Iterative Filtering Research Articles

Iterated Orthogonal Simplex Cubature Kalman Filter and Its Applications in Target Tracking

In order to increase a nonlinear system’s state estimate precision, an iterated orthogonal simplex cubature Kalman filter (IOSCKF) is presented in this study for target tracking. The Gaussian-weighted integral is decomposed into a spherical integral and a radial integral, which are approximated using the spherical simplex-radial rule and second-order Gauss–Laguerre quadrature rule, respectively, and result in the novel simplex cubature rule. To decrease the high-order error terms, cubature points with appropriate weights are taken from the cubature rule and processed using the provided orthogonal matrix. The structure supporting the nonlinear Kalman filter incorporates the altered points and weights and the calculation steps; from this, the updated time and measurement can be inferred. The Gauss–Newton iteration is employed repeatedly to adjust the measurement update until the termination condition is met and the IOSCKF is attained. The proposed algorithms are applied in target tracking, including CV target tracking and spacecraft orbit tracking, and the simulation results reveal that the IOSCKF can achieve higher accuracy compared to the CKF, SCKF, and OSCKF. In spacecraft orbit tracking simulation, compared with the SCKF, the position tracking accuracy and velocity tracking accuracy of the OSCKF are increased by 2.21% and 1.94%, respectively, which indicates that the orthogonal transformation can improve the tracking accuracy. Furthermore, compared with the OSCKF, the position tracking accuracy and velocity tracking accuracy of the IOSCKF are increased by 2.71% and 2.97%, respectively, which indicates that the tracking accuracy can be effectively improved by introducing iterative calculation into the measurement equation, thus verifying the effectiveness of the method presented in this paper.

An Enhanced Multi-Sensor Simultaneous Localization and Mapping (SLAM) Framework with Coarse-to-Fine Loop Closure Detection Based on a Tightly Coupled Error State Iterative Kalman Filter

In order to attain precise and robust transformation estimation in simultaneous localization and mapping (SLAM) tasks, the integration of multiple sensors has demonstrated effectiveness and significant potential in robotics applications. Our work emerges as a rapid tightly coupled LIDAR-inertial-visual SLAM system, comprising three tightly coupled components: the LIO module, the VIO module, and the loop closure detection module. The LIO module directly constructs raw scanning point increments into a point cloud map for matching. The VIO component performs image alignment by aligning the observed points and the loop closure detection module imparts real-time cumulative error correction through factor graph optimization using the iSAM2 optimizer. The three components are integrated via an error state iterative Kalman filter (ESIKF). To alleviate computational efforts in loop closure detection, a coarse-to-fine point cloud matching approach is employed, leverging Quatro for deriving a priori state for keyframe point clouds and NanoGICP for detailed transformation computation. Experimental evaluations conducted on both open and private datasets substantiate the superior performance of the proposed method compared to similar approaches. The results indicate the adaptability of this method to various challenging situations.

ICESat-2 noise filtering using a point cloud neural network

The ATLAS sensor onboard the ICESat-2 satellite is a photon-counting lidar (PCL) with a primary mission to map Earth's ice sheets. A secondary goal of the mission is to provide vegetation and terrain elevations, which are essential for calculating the planet's biomass carbon reserves. A drawback of ATLAS is that the sensor does not provide reliable terrain height estimates in dense, high-closure forests because only a few photons reach the ground through the canopy and return to the detector. This low penetration translates into lower accuracy for the resultant terrain model. Tropical forest measurements with ATLAS have an additional problem estimating top of canopy because of frequent atmospheric phenomena such as fog and low clouds that can be misinterpreted as top of the canopy. To alleviate these issues, we propose using a ConvPoint neural network for 3D point clouds and high-density airborne lidar as training data to classify vegetation and terrain returns from ATLAS. The semantic segmentation network provides excellent results and could be used in parallel with the current ATL08 noise filtering algorithms, especially in areas with dense vegetation. We use high-density airborne lidar data acquired along ICESat-2 transects in Central American forests as a ground reference for training the neural network to distinguish between noise photons and photons lying between the terrain and the top of the canopy. Each photon event receives a label (noise or signal) in the test phase, providing automated noise-filtering of the ATL03 data. The terrain and top of canopy elevations are subsequently aggregated in 100 m segments using a series of iterative smoothing filters. We demonstrate improved estimates for both terrain and top of canopy elevations compared to the ATL08 100 m segment estimates. The neural network (NN) noise filtering reliably eliminated outlier top of canopy estimates caused by low clouds, and aggregated root mean square error (RMSE) decreased from 7.7 m for ATL08 to 3.7 m for NN prediction (18 test profiles aggregated). For terrain elevations, RMSE decreased from 5.2 m for ATL08 to 3.3 m for the NN prediction, compared to airborne lidar reference profiles.

