双层无迹卡尔曼滤波

针对无迹卡尔曼滤波（Unscented Kalman fllter,UKF）在强非线性系统中估计效果差的问题,提出了双层无迹卡尔曼滤波（Double layer unscented Kalman filter,DLUKF）算法,该算法用带权值的采样点表征先验分布,而后用内层UKF算法对每个采样点进行更新,最后引入外层UKF算法的更新机制得到估计值和估计协方差.仿真结果表明,相比于传统算法,所提的DLUKF算法可以在较低计算负载下获得较高滤波估计精度.     

改进型平方根无迹卡尔曼滤波及其在无轴承永磁同步电机无速度传感器运行中的应用

平方根无迹卡尔曼滤波(SRUKF)解决了标准无迹卡尔曼滤波(UKF)中由于误差协方差阵负定而引起的滤波发散问题, 保证了算法的数值稳定性, 但仍存在对模型参数变化的鲁棒性差、收敛速度慢及对突变状态的跟踪能力低等缺陷. 因此, 本文提出一种改进SRUKF滤波, 通过引入时变渐消因子和弱化因子, 实时修正滤波增益矩阵和误差协方差平方根矩阵, 实现残差序列正交, 确保SRUKF滤波保持对目标实际状态的准确跟踪. 将该算法在无轴承永磁同步电机无速度传感器矢量控制系统中进行仿真研究. 结果表明: 改进SRUKF非线性近似精度、数值稳定性及滤波精度更高, 在系统状态突变或负载扰动时, 鲁棒性更强, 能够有效实现转速及转子角度的准确估计, 确保转子稳定悬浮运行.     

基于四元数衍生无迹卡尔曼滤波的二段式多旋翼无人机姿态估计算法

对传统多旋翼无人机姿态估计算法难以兼顾高精度、强实时性以及抗干扰能力差的问题,首先基于一种计算量较小的衍生无迹卡尔曼滤波算法,在量测更新中,将加速度数据和磁力计数据分为两个阶段进行姿态四元数校正处理,然后从旋转四元数的本质出发,推测出四元数各元素分别包含着不同的姿态角信息,最后将校正四元数分别乘上为降低校正过程中的相互干扰所设计的系数,提出一种基于四元数衍生无迹卡尔曼滤波的二段式多旋翼无人机姿态估计算法.通过使用PIXHAWK飞控数据,与传统姿态估计算法进行仿真实验对比,实验表明,本文提出算法与传统使用扩展卡尔曼滤波(EKF)或无迹卡尔曼滤波(UKF)的姿态估计算法相比,在实时性、解算精度和抗干扰能力方面有较大提升.     

迭代平方根UKF

针对无迹卡尔曼滤波器（UKF）测量更新方法的不足,提出了一种对UKF 进行迭代测量更新的 方法,用于提高非线性系统状态估计的近似精度．利用平方根UKF 算法确保了迭代UKF 的数值稳定性．理论 分析与实验结果表明,迭代平方根UKF 算法不仅具有无需计算雅可比矩阵的优点,而且具有较高的非线性近 似精度、较强的数值稳定性和较高的运算效率;在相同数量级运算时间的条件下,其估计性能明显优于扩展 卡尔曼滤波器（extended Kalman filter,EKF）、UKF 和迭代UKF 等非线性滤波器．     

用四元数状态切换无迹卡尔曼滤波器估计的飞行器姿态

在较大初始姿态误差角下, 针对捷联惯导/CCD星敏感器(strap-intertial navigation system/CCD star sensor, SINS/CCD)姿态估计系统扩展卡尔曼滤波(extended Kalman filter, EKF)算法精度下降的问题, 提出了基于四元数的状态切换无迹卡尔曼滤波算法. 通过状态实时切换降低了全维无迹卡尔曼滤波(unscented Kalman filter, UKF)的维数, 减小了计算复杂度, 提高了系统的实时性. 文中采用基于特征向量求解的代价函数法计算四元数均值避免了UKF算法中四元数规范化的限制; 利用乘性误差四元数表示姿态更新点与估计点之间的距离, 解决了四元数协方差阵奇异性问题. 仿真实验结果表明: 与EKF相比, 该算法在精度上有较大提高; 与全维UKF算法和修正罗德里格斯参数UKF算法相比, 该算法精度相当但估计时间均有不同程度的减少.     

