Cubature粒子滤波

非线性非高斯下后验概率密度函数解析值无法获得,需设计合理的重要性密度函数进行逼近。传统粒子滤波(particle filter,PF)直接采用未含最新量测信息的状态转移先验分布函数作为重要性密度函数来逼近后验概率密度函数。针对PF缺乏量测信息的问题,提出一种基于Cubature卡尔曼滤波(Cubature Kalman filter,CKF)重采样的Cubature粒子滤波新算法(Cubature particle filter,CPF)。该算法在先验分布更新阶段融入了最新的观测数据,通过CKF设计重要性密度函数,使其更加接近系统状态后验概率密度。仿真表明CPF估计精度高于PF和扩展卡尔曼滤波(extended particle filter,EPF),与无轨迹粒子滤波(unscented particle filter,UPF)相比,其精度相当,但算法运行时间降低了约20%。

基于Cubature卡尔曼滤波的强跟踪滤波算法

提出一种新的基于Cubature卡尔曼滤波的强跟踪滤波算法(CKF-STF)。该算法基于强跟踪滤波的理论框架,采用三阶Cubature采样积分代替传统强跟踪滤波中的雅可比矩阵求解,并给出了适用于一般非线性系统的强跟踪滤波算法的线性等价描述。新算法不仅具有强跟踪滤波鲁棒性强的优点,而且继承了CKF算法处理非线性系统的能力。采用具有实际应用背景的目标纯方位跟踪仿真实例验证CKF-STF算法,结果表明该算法不仅精度高,而且实现简单。

基于奇异值分解的多重渐消鲁棒Cubature卡尔曼滤波及在组合导航中的应用(英文)

为了提高标准Cubature卡尔曼滤波(CKF)的稳定性和鲁棒性,提出一种改进的多重渐消H∞滤波Cubature卡尔曼滤波算法。首先基于系统状态的可观测性给出多重渐消因子矩阵求解过程,提高滤波算法的稳定性,抑制滤波发散;其次,引入H∞鲁棒思想,构造多重渐消H∞滤波Cubature卡尔曼滤波器;最后,提出采用一种奇异值分解的矩阵分解策略代替标准Cubature卡尔曼滤波中的Cholesky分解,进一步提高算法的数值稳定性。实际GPS/INS组合导航实验表明,改进的多重渐消H∞滤波Cubature卡尔曼滤波算法不仅能有效抑制滤波发散提高算法的稳定性,而且对观测野值具有更高的鲁棒性;提出的新算法与标准CKF算法相比,XYZ三个方向的位置精度分别提高了55.8%,46.6%和39.7%。

WSN中利用改进FOA-GRNN和迭代Cubature卡尔曼滤波的实时目标跟踪方法

针对传统无线传感器网络(Wireless Sensor Network,WSN)对运动目标的定位和跟踪容易产生明显误差的问题,提出利用改进FOA-GRNN和迭代Cubature卡尔曼滤波的实时目标跟踪方法。基于改进FOA-GRNN法,利用从锚点接收到的运动目标的模拟(RSSI)值和相应的实际目标二维位置对GRNN进行训练,从而获得单个目标在二维运动时的准确初始位置;利用迭代Cubature卡尔曼滤波法对实时目标进行精准定位和测距,获得实时目标的准确定位和跟踪信息;将改进的FOA-GRNN法和迭代Cubature卡尔曼滤波法相结合用于WSN中实时目标跟踪和定位,在提高初始位置精度的同时,还提高了实时目标定位和跟踪信息的准确度。实验结果表明,相比其他几种较新的方法,该方法改善了WSN中实时目标的跟踪性能,降低了误差,提高了跟踪精度。

Determination of isolation layer thickness for undersea mine based on differential cubature solution to irregular Mindlin plate

The differential cubature solution to the problem of a Mindlin plate lying on the Winkler foundation with two simply supported edges and two clamped edges was derived.Discrete numerical technology and shape functions were used to ensure that the solution is suitable to irregular shaped plates.Then,the mechanical model and the solution were employed to model the protection layer that isolates the mining stopes from sea water in Sanshandao gold mine,which is the first subsea mine of China.Furthermore,thickness optimizations for the protection layers above each stope were conducted based on the maximum principle stress criterion,and the linear relations between the best protection layer thickness and the stope area under different safety factors were regressed to guide the isolation design.The method presented in this work provides a practical way to quickly design the isolation layer thickness in subsea mining.