A single-building damage detection model based on multi-feature fusion: A case study in Yangbi

Accurate and effective identification, determination of the location, and classification of damaged buildings are essential after destructive earthquakes. However, the accuracy of image change detection is limited because of the many texture features and changes in non-building information. In this context, a model for single-building damage detection based on multi-feature fusion is proposed. First, the normalized Digital Surface Model (nDSM) was extracted from the DSM through iterative filtering and point cloud thinning, followed by the extraction of building contour information. Next, single-building images were generated from different data sources through the region of interest (ROI), and the optimal texture feature parameters were extracted for fusion. Afterward, principal-component analysis (PCA) was conducted to suppress multi-feature correlation-induced information redundancy. Finally, the damage to buildings was quantitatively evaluated, and the model was compared with 13 models. The results confirmed the practicability of the model for the Yangbi MS6.4 and Honghe MS5.0 earthquakes.

Rotating 3D laser mapping system for multi-rotor drones

Accurate positional estimation is an essential prerequisite for the regular operation of an autonomous rotary-wing Unmanned Aerial Vehicles (UAV). However, the field of view (FOV) limitation problem of lidar makes it more challenging to locate the rotary-wing UAV in an unknown environment. To address rotor drones with an insufficient FOV and the observation blindness of lidar in complex environments, this paper designs a rotorcraft UAV system based on rotating 3D lidar and proposes a simultaneous localization and mapping algorithm for rotating 3D lidar. The algorithm distinguishes between planar and edge features based on the curvature value of the point cloud first. Then, to reduce the impact caused by the UAV motion and lidar rotation, messages about the Inertial Measurement Unit (IMU) and real-time rotation angles are used to compensate for these motions twice, while the IMU measurements are used for state prediction, and the error-sate iterative extended Kalman filter is used to update the residuals after matching line and surface features with sub-map. Finally, Smoother high-rate odometer data was obtained through IMU pre-integration and a first-order low-pass filter. The experimental results show that the proposed rotating lidar unit in indoor and outdoor conditions makes the rotorcraft UAV have a larger FOV, which not only improves the environmental perception capability and positional estimation accuracy of the rotorcraft but enhances the positioning reliability and flight stability of the rotorcraft UAV in complex environments.

WIG-Net: Wavelet-Based Defocus Deblurring with IFA and GCN

Although the existing deblurring methods for defocused images are capable of approximately recovering clear images, they still exhibit certain limitations, such as ringing artifacts and remaining blur. Along these lines, in this work, a novel deep-learning-based method for image defocus deblurring was proposed, which can be applied to medical images, traffic monitoring, and other fields. The developed approach is equipped with wavelet transform, an iterative filter adaptive module, and graph neural network and was specifically designed for handling defocused blur. Our network exhibits excellent properties in preserving the original information during the restoration of clear images, thereby enhancing its capability to spatially address varying blurriness and improving the quality of deblurring. From the acquired experimental results, the superiority of the introduced method in the context of image defocus deblurring compared to the majority of the existing algorithms was clearly demonstrated.

Improvement of the algebraic stress model for separated flows based on field inversion and machine learning

Due to the difficulties of precisely calculating the turbulence intensity within the separated shear layer using traditional turbulent models, computing strongly separated flows is a key task of considerable interest. In this paper, the Reynolds stress functional expression is improved toward an explicit algebraic stress model for separated flows that are similar to hump flows using a data-driven framework of field inversion and machine learning that can take model consistency into account during the model correction process. The iterative Kalman filter algorithm is utilized to address the inversion problem, and the inversion results are used as training data for correction models that are trained using random forest regression. For model verification and validation, we employ both the curved backward-facing step and bump cases. The findings indicate that the inversion produces favorable outcomes, and the enhanced model developed utilizing the inversion data exhibits good generalizability.