UKF与EKF在卫星姿态估计应用中的比较

针对卫星的姿态和角速度估计问题,分别给出基于Unscented卡尔曼滤波(UKF)与推广卡尔曼滤波(EKF)的估计算法,并做了相应比较.为了避免欧拉角带来的奇异问题,UKF选用Rodrigues参数而EKF选用四元数参数法来描述姿态误差.考虑卫星的非线性模型,UKF采用Unscented变换而EKF采用线性化方法对姿态误差进行估计.利用陀螺和磁强计的测量信息,KF和EKF都可得到三轴稳定卫星的姿态估计值,但UKF的收敛速度高于EKF.数值仿真结果表明,当初始姿态存在大偏差时,所给出的UKF的滤波算法性能明显优于EKF.     

Variants of extended Kalman filtering approaches for Bayesian tracking

We provide a tutorial for a number of variants of the extended Kalman filter (EKF). In these methods, so called, sigma points are employed to tackle the nonlinearity of problems. The sigma points exactly represent the mean and the variance of the state distribution function in a dynamic state equation. The initially developed EKF variant, that is, unscented Kalman filter (UKF) (also called sigma point Kalman filter) shows enhanced performance compared with that of conventional EKF in the literature. Another variant, which is not well known, is central difference Kalman filter (CDKF) whose way to approximate the nonlinearity is based on the Sterling's polynomial interpolation formula instead of the Taylor series. Endeavor to reduce the computational load resulted in the development of square root versions of both UKF and CDKF, that is, square root unscented Kalman filter and square root central difference Kalman filter (SR‐CDKF). These SR‐versions are supposed to be numerically more stable than their original versions because the state covariance is guaranteed to be positive definite by avoiding the step of matrix decomposition. In this paper, we provide the step‐by‐step algorithms of above‐mentioned EKF variants with their pros and cons. We apply these filtering methods to a number of problems in various disciplines for performance assessment in terms of both mean squared error (MSE) and processing speed. Furthermore, we show how to optimize the filters in terms of MSE performance depending on diverse scenarios. According to simulation results, CDKF and SR‐CDKF show the best MSE performance in most scenarios; particularly, SR‐CDKF shows faster processing speed than that of CDKF. Therefore, we justify that SR‐CDKF is the most efficient and the best approach among the Kalman variants including the EKF for various nonlinear problems. The motivation of this paper targets at the contribution to the disseminative usage of the Kalman variants approaches, particularly, SR‐CDKF taking advantage of its estimating performance and high processing speed. Copyright © 2016 John Wiley & Sons, Ltd.     

一种高阶无迹卡尔曼滤波方法

现有的研究中,高阶无迹变换（Unscented transform,UT）还不存在具体的解析解,因此,无法利用高阶无迹变换获得具备更高精度的高阶无迹卡尔曼滤波器（Unscented Kalman filter,UKF）.为了解决这一问题,本文在五阶容积变换（Cubature transform,CT）的基础上,通过引入一个自由参数κ,得到高阶无迹变换的解析解,从而获得了高阶无迹卡尔曼滤波器（Unscented Kalman filter,UKF）.同时验证了现有的五阶容积变换和五阶无迹变换分别是本文所提出的高阶无迹变换在κ=2和κ=6-n时的两个特例.进而分析和讨论了高阶无迹卡尔曼滤波器在系统不同维数条件下κ值的最优选取,并讨论了其稳定性.纯方位跟踪模型和弹道目标再入模型仿真验证了本文方法的正确性,且与现有方法相比具有更高的精度.     

