-CKF is exploited to handle the nonlinear initial alignment model, and the VB approach is utilized to iteratively estimate the sufficient statistics of the measurement noise. Mathematical simulation, turntable, and vehicle experiments are performed to demonstrate the effectiveness and the superiority of the proposed approaches.

Parameter selection method for support vector machine based on adaptive fusion of multiple kernel functions and its application in fault diagnosis

A new model parameter selection method for support vector machine based on adaptive fusion of multiple kernel functions is proposed in this paper. Characteristics of local kernels, global kernels, mixtures of kernels and multiple kernels were analyzed. Fusion coefficients of the multiple kernel function, kernel function parameters and regression parameters are combined to form the parameters of the state vector. Thus, the model selection problem is transformed into a nonlinear system state estimation problem. Then, we use a fifth-degree cubature Kalman filter to estimate the parameters. In this way, we realize adaptive selection of the multiple kernel function weighted coefficient, the kernel parameters and the regression parameters. A simulation experiment was performed to interpret the PE process for fault diagnosis.

Adaptive Fifth-Degree Cubature Information Filter for Multi-Sensor Bearings-Only Tracking.

Standard Bayesian filtering algorithms only work well when the statistical properties of system noises are exactly known. However, this assumption is not always plausible in real target tracking applications. In this paper, we present a new estimation approach named adaptive fifth-degree cubature information filter (AFCIF) for multi-sensor bearings-only tracking (BOT) under the condition that the process noise follows zero-mean Gaussian distribution with unknown covariance. The novel algorithm is based on the fifth-degree cubature Kalman filter and it is constructed within the information filtering framework. With a sensor selection strategy developed using observability theory and a recursive process noise covariance estimation procedure derived using the covariance matching principle, the proposed filtering algorithm demonstrates better estimation accuracy and filtering stability. Simulation results validate the superiority of the AFCIF.

A Novel Fifth-Degree Strong Tracking Cubature Kalman Filter for Two-Dimensional Maneuvering Target Tracking

A novel fifth-degree strong tracking cubature Kalman filter is put forward to improve the two-dimensional maneuvering target tracking accuracy. First, a new fifth-degree cubature rule, with only one point more than the theoretical lower bound, is used to approximate the intractable nonlinear Gaussian weighted integral in the nonlinear Kalman filtering framework, and a novel fifth-degree cubature Kalman filter is proposed. Then, the suboptimal fading factor is designed for the filter to adjust the filtering gain matrix online and force the residual sequences mutually orthogonal, thus improving the ability of the filter to track the mutation state, and the fifth-degree strong tracking cubature Kalman filter is derived. The suboptimal fading factor is calculated in a new method, which reduces the number of calculations for the cubature points from three times to twice without calculating the Jacobian matrix. The simulation results indicate that the proposed filter has the ability to track the maneuvering target and achieve higher target tracking accuracy and thus verifies the effectiveness of the proposed filter.

Application of Improved 5th-Cubature Kalman Filter in Initial Strapdown Inertial Navigation System Alignment for Large Misalignment Angles

In view of the fact the accuracy of the third-degree Cubature Kalman Filter (CKF) used for initial alignment under large misalignment angle conditions is insufficient, an improved fifth-degree CKF algorithm is proposed in this paper. In order to make full use of the innovation on filtering, the innovation covariance matrix is calculated recursively by an innovative sequence with an exponent fading factor. Then a new adaptive error covariance matrix scaling algorithm is proposed. The Singular Value Decomposition (SVD) method is used for improving the numerical stability of the fifth-degree CKF in this paper. In order to avoid the overshoot caused by excessive scaling of error covariance matrix during the convergence stage, the scaling scheme is terminated when the gradient of azimuth reaches the maximum. The experimental results show that the improved algorithm has better alignment accuracy with large misalignment angles than the traditional algorithm.

A Novel Fifth-Degree Cubature Kalman Filter Approaching the Lower Bound on the Number of Cubature Points

In this paper, a novel fifth-degree cubature Kalman filter, which approaches the lower bound on the number of cubature points, is proposed to reduce the computational complexity while maintaining the fifth-degree filtering accuracy. The Gaussian weighted integral of a nonlinear function is approximated using a numerical cubature rule, whose number of cubature points needed is only one more than the theoretical lower bound, and the filter is deduced under the Bayesian filtering framework by this rule. Furthermore, a square-root version of the proposed fifth-degree cubature Kalman filter is given, and it acquires higher computational efficiency and ensures the numerical stability of the filter. Three numerical simulations are taken, and the results show that the proposed filters maintain the fifth-degree filtering accuracy, while needing the least amount of computation and achieving the best real-time performance.