Underwater Doppler-bearing maneuvering target motion analysis based on joint estimated adaptive unscented Kalman filter

Noncooperative maneuvering target motion analysis is one of the challenging tasks in the field of underwater target localization and tracking for passive sonar. Underwater noncooperative targets often perform various maneuvers, and the targets are commonly modeled as a combination of constant-velocity models and coordinate-turn models with unknown turning rates. Traditional algorithms for Doppler-bearing target motion analysis are incapable of processing noncooperative maneuvering targets because the algorithms rely on a priori information of the turning rate and the center frequency. To address these shortcomings, this paper proposes the joint estimated adaptive unscented Kalman filter (JE-AUKF) algorithm. The JE-AUKF places the center frequency and turning rate into the state vector and constructs a time-varying state model that self-adapts to a maneuvering target. The JE-AUKF also introduces a time-varying fading factor into the process noise covariance matrix to improve the tracking performance. Simulations and sea trials are conducted to compare the performance of the JE-AUKF with the iterative unscented Kalman filter, the interacting multiple model-unscented Kalman filter, the interacting multiple model-iterative unscented Kalman filter, and the interacting multiple model-joint estimated unscented Kalman filter. The result shows that the JE-AUKF achieves better tracking performance for noncooperative maneuvering targets.

An iterative augmented unscented Kalman filter for simultaneous state-parameter-input estimation for systems with/without direct feedthrough

A few unscented Kalman filters (UKFs) have been developed for simultaneous state-parameter-input estimation, however, these UKFs often have at least one of these limitations: requiring ad hoc procedures; requiring displacement measurements, which may be inconvenient to obtain, to fuse with acceleration measurements to avoid the “drift phenomenon” in the estimated displacement responses and unknown inputs; requiring a (full-rank) feedthrough matrix; and undesirable computational efficiency. To overcome these limitations, a novel iterative augmented unscented Kalman filter (IAUKF) for simultaneous state-parameter-input estimation for systems either with or without direct feedthrough is developed and examined in this study. The IAUKF closely follows the Bayes’ theorem with no ad hoc procedure, featuring theoretical simplicity. An iterative strategy is introduced to improve the accuracy of the unknown input estimation at each time step, so that the “drift phenomenon” encountered by the existing augmented Kalman filter is avoided. Further, the IAUKF requires no prior models for the time histories of the unknown inputs and applies to systems with/without direct feedthrough. With realistic assumptions, the steps for computationally efficient implementations of the IAUKF are proposed. Numerical examples of nonlinear structural systems with/without direct feedthrough of the unknown inputs are adopted to examine the effectiveness and the computational efficiency of the IAUKF.

NSCKL: Normalized Spectral Clustering With Kernel-Based Learning for Semisupervised Hyperspectral Image Classification.

Spatial-spectral classification (SSC) has become a trend for hyperspectral image (HSI) classification. However, most SSC methods mainly consider local information, so that some correlations may not be effectively discovered when they appear in regions that are not contiguous. Although many SSC methods can acquire spatial-contextual characteristics via spatial filtering, they lack the ability to consider correlations in non-Euclidean spaces. To address the aforementioned issues, we develop a new semisupervised HSI classification approach based on normalized spectral clustering with kernel-based learning (NSCKL), which can aggregate local-to-global correlations to achieve a distinguishable embedding to improve HSI classification performance. In this work, we propose a normalized spectral clustering (NSC) scheme that can learn new features under a manifold assumption. Specifically, we first design a kernel-based iterative filter (KIF) to establish vertices of the undirected graph, aiming to assign initial connections to the nodes associated with pixels. The NSC first gathers local correlations in the Euclidean space and then captures global correlations in the manifold. Even though homogeneous pixels are distributed in noncontiguous regions, our NSC can still aggregate correlations to generate new (clustered) features. Finally, the clustered features and a kernel-based extreme learning machine (KELM) are employed to achieve the semisupervised classification. The effectiveness of our NSCKL is evaluated by using several HSIs. When compared with other state-of-the-art (SOTA) classification approaches, our newly proposed NSCKL demonstrates very competitive performance. The codes will be available at https://github.com/yuanchaosu/TCYB-nsckl.