Robust range-parameterized cubature Kalman filter for bearings-only tracking

In order to improve tracking accuracy when initial estimate is inaccurate or outliers exist,a bearings-only tracking approach called the robust range-parameterized cubature Kalman filter(RRPCKF)was proposed.Firstly,the robust extremal rule based on the pollution distribution was introduced to the cubature Kalman filter(CKF)framework.The improved Turkey weight function was subsequently constructed to identify the outliers whose weights were reduced by establishing equivalent innovation covariance matrix in the CKF.Furthermore,the improved range-parameterize(RP)strategy which divides the filter into some weighted robust CKFs each with a different initial estimate was utilized to solve the fuzzy initial estimation problem efficiently.Simulations show that the result of the RRPCKF is more accurate and more robust whether outliers exist or not,whereas that of the conventional algorithms becomes distorted seriously when outliers appear.

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.

Low-cost adaptive square-root cubature Kalman filter forsystems with process model uncertainty

A novel low-cost adaptive square-root cubature Kalmanfilter （LCASCKF） is proposed to enhance the robustness of processmodels while only increasing the computational load slightly.It is well-known that the Kalman filter cannot handle uncertainties ina process model, such as initial state estimation errors, parametermismatch and abrupt state changes. These uncertainties severelyaffect filter performance and may even provoke divergence. Astrong tracking filter （STF）, which utilizes a suboptimal fading factor,is an adaptive approach that is commonly adopted to solvethis problem. However, if the strong tracking SCKF （STSCKF）uses the same method as the extended Kalman filter （EKF） tointroduce the suboptimal fading factor, it greatly increases thecomputational load. To avoid this problem, a low-cost introductorymethod is proposed and a hypothesis testing theory is applied todetect uncertainties. The computational load analysis is performedby counting the total number of floating-point operations and it isfound that the computational load of LCASCKF is close to that ofSCKF. Experimental results prove that the LCASCKF performs aswell as STSCKF, while the increase in computational load is muchlower than STSCKF.

Stochastic convergence analysis of cubature Kalman filter with intermittent observations

The stochastic convergence of the cubature Kalmanfilter with intermittent observations （CKFI） for general nonlinearstochastic systems is investigated. The Bernoulli distributed ran-dom variable is employed to describe the phenomenon of intermit-tent observations. According to the cubature sample principle, theestimation error and the error covariance matrix （ECM） of CKFIare derived by Taylor series expansion, respectively. Afterwards, itis theoretically proved that the ECM will be bounded if the obser-vation arrival probability exceeds a critical minimum observationarrival probability. Meanwhile, under proper assumption corre-sponding with real engineering situations, the stochastic stabilityof the estimation error can be guaranteed when the initial estima-tion error and the stochastic noise terms are sufficiently small. Thetheoretical conclusions are verified by numerical simulations fortwo illustrative examples; also by evaluating the tracking perfor-mance of the optical-electric target tracking system implementedby CKFI and unscented Kalman filter with intermittent observa-tions （UKFI） separately, it is demonstrated that the proposed CKFIslightly outperforms the UKFI with respect to tracking accuracy aswell as real time performance.

Generalized cubature quadrature Kalman filters:derivations and extensions

A new Gaussian approximation nonlinear filter called generalized cubature quadrature Kalman filter (GCQKF) is introduced for nonlinear dynamic systems. Based on standard GCQKF, two extensions are developed, namely square root generalized cubature quadrature Kalman filter (SR-GCQKF) and iterated generalized cubature quadrature Kalman filter (I-GCQKF). In SR-GCQKF, the QR decomposition is exploited to alter the Cholesky decomposition and both predicted and filtered error covariances have been propagated in square root format to make sure the numerical stability. In I-GCQKF, the measurement update step is executed iteratively to make full use of the latest measurement and a new terminal criterion is adopted to guarantee the increase of likelihood. Detailed numerical experiments demonstrate the superior performance on both tracking stability and estimation accuracy of I-GCQKF and SR-GCQKF compared with GCQKF.

Underwater square-root cubature attitude estimator by use of quaternion-vector switching and geomagnetic field tensor

This paper presents a kind of attitude estimation algorithm based on quaternion-vector switching and square-root cubature Kalman filter for autonomous underwater vehicle(AUV).The filter formulation is based on geomagnetic field tensor measurement dependent on the attitude and a gyro-based model for attitude propagation. In this algorithm, switching between the quaternion and the three-component vector is done by a couple of the mathematical transformations. Quaternion is chosen as the state variable of attitude in the kinematics equation to time update, while the mean value and covariance of the quaternion are computed by the three-component vector to avoid the normalization constraint of quaternion. The square-root forms enjoy a continuous and improved numerical stability because all the resulting covariance matrices are guaranteed to stay positively semidefinite. The entire square-root cubature attitude estimation algorithm with quaternion-vector switching for the nonlinear equality constraint of quaternion is given. The numerical simulation of simultaneous swing motions in the three directions is performed to compare with the three kinds of filters and the results indicate that the proposed filter provides lower attitude estimation errors than the other two kinds of filters and a good convergence rate.

