High-degree Cubature Kalman Filter Research Articles

Vehicle State Joint Estimation Based on Lateral Stiffness.

In this study, a vehicle state joint estimation method based on lateral stiffness was applied to estimate the running states of electric vehicles driven by rear-drive, in-wheel motors. Different from the estimation methods used in other research, the joint estimator designed in this study uses the least-squares (LS) algorithm to estimate the lateral stiffness of the front and rear axles of the vehicle, deploying the high-degree cubature Kalman filter algorithm to estimate the vehicle state. We establish a three-degree-of-freedom nonlinear vehicle model with longitudinal velocity, lateral velocity, and yaw rate, and the lateral stiffness of the front and rear axles as the principal parameters. For the low-speed running state of the vehicle, a linearized magic tire model with high fitting accuracy was used to calculate the lateral force of the entire vehicle. The LS algorithm with a forgetting factor was used to design a lateral stiffness estimator to assess the front-axle and rear-axle lateral stiffness of the entire vehicle. The generalized high-degree cubature Kalman filter (GHCKF) algorithm was used to design the vehicle state estimator and further improve the GHCKF algorithm. A vehicle state estimator, using the square root generalized high-degree cubature Kalman filter (SRGHCKF), was designed. Therefore, the joint estimator, comprising a lateral stiffness estimator and a vehicle state estimator, adopts the LS-GHCKF/SRGHCKF algorithm and enables the estimation of the lateral stiffness, the longitudinal velocity, the lateral velocity, and the yaw rate of the entire vehicle during the driving process. A double lane change and slalom simulation were performed to analyze the feasibility and accuracy of the joint estimation algorithm and verify the results of the LS-GHCKF algorithm and the LS-SRGHCKF algorithm. Further, a low-speed driving experiment was carried out for electric vehicles driven by rear in-wheel motors. The inertial navigation system (INS), the global positioning system (GPS), the real-time kinematic (RTK), and an angle sensor were used to collect real-time vehicle data. The results were compared to verify the feasibility of the joint estimator and the progressiveness of the algorithm. The experimental verification and simulation both show that the vehicle state joint estimator, designed based on the LS-GHCKF/SRGHCKF algorithm, can accurately estimate the real-time state of the vehicle. Additionally, the LS-SRGHCKF algorithm shows better effectiveness and robustness than the LS-GHCKF algorithm.

Improved Robust High-Degree Cubature Kalman Filter Based on Novel Cubature Formula and Maximum Correntropy Criterion with Application to Surface Target Tracking

Robust nonlinear filtering is an important method for tracking maneuvering targets in non-Gaussian noise environments. Although there are many robust filters for nonlinear systems, few of them have ideal performance for mixed Gaussian noise and non-Gaussian noise (such as scattering noise) in practical applications. Therefore, a novel cubature formula and maximum correntropy criterion (MCC)-based robust cubature Kalman filter is proposed. First, the fully symmetric cubature criterion and high-order divided difference are used to construct a new fifth-degree cubature formula using fewer symmetric cubature points. Then, a new cost function is obtained by combining the weighted least-squares method and the MCC loss criterion to deal with the abnormal values of non-Gaussian noise, which enhances the robustness; and statistical linearization methods are used to calculate the approximate result of the measurement process. Thus, the final fifth-degree divided difference–maximum correntropy cubature Kalman filter (DD-MCCKF) framework is constructed. A typical surface-maneuvering target-tracking simulation example is used to verify the tracking accuracy and robustness of the proposed filter. Experimental results indicate that the proposed filter has a higher tracking accuracy and better numerical stability than other common nonlinear filters in non-Gaussian noise environments with fewer cubature points used.

