Standard Cubature Kalman Filter Research Articles

An Investigation of Extended-Dimension Embedded CKF-SLAM Based on the Akaike Information Criterion.

Simultaneous localization and mapping (SLAM) faces significant challenges due to high computational costs, low accuracy, and instability, which are particularly problematic because SLAM systems often operate in real-time environments where timely and precise state estimation is crucial. High computational costs can lead to delays, low accuracy can result in incorrect mapping and localization, and instability can make the entire system unreliable, especially in dynamic or complex environments. As the state-space dimension increases, the filtering error of the standard cubature Kalman filter (CKF) grows, leading to difficulties in multiplicative noise propagation and instability in state estimation results. To address these issues, this paper proposes an extended-dimensional embedded CKF based on truncated singular-value decomposition (TSVD-AECKF). Firstly, singular-value decomposition (SVD) is employed instead of the Cholesky decomposition in the standard CKF to mitigate the non-positive definiteness of the state covariance matrix. Considering the effect of small singular values on the stability of state estimation, a method is provided to truncate singular values by determining the truncation threshold using the Akaike information criterion (AIC). Furthermore, the system noise is embedded into the state variables, and an embedding volume criterion is used to improve the conventional CKF while extending the dimensionality. Finally, the proposed algorithm was validated and analyzed through both simulations and real-world experiments. The results indicate that the proposed method effectively mitigates the increase in localization error as the state-space dimension grows, enhancing time efficiency by 55.54%, and improving accuracy by 35.13% compared to the standard CKF algorithm, thereby enhancing the robustness and stability of mapping.

Improved maximum correntropy cubature Kalman and information filters with application to target tracking under non‐Gaussian noise

SummaryThe cubature Kalman filter (CKF) based on the maximum correntropy criterion (MCC) has been widely used in the target tracking. However, numerical problems usually occur when there are outliers in the measurement noise. In order to solve the problems of state estimation under the non‐Gaussian measurement noise, a new combined cost function is defined based on the weighted least squares (WLS) method and MCC. In addition, a new method is also used to adaptively adjust the kernel size, then the improved maximum correntropy CKF (PMCCKF) and its corresponding improved maximum correntropy cubature information filter (PMCCIF) are proposed. Compared with the existing algorithms, the new method can not only obtain similar or even better estimation performance, but also avoid numerical problems. Moreover, when the kernel size is infinite, the performance of proposed algorithms will reduce to the standard CKF and corresponding cubature information filter (CIF) respectively, but the classical maximum correntropy CIF (MCCIF) will not, and even the performance is poor in this case. The advantages of the proposed algorithms are verified by four classical nonlinear models.

State of Charge Estimation of Lithium-Ion Batteries Based on Vector Forgetting Factor Recursive Least Square and Improved Adaptive Cubature Kalman Filter

Accurate online parameter identification and state of charge (SOC) estimation are both very crucial for ensuring the operating safety of lithium-ion batteries and usually the former is a base of the latter. To achieve accurate and stable SOC estimation results, this paper proposes a model-based method, which incorporates a vector forgetting factor least square (VFFLS) algorithm and an improved adaptive cubature Kalman filter (IACKF). Firstly, considering it is difficult for the traditional forgetting factor recursive least square (FFRLS) algorithm to balance the accuracy, convergence, and stability for multiple parameters with different time-varying periods, an improved VFFLS method is employed to determine the multiple parameters of the first-order RC battery model online. It supersedes the single forgetting factor in the FFRLS with multiple forgetting factors in a vector form for improving adaptive capability to multiple time-varying parameters. Secondly, aiming at the fact that the standard cubature Kalman filter (CKF) cannot operate properly when the error covariance matrix is non-positive definite, which is caused by disturbance, initial error, and the limit of the computer word length, the UR decomposition rather than the Cholesky decomposition is applied, thus improving the algorithm stability. In addition, an adaptive update strategy is added to the CKF to enhance accuracy and convergence speed. Finally, comparative experiments with different operating patterns, positive and non-positive definite error covariance matrices, and temperatures are carried out. Experimental results showed that the proposed method can estimate the SOC accurately and stably.