Channel Estimator and Nonlinear Detector for mmWave Beamformed OTFS Systems in High Mobility Scenarios

Real-time data-intensive applications in highly mobile environments with high data rates and low latency are one of the key requirements for beyond fifth-generation (5 G) communications. Towards this end, a combination of the millimeter wave (mmWave) band with large available spectrum and orthogonal time-frequency space (OTFS) modulation, which offers robust performance in high mobility scenarios, can be envisioned as a promising solution. However, employing OTFS in the mmWave frequency band is challenging in presence of nonlinear distortions attributed to radio frequency (RF) power amplifier (PA) and complex equalization involved in OTFS receiver in the delay-Doppler domain. The nonlinearity effect of PA is inevitable in mmWave systems due to the high operating frequency and large bandwidth. The nonlinear distortions induce interference and make the target distribution analytically intractable. In this paper, we propose a receiver design for mmWave OTFS systems that aims to address the challenges posed by the nonlinear distortions of the RF PA. The proposed receiver design leverages the use of a large-scale antenna array to decouple the received signal into multiple branches through beamforming. Specifically, we first derive the input-output relation in the delay-Doppler domain with nonlinearity. Then, a combination of maximal-ratio combining (MRC) and iterative particle filter (PF)-based nonlinear detector is proposed to address the issue. We also design a pilot pattern considering the additional interference from the PA impairment. Furthermore, we reformulate the threshold-based channel estimation to incorporate the nonlinearly distorted pilot in its estimation process. Extensive numerical results demonstrate the proposed algorithm's reliability and superiority over existing digital predistorters (DPD) and conventional linear detectors.

Identification of nonlinear conservation laws for multiphase flow based on Bayesian inversion

Conservation laws of the generic form c_t+f(c)_x=0 play a central role in the mathematical description of various engineering related processes. Identification of an unknown flux function f(c) from observation data in space and time is challenging due to the fact that the solution c(x, t) develops discontinuities in finite time. We explore a Bayesian type of method based on representing the unknown flux f(c) as a Gaussian random process (parameter vector) combined with an iterative ensemble Kalman filter (EnKF) approach to learn the unknown, nonlinear flux function. As a testing ground, we consider displacement of two fluids in a vertical domain where the nonlinear dynamics is a result of a competition between gravity and viscous forces. This process is described by a multidimensional Navier–Stokes model. Subject to appropriate scaling and simplification constraints, a 1D nonlinear scalar conservation law c_t+f(c)_x=0 can be derived with an explicit expression for f(c) for the volume fraction c(x, t). We consider small (noisy) observation data sets in terms of time series extracted at a few fixed positions in space. The proposed identification method is explored for a range of different displacement conditions ranging from pure concave to highly non-convex f(c). No a priori information about the sought flux function is given except a sound choice of smoothness for the a priori flux ensemble. It is demonstrated that the method possesses a strong ability to identify the unknown flux function. The role played by the choice of initial data c_0(x) as well various types of observation data is highlighted.

An Iterative Wiener Filter Based on a Fourth-Order Tensor Decomposition

This work focuses on linear system identification problems in the framework of the Wiener filter. Specifically, it addresses the challenging identification of systems characterized by impulse responses of long length, which poses significant difficulties due to the existence of large parameter space. The proposed solution targets a dimensionality reduction of the problem by involving the decomposition of a fourth-order tensor, using low-rank approximations in conjunction with the nearest Kronecker product. In addition, the rank of the tensor is controlled and limited to a known value without involving any approximation technique. The final estimate is obtained based on a combination of four (shorter) optimal filters, which are alternatively iterated. As a result, the designed iterative Wiener filter outperforms the traditional counterpart, being more robust to the accuracy of the statistics’ estimates and/or noisy conditions. In addition, simulations performed in the context of acoustic echo cancellation indicate that the proposed iterative Wiener filter that exploits this fourth-order tensor decomposition achieves better performance as compared to some previously developed solutions based on lower decomposition levels. This study could further lead to the development of computationally efficient tensor-based adaptive filtering algorithms.

Towards precise complex AM-FM signals decomposition under strong noise conditions: A novel two-level chirp mode decomposition approach