Hyperbolic‐singular‐value‐decomposition‐based square‐root accurate continuous‐discrete extended‐unscented Kalman filters for estimating continuous‐time stochastic models with discrete measurements

This paper presents novel square‐root accurate continuous‐discrete extended‐unscented Kalman filtering (ACD‐EUKF) algorithms for treating continuous‐time stochastic systems with discrete measurements. The time updates in such methods are fulfilled as those in the extended Kalman filter whereas their measurement updates are copied from the unscented Kalman filter. All this allows accurate predictions of the state mean and covariance to be combined with accurate measurement updates. The main weakness of this technique is the need for the Cholesky decomposition of predicted covariances derived in time‐update steps. Such a factorization is highly sensitive to numerical integration and round‐off errors committed, which may result in losing the covariance's positivity and, hence, failing the Cholesky decomposition. The latter problem is usually solved in the form of square‐root filtering implementations, which propagate not the covariance matrix but its square root instead. Here, we devise square‐root ACD‐EUKF methods grounded in the singular value decomposition (SVD). The SVD rooted in orthogonal transforms is applicable to any ACD‐EUKF with nonnegative weights, whereas the remaining ones, which can enjoy negative weights as well, are treated by means of the hyperbolic SVD based on J‐orthogonal transforms. The filters constructed are presented in a concise algorithmic form, which is convenient for practical use. Their two particular versions grounded in the classical and cubature unscented Kalman filtering parameterizations are examined in severe conditions of tackling a radar tracking problem, where an aircraft executes a coordinated turn. These are also compared to their non‐square‐root predecessor and other methods within the target tracking scenario with ill‐conditioned measurements.     

自适应平方根无迹卡尔曼滤波算法

将高斯过程回归融入平方根无迹卡尔曼滤波(SRUKF)算法,本文提出了一种不确定系统模型协方差自适应调节滤波算法.该算法分为学习和估计两部分:学习阶段用高斯过程对训练数据进行学习,得到系统回归模型及噪声协方差;估计阶段由回归模型代替状态方程和观测方程,相应的噪声协方差实时自适应调整.该方法克服了传统方法容易受系统动态模型不确定性和噪声协方差不准确限制的问题,仿真结果验证了算法的有效性.     

带有色量测噪声的非线性系统 Unscented 卡尔曼滤波器

传统Unscented卡尔曼滤波器(Unscented Kalman filter, UKF)要求噪声必须为高斯白噪声, 无法解 决带有色噪声的非线性系统滤波问题. 为此, 本文提出了一种带有色量测噪声的UKF滤 波新算法. 首先,基于量测信息增广和最小方差估计, 推导出一类带有色量测噪声的非 线性离散系统状态的最优滤波框架, 接着采用Unscented变换(Unscented transformation, UT)来计算最优框架中的 非线性状态后验均值和协方差, 进而得到有色量测噪声下UKF滤波递推公式. 所设 计的UKF新方法能有效地解决传统UKF在量测噪声有色情况下非线性滤波失效的问题, 数 值仿真实例验证了其可行性和有效性.     

New minimum sigma set for unscented filtering

In this work, a new minimum set of sigma points for unscented filtering is proposed along with its unscented Kalman filter in both square‐root and nonsquare‐root forms. Comparative with the other reduced sigma sets of the literature, the new sigma set is, in some cases, better defined or, in another case, a generalization. In numerical examples, the unscented transform of the new sigma set is compared with the unscented transforms of the other reduced sigma sets of the literature. In addition, the performance of the new unscented Kalman filter is studied in an aircraft target tracking scenario. Copyright © 2014 John Wiley & Sons, Ltd.     