Backward smoothing adaptive SVDCKF integrated navigation algorithm

Aiming at the problem that the Cubature Kalman filter (CKF) algorithm has a non-positive definite of the covariance matrix in GNSS/SINS integrated navigation, which leads to the failure of Cholesky decomposition and the inability to estimate GNSS measurement noise, a backward smoothing adaptive CKF integrated navigation algorithm based on Singular Value Decomposition (SVD) is proposed. Backward smoothing adaptive CKF based on Singular Value Decomposition (BS-A-SVDCKF) algorithm replaces the Cholesky decomposition in standard CKF with SVD, constructs the test threshold for backward smoothing by using the relatively stable noise of the SINS system, and adaptively estimates based on different measurement characteristics when the GNSS measurement noise changes abruptly. Experimental results show that compared with adaptive SVDCKF (A-SVDCKF), backward smoothing SVDCKF (BS-SVDCKF) and SVDCKF, the average positioning accuracy of the proposed algorithm is improved by 32.12%, 57.07% and 68.66%, the average operating efficiency is reduced by 2.19% compared with SVDCKF, and 0.57% and 3.60% higher than that of BS-SVDCKF and A-SVDCKF, which improves the positioning accuracy of the integrated navigation system and ensures real-time.

Cubature Kalman Filter With Both Adaptability and Robustness for Tightly-Coupled GNSS/INS Integration

Tightly-coupled GNSS/INS (Global Navigation Satellite System/Inertial Navigation System) integration is of importance to vehicle positioning. However, this integration technology has difficulty in achieving optimal positioning solutions for the dynamic systems involving strong nonlinearity and systematic modelling error. This paper proposes a new methodology to address the problem of tightly-coupled GNSS/INS integration. This methodology rigorously derives a novel adaptive CKF (Cubature Kalman Filter) with fading memory for kinematic modelling error and a new robust CKF with emerging memory for observation modelling error, using the concept of Mahalanobis distance without involving artificial empiricism. Based on this, a new CKF with both adaptability and robustness is further developed by fusing the results of the standard CKF, adaptive CKF and robust CKF via the principle of interacting multiple model (IMM). Simulation and experiment results together with comparison analysis prove that the proposed methodology can curb the interferences of both kinematic and observation modelling errors on state estimation, leading to improved positioning accuracy for vehicle positioning via tightly-coupled GNSS/INS integration.

A Novel Method for State of Charge Estimation of Lithiumion Batteries using Embedded Cubature Kalman Filter

The accurate estimation of state of charge (SOC) of Lithium-ion battery is one of the most crucial issues for battery management system (BMS). This paper proposes a novel method for SOC estimation using the embedded cubature Kalman filter (ECKF). ECKF computes the cubature points with a weight depends on an optional parameter, which differs from the standard cubature Kalman filter (CKF). An online model identification method for a second-order RC networks equivalent circuit model using the forgetting factor recursive least squares (FRLS) algorithm has been presented. The Dynamic cycles are used to assess the superiority in improving the accuracy and stability of proposed method compared with the widely used algorithms. Experimental results show that, with 20% initial SOC error, the maximum estimation error is within 1%, which indicates that the proposed ECKF method has higher estimation accuracy and robustness for SOC estimation.

Cubature rule-based distributed optimal fusion with identification and prediction of kinematic model error for integrated UAV navigation

Integrated MIMU/GNSS/CNS (micro-electro-mechanical system-based inertial measurement unit/global navigation satellite system/celestial navigation system) is a promising strategy for UAV (unmanned aerial vehicle) navigation. However, given its strong nonlinearity and involvement of kinematic model error, integrated MIMU/GNSS/CNS UAV navigation presents the difficulty in achieving optimal navigation solutions. This paper presents a method of cubature rule-based distributed optimal fusion combined with identification and prediction of kinematic model error to address the above problem in nonlinear integrated MIMU/GNSS/CNS. This method is in a distributed structure to simultaneously process observations from integrated MIMU/GNSS and MIMU/CNS subsystems for the subsequent global fusion. A theory for identification of kinematic model error is established using the concept of Mahalanobis distance. Further, the standard cubature Kalman filter is modified with the model prediction filter to serve as the local filters in integrated MIMU/GNSS and MIMU/CNS subsystems to hinder the disturbance of kinematic model error. Based on the above, an optimal fusion technique is developed to fuse the filtering results of each subsystem for achieving globally optimal state estimation in the sense of mean square error. Simulations and experimental results as well as comparison analysis demonstrate that the proposed distributed optimal fusion method can effectively identify and predict kinematic model error and further achieve globally optimal fusion results, leading to improved performance for integrated MIMU/GNSS/CNS UAV navigation.