Many signals in the real world contain noise, which interferes with the analysis of signals. Under harsh environments, the signal will be seriously disturbed by strong noise and overwhelm its changing features. Decomposing the signal with strong noise can effectively reveal the actual change law of original signal. However, the current methods cannot effectively process complex amplitude modulation-frequency modulation (AM-FM) signals with strong noise, and suffer from serious mode aliasing problems. In order to effectively decompose complex AM-FM signals with strong noise, a novel two-level chirp mode decomposition (TCMD) approach is proposed in this paper. TCMD can effectively solve the mode aliasing problem caused by strong noise interference and accurately decompose each sub-signal from original signal with strong noise. TCMD mainly includes two core parts: instantaneous frequency (IF) extraction and signal decomposition. Firstly, we establish a new IF extraction model for strong noise signals called improved parameterized time–frequency transform (IPTFT), which overcomes the shortcomings of current PTFT that cannot effectively handle multi-component signals. IPTFT can effectively filter out the interference of strong noise and accurately extract the initial IFs of multi-component signal. Then, we propose a novel adaptive iterative envelope filter tracking (AIETF) for signal decomposition that adopts the adaptive decomposition idea. AIETF can effectively decompose complex multi-component signals with strong noise, accurately estimate the IFs and reconstruct sub-signals. Decomposition experiments of both simulated and experimental signals with strong noise were carried out to verify proposed TCMD. The results show that TCMD can effectively decompose strong noisy AM-FM signals with close IFs, fast-changing IFs and crossing IFs. Meanwhile, TCMD can also effectively decompose the vibration signals with strong noise of wind turbine gearboxes, aero-engine gearboxes, and faulty bearings under non-stationary conditions to obtain high-quality TFR. Furthermore, TCMD achieves better decomposition effect for strong noise signals than the existing decomposition and TFT methods.

State of charge estimation for Li-ion batteries based on iterative Kalman filter with adaptive maximum correntropy criterion

Nonlinear filter methods are commonly employed for model-driven based battery state of charge (SOC) estimation because of their ability to suppress Gaussian noise. Due to the uncertainty of complicated operational conditions, the minimum mean square error (MMSE) criterion based Kalman filter (KF) methods may not guarantee sufficient SOC estimation accuracy in the presence of non-Gaussian noise. To enhance the anti-interference performance of the KFs based SOC estimation method, a novel methodology that combines the adaptive kernel width based maximum correntropy criterion (AMCC) with the Levenberg-Marguardt (L-M) principle based adaptive iterative extended KF (AIEKF) is proposed. Firstly, the MMSE criterion of the EKF is replaced by the MCC criterion and an adaptive kernel width strategy is designed to weaken the effect of anomalous data. Then, the AMCC-corrected covariance matrix and state are updated using an L-M optimized multi-step iterative filter during the measurement update phase. Finally, the combination of AMCC and AIEKF is devised to execute SOC estimation. Several sets of data from different temperatures and operational conditions are used to verify the validity of the proposed AMCC-AIEKF method. Compared with the adaptive EKF (AEKF) and the adaptive iterative EKF (AIEKF) method, the AMCC-AIEKF method has the advantage in terms of SOC estimation accuracy and robustness under different operational conditions with noiseless, Gaussian noise and non-Gaussian noise.

Improving quantification of methane point source emissions from imaging spectroscopy

The matched filter (MF) method is widely used for hyperspectral imaging spectrometers to detect and quantify methane point sources due to its high computational efficiency. However, it would result in an unavoidable underestimation, especially for large concentration enhancements. The lognormal matched filter (LMF) has provided a unique and not limited to weak methane plumes theory through lognormal background radiances modeling, but validations on simulated experiments and applications to real data remain to be explored. Moreover, the covariance contamination caused by the enhanced pixels and the surface heterogeneity have detrimental effects on the detection methods for real data application. In this study, we propose the iterative lognormal matched filter (ILMF) method to address these challenges. We evaluate the performance of the ILMF with two ideal simulations and the end-to-end simulation. The results of random simulated enhancement retrievals show that the retrieved enhancement by the ILMF method agrees well with the true enhancement, with an R2 of 0.984 and a small root-mean-square error (RMSE) of 55.856 ppb. The ILMF method reduce the RMSE of retrieved enhancement by 80% compared with the MF method. The results of end-to-end simulations show the underestimation of MF method at different sites regarding the emission rate, as well as the improvement of the ILMF method. Further, we apply the ILMF method to detect and quantify point sources in some methane hotspot regions. Our study reports the underestimation in traditional MF method and provides an unbiased and robust method for quantifying methane emissions.