Signal reconstruction in wireless sensor networks based on a cubature Kalman particle filter

For solving the issues of the signal reconstruction of nonlinear non-Gaussian signals in wireless sensor networks （WSNs）, a new signal reconstruction algorithm based on a cubature Kalman particle filter （CKPF） is proposed in this paper. We model the reconstruction signal first and then use the CKPF to estimate the signal. The CKPF uses a cubature Kalman filter （CKF） to generate the importance proposal distribution of the particle filter and integrates the latest observation, which can approximate the true posterior distribution better. It can improve the estimation accuracy. CKPF uses fewer cubature points than the unscented Kalman particle filter （UKPF） and has less computational overheads. Meanwhile, CKPF uses the square root of the error covariance for iterating and is more stable and accurate than the UKPF counterpart. Simulation results show that the algorithm can reconstruct the observed signals quickly and effectively, at the same time consuming less computational time and with more accuracy than the method based on UKPF.

Maneuvering target tracking algorithm based on cubature Kalman filter with observation iterated update

Reasonable selection and optimization of a filter used in model estimation for a multiple model structure is the key to improve tracking accuracy of maneuvering target.Combining with the cubature Kalman filter with iterated observation update and the interacting multiple model method,a novel interacting multiple model algorithm based on the cubature Kalman filter with observation iterated update is proposed.Firstly,aiming to the structural features of cubature Kalman filter,the cubature Kalman filter with observation iterated update is constructed by the mechanism of iterated observation update.Secondly,the improved cubature Kalman filter is used as the model filter of interacting multiple model,and the stability and reliability of model identification and state estimation are effectively promoted by the optimization of model filtering step.In the simulations,compared with classic improved interacting multiple model algorithms,the theoretical analysis and experimental results show the feasibility and validity of the proposed algorithm.

基于Cubature粒子滤波检测前跟踪方法

为了改善低信噪比条件下弱目标的检测实时性与跟踪精度,提出了基于Cubature粒子滤波检测前跟踪方法。该方法直接使用原始传感器数据,使用Cubature卡尔曼滤波构造粒子滤波的重要性建议函数来估计目标的运动状态,充分利用了系统的状态模型以及加入了新的观测信息。仿真实验表明,该算法对低信噪比条件下雷达弱目标具有良好的实时检测和跟踪性能。

Cubature Kalman Filter Under Minimum Error Entropy With Fiducial Points for INS/GPS Integration

Traditional cubature Kalman filter(CKF)is a preferable tool for the inertial navigation system(INS)/global positioning system(GPS)integration under Gaussian noises.The CKF,however,may provide a significantly biased estimate when the INS/GPS system suffers from complex non-Gaussian disturbances.To address this issue,a robust nonlinear Kalman filter referred to as cubature Kalman filter under minimum error entropy with fiducial points(MEEF-CKF)is proposed.The MEEF-CKF behaves a strong robustness against complex nonGaussian noises by operating several major steps,i.e.,regression model construction,robust state estimation and free parameters optimization.More concretely,a regression model is constructed with the consideration of residual error caused by linearizing a nonlinear function at the first step.The MEEF-CKF is then developed by solving an optimization problem based on minimum error entropy with fiducial points(MEEF)under the framework of the regression model.In the MEEF-CKF,a novel optimization approach is provided for the purpose of determining free parameters adaptively.In addition,the computational complexity and convergence analyses of the MEEF-CKF are conducted for demonstrating the calculational burden and convergence characteristic.The enhanced robustness of the MEEF-CKF is demonstrated by Monte Carlo simulations on the application of a target tracking with INS/GPS integration under complex nonGaussian noises.

Cubature Kalman filters: Derivation and extension

This paper focuses on the cubature Kalman filters （CKFs） for the nonlinear dynamic systems with additive process and measurement noise. As is well known, the heart of the CKF is the third-degree spherical–radial cubature rule which makes it possible to compute the integrals encountered in nonlinear filtering problems. However, the rule not only requires computing the integration over an n-dimensional spherical region, but also combines the spherical cubature rule with the radial rule, thereby making it difficult to construct higher-degree CKFs. Moreover, the cubature formula used to construct the CKF has some drawbacks in computation. To address these issues, we present a more general class of the CKFs, which completely abandons the spherical–radial cubature rule. It can be shown that the conventional CKF is a special case of the proposed algorithm. The paper also includes a fifth-degree extension of the CKF. Two target tracking problems are used to verify the proposed algorithm. The results of both experiments demonstrate that the higher-degree CKF outperforms the conventional nonlinear filters in terms of accuracy.