State of charge estimation of Lithium-ion batteries based on the probabilistic fusion of two kinds of cubature Kalman filters

The accurate simulation of batteries’ dynamic characteristics is important for improving the State of Charge (SoC) estimation performance. However, an equivalent circuit model of batteries tends to have changing simulation accuracy of the battery’s dynamic characteristics during SoC estimation. Although the adaptive high-degree cubature Kalman filter (AHCKF) has a more accurate estimation of the dynamic characteristics simulated by the battery model than the adaptive cubature Kalman filter (ACKF) does, AHCKF may have lower estimation accuracy of the battery’s real dynamic characteristics. Therefore, AHCKF does not always outperform ACKF at each time step during SoC estimation.To improve SoC estimation accuracy, this paper proposes an estimation method based on the probabilistic fusion of ACKF and AHCKF. Furthermore, the impact of the filters’ initial weights on the accuracy of the fusion method is analyzed. Under the dynamic stress test, when ACKF’s initial weight is 0.5, the mean absolute error and the root mean square error of SoC estimation based on the proposed method decrease by 22.35% and 6.36% compared with ACKF, respectively, as well as 7.04% and 4.63% compared with AHCKF, respectively. The result shows that the proposed method can improve the accuracy of SoC estimation when the filters’ initial weights are appropriately set.

A Nonlinear System State Estimation Method Based on Adaptive Fusion of Multiple Kernel Functions

With the development of the industry, the physical model of controlled object tends to be complicated and unknown. It is particularly important to estimate the state variables of a nonlinear system when the model is unknown. This paper proposes a state estimation method based on adaptive fusion of multiple kernel functions to improve the accuracy of system state estimation. First, a dynamic neural network is used to build the system state model, where the kernel function node is constructed by a weighted linear combination of multiple local kernel functions and global kernel functions. Then, the state of the system and the weight of the kernel functions are put together to form an augmented state vector, which can be estimated in real time by using high-degree cubature Kalman filter. The high-degree cubature Kalman filter performs adaptive fusion of the kernel function weights according to specific samples, which makes the neural network function approximate the real system model, and the state estimate follows the real value. Finally, the simulation results verify the feasibility and effectiveness of the proposed algorithm.

Improved high-degree cubature Kalman filter based on resampling-free sigma-point update framework and its application for inertial navigation system-based integrated navigation

Sigma-point based Kalman filters have been widely applied to nonlinear filtering problems, where cubature Kalman filter (CKF) is considered to be an ideal solution to tightly coupled GNSS/INS. Different sigma-point update frameworks (SUFs) for CKF have been reported in our previous work, where instead of being abandoned after the moment matching, the instantiated points are used to approximate the likelihood function and update the posterior sigma points directly, which leads to a resampling-free SUF. By propagating more cubature points through the filtering model, the High-degree CKF (HCKF) achieves more accurate moments approximation result than CKF. In this paper, the performance of different SUFs for HCKF is analyzed by simulation on tightly coupled GNSS/INS, and observations missing during different maneuvers are simulated to evaluate the performance of resampling-free HCKF. The simulation result gives an insight into the uncertainty propagation in the two-stage update of Gaussian filter for GNSS/INS application.

A novel optimal data fusion algorithm and its application for the integrated navigation system of missile

For Inertial Navigation System (INS)/Celestial Navigation System (CNS)/Global Navigation Satellite System (GNSS) integrated navigation system of the missile, the performance of data fusion algorithms based on the Cubature Kalman Filter (CKF) is seriously degraded when there are non-Gaussian noise and process-modeling errors in the system model. Therefore, a novel method is proposed, which is called Optimal Data Fusion algorithm based on the Adaptive Fading maximum Correntropy generalized high-degree CKF (AFCCKF-ODF). First, the Adaptive Fading maximum Correntropy generalized high-degree CKF (AFCCKF) is proposed and used as the local filter for the INS/GNSS and INS/CNS subsystems to improve the robustness of local state estimation. Then, the local state estimation is fused based on the minimum variance principle and high-degree cubature criterion to get the globally optimal state. Finally, the experimental results verify that the proposed algorithm can significantly improve the robustness of the missile-borne INS/CNS/GNSS integrated navigation system to non-Gaussian noise and process modeling error and obtain the global optimal navigation information.