Control Optimization of Stochastic Systems Based on Adaptive Correction CKF Algorithm

Standard cubature Kalman filter (CKF) algorithm has some disadvantages in stochastic system control, such as low control accuracy and poor robustness. This paper proposes a stochastic system control method based on adaptive correction CKF algorithm. Firstly, a nonlinear time-varying discrete stochastic system model with stochastic disturbances is constructed. The control model is established by using the CKF algorithm, the covariance matrix of standard CKF is optimized by square root filter, the adaptive correction of error covariance matrix is realized by adding memory factor to the filter, and the disturbance factors in nonlinear time-varying discrete stochastic systems are eliminated by multistep feedback predictive control strategy, so as to improve the robustness of the algorithm. Simulation results show that the state estimation accuracy of the proposed adaptive cubature Kalman filter algorithm is better than that of the standard cubature Kalman filter algorithm, and the proposed adaptive correction CKF algorithm has good control accuracy and robustness in the UAV control test.

State estimation based on improved cubature Kalman filter algorithm

During the recursive calculate process of the cubature Kalman filter (CKF), the covariance matrix tends to lose positive definiteness and noise statistical characteristics are inaccurate, which results in inaccurate filtering or even filter divergence. This study presents an improved algorithm based on the CKF. The algorithm combines the square root filter algorithm and the Sage–Husa maximum a posterior noise estimator, which can ensure the non-negative determination and symmetry of the covariance matrix and has the ability to deal with unknown and time-varying noise statistical characteristics in the filtering process adaptively. In the multi-dimension system, the noise covariance matrix may dissatisfy non-negative definiteness and result in filter divergence, and then the noise covariance matrix estimator is improved. The analysis is verified by state estimation example of the non-linear system, compared with the standard CKF, the accuracy of the adaptive square root CKF (ASRCKF) state estimation is increased by 63.13, 63.88, and 42.71%, respectively. Finally, the effectiveness of the ASRCKF is verified by the state estimation of the permanent magnet linear synchronous motor.

Robust cubature Kalman filter method for the nonlinear alignment of SINS

Nonlinear initial alignment is a significant research topic for strapdown inertial navigation system (SINS). Cubature Kalman filter (CKF) is a popular tool for nonlinear initial alignment. Standard CKF assumes that the statics of the observation noise are pre-given before the filtering process. Therefore, any unpredicted outliers in observation noise will decrease the stability of the filter. In view of this problem, improved CKF method with robustness is proposed. Multiple fading factors are introduced to rescale the observation noise covariance. Then the update stage of the filter can be autonomously tuned, and if there are outliers exist in the observations, the update should be less weighted. Under the Gaussian assumption of KF, the Mahalanobis distance of the innovation vector is supposed to be Chi-square distributed. Therefore a judging index based on Chi-square test is designed to detect the noise outliers, determining whether the fading tune are required. The proposed method is applied in the nonlinear alignment of SINS, and vehicle experiment proves the effective of the proposed method.

A Robust Cubature Kalman Filter with Abnormal Observations Identification Using the Mahalanobis Distance Criterion for Vehicular INS/GNSS Integration.

INS/GNSS (inertial navigation system/global navigation satellite system) integration is a promising solution of vehicle navigation for intelligent transportation systems. However, the observation of GNSS inevitably involves uncertainty due to the vulnerability to signal blockage in many urban/suburban areas, leading to the degraded navigation performance for INS/GNSS integration. This paper develops a novel robust CKF with scaling factor by combining the emerging cubature Kalman filter (CKF) with the concept of Mahalanobis distance criterion to address the above problem involved in nonlinear INS/GNSS integration. It establishes a theory of abnormal observations identification using the Mahalanobis distance criterion. Subsequently, a robust factor (scaling factor), which is calculated via the Mahalanobis distance criterion, is introduced into the standard CKF to inflate the observation noise covariance, resulting in a decreased filtering gain in the presence of abnormal observations. The proposed robust CKF can effectively resist the influence of abnormal observations on navigation solution and thus improves the robustness of CKF for vehicular INS/GNSS integration. Simulation and experimental results have demonstrated the effectiveness of the proposed robust CKF for vehicular navigation with INS/GNSS integration.