Interacting Multiple Model (IMM) Fifth-Degree Spherical Simplex-Radial Cubature Kalman Filter for Maneuvering Target Tracking.

For improving the tracking accuracy and model switching speed of maneuvering target tracking in nonlinear systems, a new algorithm named the interacting multiple model fifth-degree spherical simplex-radial cubature Kalman filter (IMM5thSSRCKF) is proposed in this paper. The new algorithm is a combination of the interacting multiple model (IMM) filter and the fifth-degree spherical simplex-radial cubature Kalman filter (5thSSRCKF). The proposed algorithm makes use of Markov process to describe the switching probability among the models, and uses 5thSSRCKF to deal with the state estimation of each model. The 5thSSRCKF is an improved filter algorithm, which utilizes the fifth-degree spherical simplex-radial rule to improve the filtering accuracy. Finally, the tracking performance of the IMM5thSSRCKF is evaluated by simulation in a typical maneuvering target tracking scenario. Simulation results show that the proposed algorithm has better tracking performance and quicker model switching speed when disposing maneuver models compared with the interacting multiple model unscented Kalman filter (IMMUKF), the interacting multiple model cubature Kalman filter (IMMCKF) and the interacting multiple model fifth-degree cubature Kalman filter (IMM5thCKF).

A novel algorithm of fifth-degree cubature Kalman filter for orbit determination at the lower bound approaching to the number of cubature points

With more satellites launched into orbits during recent years, monitoring and cataloging of satellites play an important role in improving the utilization rate of space resource and alleviating the pressure of orbit resource. Groundbased radar, a kind of sensor in space surveillance system, does not consider the influences of the weather and other special circumstances. And it is a key technology in space target tracking by using the measurement data for real-time orbit determination. Due to the influence of orbital perturbation, the satellite orbital dynamic model is a nonlinear system. The optimal estimation of the orbital state can be achieved by means of nonlinear filtering based on the measured ranging, velocity and angle data with measurement noise, which is the essence of real time orbit determination and has important research value. The extended Kalman filter (EKF) and unscented Kalman filter (UKF) are most widely used nonlinear Kalman filters. However, the first-order Taylor expansion of nonlinear function in EKF degrades the filtering accuracy. And the weight value in UKF might be negative for the high-dimensional system, which may directly affect the filtering stability. As an important method in nonlinear filtering, cubature Kalman filter (CKF) has better accuracy and stability than UKF. However, CKF only has third-degree filtering accuracy. In order to improve the filtering accuracy further, some fifth-degree cubature Kalman filters are proposed, mainly including the fifth-degree cubature Kalman filter and the fifth-degree spherical simplex-radial cubature Kalman filter. The optimality of the radial integral cannot be guaranteed by using the moment matching method in these fifth-degree filters, so a high-degree cubature quadrature Kalman filter (HDCQKF) is proposed. The radial integral is calculated using the high-degree Gauss-Laguerre formula in HDCQKF. However, the aforementioned filtering algorithm leads to an increase in the number of cubature points, thereby improving the accuracy, and the number of cubature points increases polynomially with the increase of system dimension. Once the algorithm is applied to a high-dimensional system, or the processor has a relatively poor performance, it may impose a heavier computing burden, thus the real-time performance decreases. Therefore, it is necessary to study how to reduce the computational complexity of the fifth-degree filtering algorithm. In order to improve the real-time performance of orbit determination on condition that the accuracy of orbit determination is kept, a novel fifth-degree cubature Kalman filter for orbit determination is proposed at the lower bound approaching to the number of cubature points. The key problem in the nonlinear Kalman filter is to calculate the multidimensional integral in the form of nonlinear functionGaussian probability density function, and the integral is approximated using a fifth-degree numerical cubature rule, in which the number of cubature points required is only one more than the theoretical lower bound. The abovementioned cubature rule is embedded into the nonlinear Kalman filtering framework, from which the update steps of the novel cubature Kalman filter are derived. Then, the equations of state and measurement for real-time orbit determination are obtained. The J2 perturbation and atmospheric drag perturbation are taken into account in the state equation, and the coordinate transformation is used to derive the nonlinear relationship between the orbital state and measurement element. The simulation results show that the proposed fifth-degree cubature Kalman filter can achieve a higher filtering accuracy than the CKF and the same accuracy as the existing fifth-degree filters, but has the fewest cubature points and the best real-time performance, which proves the effectiveness of the proposed algorithm.