Variational Bayesian adaptive high‐degree cubature Huber‐based filter for vision‐aided inertial navigation on asteroid missions

Vision-aided inertial navigation (VAIN) is a prospective technique for determining the pose of the spacecraft during asteroid missions. The VAIN system can fuse the inertial and visual data by employing the high-degree cubature Kalman filter (HCKF) because it can accurately handle non-linear problems. However, the visual measurements can be corrupted by non-Gaussian noise with unknown time-varying covariance, resulting in severe degradation of the HCKF. To improve the navigational accuracy of the spacecraft in these situations, the authors propose a novel adaptive robust HCKF known as variational Bayesian (VB) adaptive high-degree cubature Huber-based filter (VB-AHCHF). In the novel algorithm, the fifth-degree cubature rule and VB theory are combined to estimate the state and track the non-stationary statistical characteristics of the measurement noise. In addition, utilising the M-estimation, which is defined as the Huber technique, it modifies the update step of the formal Bayesian filtering. Therefore, the VB-AHCHF can exhibit adaptability and robustness to the covariance uncertainty and non-Gaussianity of the measurement noise. Their simulation results show that the estimation accuracy of VB-AHCHF, as well as its adaptability and robustness, is superior to all state-of-the-art algorithms, e.g. HCKF, high-degree cubature Huber-based filter, and the VB adaptive HCKF.

Multiple-Step Randomly Delayed Adaptive Robust Filter With Application to INS/VNS Integrated Navigation on Asteroid Missions

This paper develops a novel nonlinear adaptive robust filter called the multiple-step randomly delayed variational Bayesian adaptive high-degree cubature Huber-based filter (MRD-VBAHCHF) for a class of nonlinear stochastic systems whose measurements are randomly delayed by multiple sampling times and corrupted by contaminated Gaussian noise with unknown covariance. First, a system with randomly delayed measurement is modeled in terms of multiple Bernoulli random variables. Then, the multiple-step randomly delayed high-degree cubature Kalman filter (MRD-HCKF) is derived by employing the fifth-degree cubature rule to compute the mean and covariance of the nonlinear equations in the system model. Next, the MRD-HCKF is modified to the MRD-VBAHCHF by incorporating the variational Bayesian theory and Huber technique for estimating the measurement noise covariance online and suppressing the influence of non-Gaussian noise. Consequently, the proposed filter is not only adaptive to unknown measurement noise statistics but also robust to random measurement delays and non-Gaussian noise. Finally, the MRD-VBAHCHF is verified for use in inertial navigation system/visual navigation system (INS/VNS) integrated navigation on asteroid missions, and the results of Monte Carlo simulations demonstrate that the MRD-VBAHCHF outperforms the high-degree cubature Kalman filter (HCKF), the MRD-HCKF and the variational Bayesian adaptive high-degree cubature Huber-based filter (VBAHCHF), thus showing the superiority of the proposed filter.

Maximum correntropy generalized high-degree cubature Kalman filter with application to the attitude determination system of missile

For SINS/CNS integrated navigation system, the CKF can perform well for state estimation in Gaussian noise. However, its performances are likely to degrade significantly under non-Gaussian noise conditions. To improve the robustness of the CKF against non-Gaussian noise, we propose an improved cubature Kalman filter, called the maximum correntropy generalized high-degree CKF (MCGHCKF). In the MCGHCKF, the generalized high-degree cubature rule is used to improve the filtering performance, and the maximum correntropy criterion is utilized to reduce the influence of non-Gaussian noise on state estimation. Simulation experiments illustrate the effectiveness and robustness of our algorithm.

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.

Adaptive high‐degree cubature Kalman filter in the presence of unknown measurement noise covariance matrix

Here, the authors address the state estimation problem of non-linear systems in the presence of unknown measurement noise (MN) covariance matrix. Recently, a high-degree cubature Kalman filter (HCKF) has been successfully used in the non-linear-state estimation problem with arbitrary degrees of accuracy in computing the spherical and radial integrals. However, the efficiency of the HCKF depends on a priori knowledge of the MN. To improve the performance of HCKF for non-linear systems with unknown MN covariance matrix, the authors proposed an adaptive HCKF, which combines the high-degree cubature rule with the variational Bayesian (VB) method to jointly estimate the system state and the unknown covariance matrix online. Experimental results demonstrate the effectiveness of the proposed filter.

Accuracy analysis of three deterministic sampling nonlinear filtering algorithms

To study the estimation accuracy of nonlinear Kalman filter in transfer alignment system, the Taylor expansion of multivariable function is utilized to analyze the estimation accuracy of the three typical deterministic sampling nonlinear Kalman filters, such as high-degree cubature Kalman filter (HCKF), cubature Kalman filter (CKF) and unscented Kalman filter (UKF). The Taylor expansion analyses demonstrate that CKF and UKF produce truncation error since the fourth degree term, while HCKF can capture fifth degree term of Taylor series expansion. The three nonlinear filtering algorithms are applied to carrier-aircraft transfer alignment system under large misalignment angle. The simulation results show that CKF and UKF have the same accuracy, and HCKF has higher estimation accuracy than the other two, which is consistent with the theoretical analysis results.