Cooperative Localization of Connected Vehicles: Integrating GNSS With DSRC Using a Robust Cubature Kalman Filter

Cooperative localization of the connected vehicles is significant for many advanced intelligent transportation system (ITS) applications. Vehicle-to-vehicle communication using dedicated short-range communication (DSRC) has great potential to enhance global navigation satellite systems (GNSSs) for the capability of cooperative localization. In the integration of DSRC and GNSS, the tolerance against the unknown and time-varying observation conditions is a key factor to fulfill the requirements of several specific ITS applications. Under a GNSS/DSRC integrated architecture for cooperative localization, a novel robust cubature Kalman filter (CKF) is proposed in this paper to improve the performance of the data fusion under uncertain sensor observation environments. In the proposed solution, the structure of the standard CKF is enhanced using the Huber M-estimation technique, in which the original measurement update in the CKF is modified considering the probable anomalies in state estimation. Furthermore, based on the investigation of the adjustment effect from the constraint factor, an adaptive strategy for this parameter is introduced to optimize the performance comprehensively. The proposed method is validated using a specific simulation system. Results of experiment and simulations demonstrate the capability of improving the robustness and adaptive performance over the original filters under the unknown operation conditions.

Alternative framework of the Gaussian filter for non‐linear systems with synchronously correlated noises

To conveniently deduce the Gaussian filter (GF), it is based on the assumption that the process and measurement noises are independent. However, it does not always satisfy this assumption in practice. In this study, a novel framework of GF for non-linear system in which the process and measurement noises are correlated at the same time is proposed. The novel framework of GF takes the type of synchronous correlation into account. It is inspired by the fact that the probability of process noises conditioned on measurement noises has much more correlation information than its original probability in the situation of measurements data received, and exploits the conditional Gaussian distributions to improve the GF accuracy. In addition, the issue about multi-dimensional integrals in the implementation of novel framework of GF is solved by the spherical-radial cubature rule. The cubature Kalman filter with synchronously correlated noises (CKF-SCN) is derived. The results of simulations on a non-linear example show better performance of the CKF-SCN in contrast to the extended Kalman filter with correlation noises, the unscented Kalman filter with correlation noises, the Gaussian approximation recursive filter, and the standard cubature Kalman filter.

Feedback Robust Cubature Kalman Filter for Target Tracking Using an Angle Sensor

The direction of arrival (DOA) tracking problem based on an angle sensor is an important topic in many fields. In this paper, a nonlinear filter named the feedback M-estimation based robust cubature Kalman filter (FMR-CKF) is proposed to deal with measurement outliers from the angle sensor. The filter designs a new equivalent weight function with the Mahalanobis distance to combine the cubature Kalman filter (CKF) with the M-estimation method. Moreover, by embedding a feedback strategy which consists of a splitting and merging procedure, the proper sub-filter (the standard CKF or the robust CKF) can be chosen in each time index. Hence, the probability of the outliers’ misjudgment can be reduced. Numerical experiments show that the FMR-CKF performs better than the CKF and conventional robust filters in terms of accuracy and robustness with good computational efficiency. Additionally, the filter can be extended to the nonlinear applications using other types of sensors.

A multi-agent-based power system hybrid dynamic state estimator

This article proposes a multi-agent-based power system hybrid dynamic state estimator (PSHDSE) that uses field measurements from remote terminal units (RTUs) as well as phasor measurement units (PMUs). The standard cubature Kalman filter (CKF) process is modified to make it suitable for PSHDSE execution, and software agents are utilized to receive data and run PSHDSE for the RTU and PMU measurements separately. PSHDSE is solved by utilizing the CKF, the extended Kalman filter (EKF), and the unscented Kalman filter (UKF). The relative performances of the EKF, UKF, and CKF in executing PSHDSE are established through simulations on the IEEE 30 bus and practical 246-bus Indian test systems.

Robust square-root cubature Kalman filter based on Huber’s M-estimation methodology

In practical engineering applications, the performance of the standard cubature Kalman filter (CKF) and its square-root version can be severely degraded due to outliers in measurement or contaminated distribution. In order to address the problem, a robust version of CKF is presented using Huber’s M-estimation methodology and square-root filtering framework. By making use of the Huber technique to reformulate the measurement update of CKF in square-root filtering framework, the proposed filter can exhibit robustness and numerical stability against deviation from Gaussian distribution assumption. In simulation tests, four versions of CKF—the standard, the square-root, Huber-based, and the proposed are evaluated in terms of estimation accuracy, numerical stability, and robustness under Gaussian and non-Gaussian distribution. The results are concluded that the square-root version outperforms the others under Gaussian distribution, whereas the proposed filter has improved performance in maintaining the robustness and numerical stability under non-Gaussian distribution. The investigated robust framework can be extended to other Gaussian filtering algorithms and the study is expected to facilitate applications of CKF in practical engineering as well.