Multi-sensor Hybrid Fusion Algorithm Based on Adaptive Square-root Cubature Kalman Filter

In the normal operation condition, a conventional square-root cubature Kalman filter （SRCKF） gives sufficiently good estimation results. However, if the measurements are not reliable, the SRCKF may give inaccurate results and diverges by time. This study introduces an adaptive SRCKF algorithm with the filter gain correction for the case of measurement malfunctions. By proposing a switching criterion, an optimal filter is selected from the adaptive and conventional SRCKF according to the measurement quality. A subsystem soft fault detection algorithm is built with the filter residual. Utilizing a clear subsystem fault coefficient, the faulty subsystem is isolated as a result of the system reconstruction. In order to improve the performance of the multi-sensor system, a hybrid fusion algorithm is presented based on the adaptive SRCKF. The state and error covariance matrix are also predicted by the priori fusion estimates, and are updated by the predicted and estimated information of subsystems. The proposed algorithms were applied to the vessel dynamic positioning system simulation. They were compared with normal SRCKF and local estimation weighted fusion algorithm. The simulation results show that the presented adaptive SRCKF improves the robustness of subsystem filtering, and the hybrid fusion algorithm has the better performance. The simulation verifies the effectiveness of the proposed algorithms.

Cubature Kalman Filter Based Millimeter Wave Beam Tracking for OTFS Systems

In this paper,a Millimeter wave(mmWave)beam tracking problem is studied in orthogonal time frequency space(OTFS)systems.Considering the nonlinearity of beamforming and the constraints of existing Kalman-filtering based beam tracking schemes,we propose a novel Cubature Kalman Filter(CKF)framework tracking the channel state information(CSI)to manage the challenge of highspeed channel variation in single-user moving scene for OTFS systems.Aiming for low complexity for mobile settings,this paper trains only one beam pair to track a path to maintain the reliable communication link in the analog beamforming architecture.Simulation results show that our proposed method has better tracking performance to improve the accuracy of the estimated beam angle compared with prior work.

Cubature Formula and Interpolation on the Cubic Domain

Several cubature formulas on the cubic domains are derived using the discrete Fourier analysis associated with lattice tiling, as developed in [10]. The main results consist of a new derivation of the Gaussian type cubature for the product Chebyshev weight functions and associated interpolation polynomials on [-1,1]^2, as well as new results on [-1, 1]^3. In particular, compact formulas for the fundamental interpolation polynomials are derived, based on n3/4 ＋ O（n^2） nodes of a cubature formula on [-1,1]^3.

Estimation Performance for the Cubature Particle Filter under Nonlinear/Non-Gaussian Environments

This paper evaluates the state estimation performance for processing nonlinear/non-Gaussian systems using the cubature particle lter(CPF),which is an estimation algorithm that combines the cubature Kalman lter(CKF)and the particle lter(PF).The CPF is essentially a realization of PF where the third-degree cubature rule based on numerical integration method is adopted to approximate the proposal distribution.It is benecial where the CKF is used to generate the importance density function in the PF framework for effectively resolving the nonlinear/non-Gaussian problems.Based on the spherical-radial transformation to generate an even number of equally weighted cubature points,the CKF uses cubature points with the same weights through the spherical-radial integration rule and employs an analytical probability density function(pdf)to capture the mean and covariance of the posterior distribution using the total probability theorem and subsequently uses the measurement to update with Bayes’rule.It is capable of acquiring a maximum a posteriori probability estimate of the nonlinear system,and thus the importance density function can be used to approximate the true posterior density distribution.In Bayesian ltering,the nonlinear lter performs well when all conditional densities are assumed Gaussian.When applied to the nonlinear/non-Gaussian distribution systems,the CPF algorithm can remarkably improve the estimation accuracy as compared to the other particle lterbased approaches,such as the extended particle lter(EPF),and unscented particle lter(UPF),and also the Kalman lter(KF)-type approaches,such as the extended Kalman lter(EKF),unscented Kalman lter(UKF)and CKF.Two illustrative examples are presented showing that the CPF achieves better performance as compared to the other approaches.