Gaussian Approximation Filter Research Articles

A Gaussian approximation filter for nonlinear systems with compound noises

In practical applications, measurements may contain many different noise components due to complex environments and other factors. The Gaussian approximation filter, constrained by its requirement for Gaussian characteristics in measurement noises, faces challenges in obtaining a satisfactory filtering performance under compound noise environments. In this article, an improved Gaussian approximation filter is designed for nonlinear systems with compound measurement noises. Hypothesis tests on the prior and posterior residuals are designed to identify the types of noise present in the system. Subsequently, a posterior residual-based adaptive compensation is proposed to mitigate the adverse effects of compound noise on the Gaussian approximation filter. Moreover, it is demonstrated that when the measurement acceptance rate exceeds a certain critical threshold, the estimation error remains bounded. Finally, the effectiveness of the proposed algorithm and analysis is validated through simulations in the context of target tracking.

A novel resampling-free update framework-based cubature Kalman filter for robust estimation

The resampling-free update (RFU) framework avoids discarding the higher-order moment information of the state probability distribution in Gaussian approximation filters. Still, it suffers from the problem of numerical instability and estimation optimality being corrupted caused by non-closed mapping without Gaussian reconstruction. This study proposes a novel robust RFU framework-based cubature Kalman filter. The maximum correntropy criterion is adopted as the optimization cost to exploit the non-Gaussian moments caused by non-closed mapping in RFU. An RFU update is reconstructed based on the square root of a posterior error matrix to improve the numerical stability. In addition, the periodic resampling operation is implemented to mitigate the non-Gaussianity while keeping higher-order moments. The illustrative example demonstrates that the proposed method can improve the estimation robustness and consistency of the RFU framework compared to other state-of-the-art RFU-based filters.

A novel robust IMM filter for jump Markov systems with heavy-tailed process and measurement noises

In some tracking application scenarios, the performances of the traditional interacting multiple model (IMM) approach may degrade dramatically since the presence of outliers in process and measurement noises. Aiming at solving the filtering problem for jump Markov systems containing heavy-tailed process and measurement noises, a new robust student's t-based Gaussian approximate filter utilizing the IMM filtering framework is presented in this paper. Firstly, the heavy-tailed process and measurement noises are assumed to obey student's t-distributions. Secondly, the degree of freedom parameters and the scale matrices are modeled as Gamma distributions and inverse-Wishart distributions respectively. Finally, the system state, degree of freedom parameters, scale matrices and the mode probabilities are inferred simultaneously by introducing the variational Bayesian (VB) theory. Maneuvering target tracking simulation verifies that the designed algorithm achieves superior estimation performances among the state-of-the-art filters.

Design of robust Gaussian approximate filter and smoother with latency probability identification

The Kalman filter is a linear optimal estimator when it can accurately receive measurements and the noise obeys a Gaussian distribution. However, the measurement noise may have heavy-tailed characteristics because of outlier interference. Actually, real-world measurement information may have one-step randomly delayed measurements (ORDMs) with unknown latency probability (LP) due to exogenous disturbances. Moreover, systems are usually not linear. To overcome the influence of the state estimator of nonlinear systems from heavy-tailed measurement noise (HMN) and the unknown LP, this paper focuses on the robust Gaussian approximate (GA) filter and smoother for the above issues. First, the one-step predicted probability density function (PDF) is modeled as a Gaussian distribution, and the likelihood PDF is modeled as a Normal–Gamma–Beta (NGBM) distribution. Furthermore, novel robust Gaussian approximate filter and smoother are proposed based on the variational Bayesian technique. They provide general estimation frameworks for estimating state of nonlinear systems with HMN and ORDMs, and different nonlinear estimators can be implemented using different numerical techniques. In addition, the computational complexity of the proposed algorithms is calculated. Finally, taking a univariate nonstationary growth model, a passive ranging problem, and target tracking as examples, the effectiveness and superiority of the proposed robust algorithms contrasted with the existing algorithms are shown. The outcomes show that the proposed methods have higher estimation precision; however, the computational burden is marginally increased.