Multiple Robust High-degree Cubature Kalman Filter for Relative Position and Attitude Estimation of Satellite Formation

The High-degree Cubature Kalman Filter (HCKF) is proposed as a novel methodology based on the arbitrary degree spherical rule, which can achieve better performance than the traditional Kalman filter. However, it also has a large calculation burden when used in a high-dimension and high-degree of accuracy estimation system. The number of sampling points of an HCKF increases polynomially with increasing state-space dimensions, which further increases the calculation burden. The reduction of the number of the state-space dimensions is the main contribution of this study. A strategy for HCKF based on the partitioning of the state-space and orthogonal principle is introduced, referred to as the Multiple Robust HCKF (MRHCKF). It is shown that this technique can effectively reduce the calculation burden for the high-dimension system with robust performance. Numerical simulations are performed for the example of high-dimension relative position and attitude estimation to show that the proposed method can obtain nearly the same performance as the HCKF, while drastically reducing computational complexity.

Marginal Reduced High-degree CKF and its Application in Nonlinear Rapid Transfer Alignment

The error model is not perfect when the nonlinear transfer alignment is applied to the inertial navigation system of shipborne aircraft. In this paper, a new model incorporating acceleration error of lever-arm is proposed. Considering the influence of flexure deformation and dynamic lever-arm, it is suitable for the case of large azimuth misalignment The state filtering estimation selects the High-degree Cubature Kalman filter (HCKF). A reduced HCKF algorithm with marginalized-sampling-based cubature transformation is designed according to the state equation and measurement equation structure of the transfer alignment model, and it is proved. This algorithm simplifies the measurement update process, and applies the marginal sampling algorithm to the time update process, achieving the purpose of reducing the calculation amount of HCKF. In the simulation, the velocity plus attitude matching method is used. The experimental results show that compared with the transfer alignment model without considering the dynamic lever-arm, this model has a higher accuracy, and the alignment precision and time also meet the requirements.

Application of MCC-based Robust High-degree Cubature Kalman Filter in Integrated Navigation System

As the integrated navigation system is a nonlinear system, in the case of non-gaussian noise, the traditional nonlinear gaussian filtering algorithm has a serious problem of decreasing filtering precision. In this paper, a new robust high-degree Cubature Kalman filtering algorithm is proposed, which takes into account the nonlinearity of the system and non-gaussian noise. The algorithm improves the measurement updating process by using the Maximum correntropy criterion(MCC), and converts the traditional measurement updating problem into the linear regressione quation solving problem. Combines the advantages of Maximum correntropy criterion and Cubature Kalman filter to deal with non-Gaussian and nonlinear systems. The proposed algorithm is applied to the SINS/GPS integrated navigation system, the simulation results show that the proposed algorithm’s filtering performance is greatly affected by the kernel width. Under the condition of gaussian mixture noise, the new robust high-degree Cubature Kalman filter based on Maximum correntropy criterion(MCC-HCKF) is more robust and has higher filtering precision than the traditional high-degree Cubature Kalman filter(HCKF).

Incorporating delayed measurements in an improved high-degree cubature Kalman filter for the nonlinear state estimation of chemical processes

The on-line estimation of process quality variables has a large impact on the advanced monitoring and control techniques of chemical processes. The present study offers an improved high-degree cubature Kalman filter (HCKF) to solve the nonlinear state estimation problem of high-dimensional chemical processes. We substituted the Cholesky decomposition in the HCKF filter with a diagonalization transformation of the matrix. In addition, we enhanced numerical stability and estimation accuracy. On this basis, we present one nonlinear state estimation method based on the sample-state augmentation and improved HCKF to handle issues with delayed measurements. Finally, we used the nonlinear state estimation experiments for the polymerization process to validate the proposed method. The numerical results indicated the achievement of state estimation with higher accuracy and better stability following the effective utilization of the delayed measurements for nonlinear chemical processes.