4D iRIOM: 4D Imaging Radar Inertial Odometry and Mapping

Millimeter wave radar can measure distances, directions, and Doppler velocity for objects in harsh conditions such as fog. The 4D imaging radar with both vertical and horizontal data resembling an image can also measure objects' height. Previous studies have used 3D radars for ego-motion estimation. But few methods leveraged the rich data of imaging radars, and they usually omitted the mapping aspect, thus leading to inferior odometry accuracy. This paper presents a real-time imaging radar inertial odometry and mapping method, iRIOM, based on the submap concept. To deal with moving objects and multipath reflections, we use the graduated non-convexity method to robustly and efficiently estimate ego-velocity from a single scan. To measure the agreement between sparse non-repetitive radar scan points and submap points, the distribution-to-multi-distribution distance for matches is adopted. The ego-velocity, scan-to-submap matches are fused with the 6D inertial data by an iterative extended Kalman filter to get the platform's 3D position and orientation. A loop closure module is also developed to curb the odometry module's drift. To our knowledge, iRIOM based on the two modules is the first 4D radar inertial SLAM system. On our and third-party data, we show iRIOM's favorable odometry accuracy and mapping consistency against the FastLIO-SLAM and the EKFRIO. Also, the ablation study reveal the benefit of inertial data versus the constant velocity model, and scan-to-submap matching versus scan-to-scan matching.

A low‐drift and real‐time localisation and mapping method for handheld hemispherical view LiDAR‐IMU integration system

AbstractThis paper proposes a simultaneous localisation and mapping (SLAM) framework that uses a handheld hemispherical view LiDAR‐IMU integration system. Inspired by the specific characteristic of the hemispherical view LiDAR, a ground segmentation module based on seed points is designed. The ground points are then downsampled to eliminate redundant vertical constraints. The IMU data and the pre‐processed point cloud are used to perform state estimation via a tightly coupled iterative extended Kalman filter (iEKF) to obtain the pose estimation. The automatically detected loop closures provide closed‐loop constraints for the odometry, and a factor graph ensures the global consistency of the map. Data from diverse scenes are collected via a prototype system. Both qualitative and quantitative experiments are carried out to verify the framework's performance. According to the experimental results, our framework achieves low‐drift, high‐coverage and real‐time performance, outperforming the state‐of‐the‐art LiDAR SLAM methods in our handheld hemispherical view LiDAR‐IMU test sites. For the research community's benefit, the dataset is publicly provided for other researchers to compare against.

Iterative Kalman filter for biological tissue identification

Dynamic soft tissue identification plays an important role in robotic-assisted minimally invasive surgery to achieve realistic force feedback for precise and safe surgical operations. This article studies a dynamic soft tissue identification method by combination of the Hunt Crossley contact model with an iterated Kalman filter. The dynamic system equation of tool-tissue interaction is constructed by combining mechanical tool's kinematic dynamics with the Hunt Crossley contact dynamics. Upon this, an iterative Kalman filter is developed for online soft tissue identification by integrating the maximum a posteriori principle into the Kalman filtering framework to optimize the posterior state estimate to account for the strong nonlinearity of the Hunt Crossley contact model. Results and comparison analysis demonstrate that the proposed method can effectively identify the Hunt-Crossley model parameters, resulting in improved accuracy compared with the conventional recursive least square method and extended Kalman filter.

Forward modeling and inverse estimation for nonlinear filtering

AbstractThe highly accurate state estimation of nonlinear systems is needed in many different application fields. In many existing nonlinear filters, the uncompensated nonlinear approximation error and widely used numerical sampling methods both seriously affect the estimation accuracy and numerical stability. In this paper, based on the variational Bayesian framework, a novel iterative nonlinear filter consisting of forward modeling and inverse estimation is proposed. In the forward modeling stage, a linear Gaussian regression process with distributed variational parameters (VPs) is designed to fit the whole nonlinear measurement likelihood. The distributed VPs consider both first and second moments information, so as to compensate for approximation error from the whole likelihood perspective, instead of only the nonlinear function approximation, which can further improve the accuracy of Bayesian posterior update. In the linear Gaussian regression process, the original nonlinear measurement function (NMF) is just viewed as a prior and no longer directly participates in the modeling of the measurement likelihood. Hence, in the inverse estimation stage, when maximizing the evidence lower bound to update the posteriors of the state and VPs, we can obtain closed‐form solutions without numerical sampling methods to approximate the integral of NMF required in many existing methods, which can guarantee numerical stability. Moreover, to avoid the random setting of hyperparameters and to block the propagation of the accumulated error, we propose two rules, estimation and lower bound consistencies, to conduct the hyperparameters' initialization at the beginning of iteration at each sampling time and also derive an optimization method to update hyperparameters during iteration. Finally, the performance of our proposed method is demonstrated in the nonlinear orbital estimation problems.