A Novel Fifth-Degree Cubature Kalman Filter for Real-Time Orbit Determination by Radar

A novel fifth-degree cubature Kalman filter is proposed to improve the accuracy of real-time orbit determination by ground-based radar. A fully symmetric cubature rule, approaching the Gaussian weighted integral of a nonlinear function in general form, is introduced, and the specific points and weights are calculated by matching the monomials of degree not greater than five with the exact values. On the basis of the above rule, a novel fifth-degree cubature Kalman filter, which can achieve a higher accuracy than UKF and CKF, is derived under the Bayesian filtering framework. Then, to describe the nonlinear system more accurately, the orbital dynamics equation with J2 perturbation is used as the state equation, and the nonlinear relationship between the radar measurement elements and orbital states is built as the measurement equation. The simulation results show that, compared with the traditional third-degree algorithm, the proposed fifth-degree algorithm has a higher accuracy of orbit determination.

多终端五阶强跟踪容积卡尔曼滤波实时定轨

多终端五阶强跟踪容积卡尔曼滤波实时定轨

Cubature Information Filters Using High-Degree and Embedded Cubature Rules

The information form of the Kalman filter (KF) is preferred over standard covariance filters in multiple sensor fusion problems. Aiming at this issue, two types of cubature information filters (CIF) for nonlinear systems are presented in this article. The two approaches, which we have named the embedded cubature information filter (ECIF) and the fifth-degree cubature information filter (FCIF), are developed from a fifth-degree cubature Kalman filter and a newly proposed embedded cubature KF. Theoretical analysis shows that the proposed filters can achieve higher level estimation accuracy than conventional information filters, such as the CIF and the extended information filter (EIF). Performance comparisons of the proposed information filters with the conventional CIF are demonstrated via two independent multisensor tracking problems. The experimental results, presented herein, demonstrate that the proposed algorithms are more reliable and accurate than the CIF.

Gene regulatory network inference by point-based Gaussian approximation filters incorporating the prior information

The extended Kalman filter (EKF) has been applied to inferring gene regulatory networks. However, it is well known that the EKF becomes less accurate when the system exhibits high nonlinearity. In addition, certain prior information about the gene regulatory network exists in practice, and no systematic approach has been developed to incorporate such prior information into the Kalman-type filter for inferring the structure of the gene regulatory network. In this paper, an inference framework based on point-based Gaussian approximation filters that can exploit the prior information is developed to solve the gene regulatory network inference problem. Different point-based Gaussian approximation filters, including the unscented Kalman filter (UKF), the third-degree cubature Kalman filter (CKF3), and the fifth-degree cubature Kalman filter (CKF5) are employed. Several types of network prior information, including the existing network structure information, sparsity assumption, and the range constraint of parameters, are considered, and the corresponding filters incorporating the prior information are developed. Experiments on a synthetic network of eight genes and the yeast protein synthesis network of five genes are carried out to demonstrate the performance of the proposed framework. The results show that the proposed methods provide more accurate inference results than existing methods, such as the EKF and the traditional UKF.

High-degree cubature Kalman filter

The cubature Kalman filter (CKF), which is based on the third degree spherical–radial cubature rule, is numerically more stable than the unscented Kalman filter (UKF) but less accurate than the Gauss–Hermite quadrature filter (GHQF). To improve the performance of the CKF, a new class of CKFs with arbitrary degrees of accuracy in computing the spherical and radial integrals is proposed. The third-degree CKF is a special case of the class. The high-degree CKFs of the class can achieve the accuracy and stability performances close to those of the GHQF but at lower computational cost. A numerical integration problem and a target tracking problem are utilized to demonstrate the necessity of using the high-degree cubature rules to improve the performance. The target tracking simulation shows that the fifth-degree CKF can achieve higher accuracy than the extended Kalman filter, the UKF, the third-degree CKF, and the particle filter, and is computationally much more efficient than the GHQF.