Fifth‐degree continuous–discrete cubature Kalman filter for radar

In this study, the authors extend the high-degree cubature Kalman filter to operate with continuous-time non-linear stochastic systems with discrete measurements. For this purpose, they utilise two known approximations to solve the stochastic differential equation used in the modelling of continuous-time dynamics. The first approach is grounded in an ordinary differential equations solver. The second approach is based on the Ito-Taylor expansion of order 1.5. In addition, the errors presented in each approach were classified. Finally, the proposed filters were compared with the continuous-discrete cubature Kalman filter in a challenging radar-tracking experiment. The results of the experiment show an improvement in the accuracy of the proposed method, and more importantly, a better performance of the filters based on the Ito-Taylor expansion.

Arbitrary-step randomly delayed robust filter with application to boost phase tracking

The conventional filters such as extended Kalman filter, unscented Kalman filter and cubature Kalman filter assume that the measurement is available in real-time and the measurement noise is Gaussian white noise. But in practice, both two assumptions are invalid. To solve this problem, a novel algorithm is proposed by taking the following four steps. At first, the measurement model is modified by the Bernoulli random variables to describe the random delay. Then, the expression of predicted measurement and covariance are reformulated, which could get rid of the restriction that the maximum number of delay must be one or two and the assumption that probabilities of Bernoulli random variables taking the value one are equal. Next, the arbitrary-step randomly delayed high-degree cubature Kalman filter is derived based on the 5th-degree spherical-radial rule and the reformulated expressions. Finally, the arbitrary-step randomly delayed high-degree cubature Kalman filter is modified to the arbitrary-step randomly delayed high-degree cubature Huber-based filter based on the Huber technique, which is essentially an M-estimator. Therefore, the proposed filter is not only robust to the randomly delayed measurements, but robust to the glint noise. The application to the boost phase tracking example demonstrate the superiority of the proposed algorithms.

Hypersonic entry vehicle state estimation using nonlinearity-based adaptive cubature Kalman filters

Guidance, navigation, and control of a hypersonic vehicle landing on the Mars rely on precise state feedback information, which is obtained from state estimation. The high uncertainty and nonlinearity of the entry dynamics make the estimation a very challenging problem. In this paper, a new adaptive cubature Kalman filter is proposed for state trajectory estimation of a hypersonic entry vehicle. This new adaptive estimation strategy is based on the measure of nonlinearity of the stochastic system. According to the severity of nonlinearity along the trajectory, the high degree cubature rule or the conventional third degree cubature rule is adaptively used in the cubature Kalman filter. This strategy has the benefit of attaining higher estimation accuracy only when necessary without causing excessive computation load. The simulation results demonstrate that the proposed adaptive filter exhibits better performance than the conventional third-degree cubature Kalman filter while maintaining the same performance as the uniform high degree cubature Kalman filter but with lower computation complexity.

Iterative Diffusion-Based Distributed Cubature Gaussian Mixture Filter for Multisensor Estimation.

In this paper, a distributed cubature Gaussian mixture filter (DCGMF) based on an iterative diffusion strategy (DCGMF-ID) is proposed for multisensor estimation and information fusion. The uncertainties are represented as Gaussian mixtures at each sensor node. A high-degree cubature Kalman filter provides accurate estimation of each Gaussian mixture component. An iterative diffusion scheme is utilized to fuse the mean and covariance of each Gaussian component obtained from each sensor node. The DCGMF-ID extends the conventional diffusion-based fusion strategy by using multiple iterative information exchanges among neighboring sensor nodes. The convergence property of the iterative diffusion is analyzed. In addition, it is shown that the convergence of the iterative diffusion can be interpreted from the information-theoretic perspective as minimization of the Kullback–Leibler divergence. The performance of the DCGMF-ID is compared with the DCGMF based on the average consensus (DCGMF-AC) and the DCGMF based on the iterative covariance intersection (DCGMF-ICI) via a maneuvering target-tracking problem using multiple sensors. The simulation results show that the DCGMF-ID has better performance than the DCGMF based on noniterative diffusion, which validates the benefit of iterative information exchanges. In addition, the DCGMF-ID outperforms the DCGMF-ICI and DCGMF-AC when the number of iterations is limited.