Switching Gaussian-heavy-tailed distribution based robust Gaussian approximate filter for INS/GNSS integration

In inertial navigation system and global navigation satellite system (INS/GNSS) integration, the practical stochastic measurement noise may be non-stationary heavy-tailed distribution due to outlier measurements induced by multipath and/or non-line-of-sight receptions of the original GNSS signals. To address the problem, a new switching Gaussian-heavy-tailed (SGHT) distribution is presented, which models the measurement noise with the help of switching between the Gaussian and the an existing heavy-tailed distribution. Then, utilizing two auxiliary parameters satisfying categorical and Bernoulli distributions respectively, we construct the SGHT distribution as a hierarchical Gaussian presentation. Furthermore, applying variational Bayesian inference, a novel SGHT distribution based robust Gaussian approximate filter is derived. Meanwhile, to reduce the computational complexity of the filtering process, an improved fixed-point iteration method is designed. Finally, the simulation of integrated navigation for an aircraft illustrates effectiveness and superiority of the proposed filter as compared the existing robust filters.

Adaptive fuzzy neural network-aided progressive Gaussian approximate filter for GPS/INS integration navigation

To address the difficulty of high uncertainty priori information in nonlinear filtering that occurs in tightly coupled navigation and positioning systems, a novel adaptive fuzzy neural network-aided progressive Gaussian filter is proposed in this paper to further improve the robustness and accuracy of the filter. A joint estimation of the step size and measurement noise covariance matrix is performed by a variational Bayesian approach. The estimation accuracy of the progressive Gaussian approximation filter is thereby enhanced. In particular, the measurement noise covariance matrix is adjusted using an adaptive fuzzy algorithm, and the state estimation error covariance matrix is optimized exploiting a neural network model. The proposed filter is experimentally verified to have higher accuracy than the existing state-of-the-art filters.

Convergence analysis and improvement of progressive Gaussian approximation filters

This paper studies the convergence problem of progressive Gaussian approximation filters (PGAFs). The convergence analysis of the PGAF is presented based on the Lyapunov method. It is proved that the estimation error is bounded when the convergence conditions hold. Moreover, based on the convergence analysis, the filtering performance of the PGAF is improved by adjusting the strategy of the progressive measurement update. Finally, a simulation experiment of mobile target tracking is designed to verify the effectiveness of the analysis results by comparing the improved PGAF with some existing methods.

An Adaptive Fuzzy Neural Network Based On Progressive Gaussian Approximate Filter with Variable Step Size

The nonlinear filtering problem is a hot spot in robot navigation research. As an advanced method that can effectively improve the robustness and accuracy of the system, the progressive Gaussian approximate filter with variable step size (PGAFVS) still has some shortcomings, how to resolve the nonlinear filtering problem in the application of tightly coupled integration under the premise of the prior uncertainty and further promote robustness high measurement accuracy, which becomes the purpose of this paper. This paper formulates the processing of trajectory tracking measurement noise problem as a Kalman filtering procedure and the measurement noise covariance matrix in controller, is jointly estimated based on the progressive Gaussian approximate filter (PGAF), after that, PGAFVS can be deduced. Then we proposed an adaptive fuzzy and backpropagation neural network controller based on PGAFVS (AFNPGA-VS) that can improve the application of tightly coupled integration under the premise of the prior uncertainty and further promote robustness high measurement accuracy. The simulation results show that the proposed algorithm outperforms the state-of-the-art methods.

Convergence Analysis and Improvement of Progressive Gaussian Approximation Filters

Convergence Analysis and Improvement of Progressive Gaussian Approximation Filters

A new Gaussian approximate filter with colored non-stationary heavy-tailed measurement noise

The filtering issue of a nonlinear system with colored non-stationary heavy-tailed measurement noise (CNSHMN) is addressed in this study via designing a new Gaussian approximate filter. By utilizing the state expansion method and the measurement difference method, the nonlinear filtering problem with the one-step delayed state and the white non-stationary heavy-tailed measurement noise (NSHMN) after the difference is turned into the traditional nonlinear filtering problem with NSHMN. A Gaussian student t-mixed distribution (GSTM) with Bernoulli random variable is utilized to describe the differenced measurement noise. The state vector, intermediate random variables (IRV), mixed probability and Bernoulli random variable (BRV) are simultaneously inferred by introducing variational Bayesian (VB) technique. Target tracking simulation examples reveal that the proposed filter is superior to the existing methods in the nonlinear filtering issue of CNSHMN.

A critical comparison on attitude estimation: From gaussian approximate filters to coordinate‐free dual optimal control

This paper conveys attitude and rate estimation without rate sensors by performing a critical comparison, validated by extensive simulations. The two dominant approaches to facilitate attitude estimation are based on stochastic and set-membership reasoning. The first one mostly utilizes the commonly known Gaussian-approximate filters, namely the EKF and UKF. Although more conservative, the latter seems to be more promising as it considers the inherent geometric characteristics of the underline compact state space and accounts -- from first principles -- for large model errors. We address the set-theoretic approach from a control point of view, and we show that it can overcome reported deficiencies of the Bayesian architectures related to this problem, leading to coordinate-free optimal filters. Lastly, as an example, we derive a modified predictive filter on the tangent bundle of the special orthogonal group $\mathbb{TSO}(3)$.

A novel robust Student’s t-based Gaussian approximate filter with one-step randomly delayed measurements

In this paper, a novel robust Student’s t-based Gaussian approximate filter (RSTGAF) is proposed to solve the filtering problem of the nonlinear system with one-step randomly delayed measurements (ORDM) and heavy-tailed measurement noise. The conditional likelihood function is transformed into an exponential multiplication form after using state augmentation approach and introducing a Bernoulli random variable, and then the augmentation state vector, the auxiliary random variables and the Bernoulli random variable are jointly estimated based on the variational Bayesian (VB) approach. The simulation results demonstrate the superiority of the proposed filter, as compared with the existing filters, to address the ORDM and heavy-tailed measurement noise.

A Novel Progressive Gaussian Approximate Filter for Tightly Coupled GNSS/INS Integration

In this article, we focus on addressing the nonlinear filtering problem with large prior uncertainty but high measurement accuracy, which may be encountered in the application of tightly coupled global navigation satellite system (GNSS)/inertial navigation system (INS) integration. Although the existing methods, such as progressive Gaussian approximate filter (PGAF), can address this problem, it has poor estimation accuracy. To improve the estimation accuracy of PGAF, the step sizes as well as the measurement noise covariance matrix (MNCM) are jointly estimated based on the variational Bayesian approach, from which a novel PGAF with variable step size is developed. Tightly coupled GNSS/INS integration simulations illustrate that the proposed filter outperforms the existing methods both in estimation accuracy and rate of convergence.

Progressive Gaussian approximation filter with adaptive measurement update

This paper presents a progressive Gaussian approximation filter with adaptive measurement update for nonlinear systems. First, a progressive Gaussian filtering (PGF) framework with variable step-size is presented by using the Bayesian rule, and second, a progressive Gaussian approximation filter is designed under the framework of the variable step-size PGF, where the means and covariances of corresponding Gaussian distributions are calculated by using the unscented transform. Moreover, termination conditions for stopping the progressive measurement update are given, which are not only good for the computational efficiency, but also improve the adaptive ability against measurement uncertainties. Finally, by simulations of a target tracking example, it demonstrates that the proposed method has higher computational efficiency and robustness than the existing PGAF.

Entropy-based adaptive attitude estimation

Gaussian approximation filters have increasingly been developed to enhance the accuracy of attitude estimation in space missions. The effective employment of these algorithms demands accurate knowledge of system dynamics and measurement models, as well as their noise characteristics, which are usually unavailable or unreliable. An innovation-based adaptive filtering approach has been adopted as a solution to this problem; however, it exhibits two major challenges, namely appropriate window size selection and guaranteed assurance of positive definiteness for the estimated noise covariance matrices. The current work presents two novel techniques based on relative entropy and confidence level concepts in order to address the abovementioned drawbacks. The proposed adaptation techniques are applied to two nonlinear state estimation algorithms of the extended Kalman filter and cubature Kalman filter for attitude estimation of a low earth orbit satellite equipped with three-axis magnetometers and Sun sensors. The effectiveness of the proposed adaptation scheme is demonstrated by means of comprehensive sensitivity analysis on the system and environmental parameters by using extensive independent Monte Carlo simulations.

A Variational Bayesian Based Strong Tracking Interpolatory Cubature Kalman Filter for Maneuvering Target Tracking

In tracking a maneuverable target, a proper estimation method with better filtering accuracy, stronger robustness, and faster convergence speed is crucial to the tracking system. The performance of conventional nonlinear Gaussian approximate filters may decline when the target engages in abrupt state changes or the noise covariance matrix is unknown and time-varying. In order to overcome these problems, a new variational Bayesian-based strong tracking interpolatory cubature Kalman filter (VB-STICKF) is deduced in this paper. Gaussian weighted integrals in the nonlinear filter are performed using the interpolatory cubature rule, which has better numerical characteristics for maneuvering target tracking. Moreover, by introducing the strong tracking filter into ICKF, the fading factor is used to modify the predicted error covariance, and the residual sequences are forced to be orthogonal, thus the decreasing performance resulted by the states change and the uncertain process noise could be effectively prevented. Furthermore, the measurement noise can be estimated online by variational Bayesian approach based on the inverse wishart distribution, the robustness of dealing with the uncertain measurement noise is improved. The detailed derivation of VB-STICKF for the general nonlinear models is presented in the paper. A target tracking problem with model uncertainty and time-varying process and measurement noise is utilized to test the performance of the proposed filter, the experimental results of three different scenarios demonstrate the improved filtering performance of the deduced VB-STICKF algorithm.

A New Outlier-Robust Student's t Based Gaussian Approximate Filter for Cooperative Localization

In this paper, a new outlier-robust Student's t based Gaussian approximate filter is proposed to address the heavy-tailed process and measurement noises induced by the outlier measurements of velocity and range in cooperative localization of autonomous underwater vehicles (AUVs). The state vector, scale matrices, and degrees of freedom (DOF) parameters are jointly estimated based on the variational Bayesian approach by using the constructed Student's t based hierarchical Gaussian state-space model. The performances of the proposed filter and existing filters are tested in the cooperative localization of an AUV through a lake trial. Experimental results illustrate that the proposed filter has better localization accuracy and robustness than existing state-of-the-art outlier-robust filters.

Design of High-Degree Student’s t-Based Cubature Filters

The Student’s t-based nonlinear filter (STNF) can cope with the filtering problem of nonlinear systems with heavy-tailed process and measurement noises. The key problem in the design of a STNF is how to calculate the Student’s t weighted integral, and the performance of the STNF depends heavily on the used numerical integration technique. In this paper, new high-degree Student’s t spherical-radial cubature rules (STSRCRs) for the Student’s t weighted integral are proposed based on spherical-radial transformation and moment matching methods, from which new high-degree Student’s t-based cubature filters (STCFs) are developed. The proposed high-degree STSRCRs can achieve better approximation to the Student’s t weighted integral as compared with the existing third-degree Student’s t integral rules. As a result, the proposed high-degree STCFs have higher estimation accuracy than the existing STNFs. Simulation results illustrate that the proposed filters can achieve higher estimation accuracy than the existing Gaussian approximate filters, Huber-based nonlinear Kalman filters and STNFs with slightly increased computational complexities, and are computationally much more efficient than the existing Gaussian sum filter and particle filter.

Tracking of group targets using multiple models GGIW-PHD algorithm based on best-fitting Gaussian approximation and strong tracking filter

Gamma Gaussian inverse Wishart probability hypothesis density (GGIW-PHD) filter algorithm is always used for tracking group targets with unknown number and variable measurement rates in the presenc...

Robust Student’s t-Based Stochastic Cubature Filter for Nonlinear Systems With Heavy-Tailed Process and Measurement Noises

In this paper, a new robust Student's t-based stochastic cubature filter (RSTSCF) is proposed for a nonlinear state-space model with heavy-tailed process and measurement noises. The heart of the RSTSCF is a stochastic Student's t-spherical radial cubature rule (SSTSRCR), which is derived based on the third-degree unbiased spherical rule and the proposed third-degree unbiased radial rule. The existing stochastic integration rule is a special case of the proposed SSTSRCR when the degrees of freedom parameter tends to infinity. The proposed filter is applied to a maneuvering bearings-only tracking example, in which an agile target is tracked and the bearing is observed in clutter. Simulation results show that the proposed RSTSCF can achieve higher estimation accuracy than the existing Gaussian approximate filter, Gaussian sum filter, Huber-based nonlinear Kalman filter, maximum correntropy criterion-based Kalman filter, and robust Student's t-based nonlinear filters, and is computationally much more efficient than the existing particle filter.