有限时间一致无迹Kalman滤波器

本文研究多个传感器测量非线性系统时的分布式无迹Kalman滤波器(Unscented Kalman filter, UKF)的设计问题.借助离散多智能体系统有限时间平均一致算法的思想, 针对无向通信和有向通信网络分别设计了两种不同的滤波算法.对于无向连通的通信拓扑, 利用节点存储的一致性算法的迭代值构造差向量, 由该差向量构成的Hankel矩阵的核来得到分布式无迹Kalman滤波器, 并通过利用误差协方差矩阵的逆来构造Lyapunov函数, 基于随机稳定性引理证明了该有限时间一致无迹Kalman滤波器的稳定性.对于有向强连通的通信拓扑, 结合比率一致和Hankal矩阵的核来设计分布式无迹Kalman滤波器, 该滤波器的稳定性与无向通信拓扑的滤波器相同.最后, 通过仿真例子来验证所提滤波器的跟踪效果.     

Unscented Kalman filter for continuous‐time nonlinear fractional‐order systems with process and measurement noises

This study proposes the design of unscented Kalman filter for a continuous‐time nonlinear fractional‐order system involving the process noise and the measurement noise. The nonlinear fractional‐order system is discretized to get the difference equation. According to the unscented transformation, the design method of unscented Kalman filter for a continuous‐time nonlinear fractional‐order system is provided. Compared with the extended Kalman filter, the proposed method can obtain a more accurate estimation effect. For fractional‐order systems containing non‐differentiable nonlinear functions, the method proposed in this paper is still effective. The unknown parameters are also discussed by the augmented vector method to achieve the state estimation and parameter identification. Finally, two examples are offered to verify the effectiveness of the proposed unscented Kalman filter for nonlinear fractional‐order systems.     

基于Gauss-Hermite逼近的非线性加权观测融合无迹Kalman滤波器

对非线性多传感器系统,基于Gauss-Hermite逼近方法和加权最小二乘法,提出了一种具有普适性的非线性加权观测融合算法.该算法可将一个高维观测压缩为一个低维观测.在此基础上,结合无迹Kalman滤波器（Unscented Kalman filter,UKF）,提出了非线性加权观测融合无迹Kalman滤波器（WMF（Weighted measurement fusion）-UKF）.与集中式融合UKF（CMF（Centralized measurement fusion）-UKF）相比,该算法计算负担小且具有逼近的估计精度.特别是在传感器数量较大时,该算法在计算量上的优势更加明显.仿真例子验证了算法的有效性.     

基于无迹卡尔曼滤波器的改进SLAM问题求解方法

目前在即时定位与地图构建（Simultaneous Localization And Mapping,SLAM）的研究中已经使用局部取样策略来降低无迹卡尔曼滤波（Unscented Kalman Filter,UKF）的计算复杂度至状态向量维数的平方等级.但是在大规模的SLAM中平方复杂度仍然难以满足实时性需求.为了解决这个问题,提出了一种收缩无迹卡尔曼滤波器（Shrink Unscented Kalman Filter,S-UKF）,并应用于SLAM问题中.首先证明了解耦非线性系统中的部分取样策略和全取样策略的等价性.然后提出了一个通过重构公式中相关项的收缩方式来降低计算复杂度.最后,仿真实验的结果和基于真实环境数据集的实验结果证明了该方法的有效性.     

基于无迹卡尔曼滤波的机器人手眼标定

提出一种基于无迹卡尔曼滤波(UKF)的机器人在线手眼标定算法来求解齐次变换矩阵方程AX =XB.建立手眼标定的隐式马尔可夫模型(HMM),并对它进行无迹卡尔曼滤波,从而对标定参数的状态进行递归贝叶斯估计和实时可视化处理.蒙特卡洛仿真结果表明,在小高斯噪声、较大高斯噪声以及非等方向性高斯噪声模型下,本文算法估计结果的精确...     

Maximum correntropy unscented filter

The unscented transformation (UT) is an efficient method to solve the state estimation problem for a non-linear dynamic system, utilising a derivative-free higher-order approximation by approximating a Gaussian distribution rather than approximating a non-linear function. Applying the UT to a Kalman filter type estimator leads to the well-known unscented Kalman filter (UKF). Although the UKF works very well in Gaussian noises, its performance may deteriorate significantly when the noises are non-Gaussian, especially when the system is disturbed by some heavy-tailed impulsive noises. To improve the robustness of the UKF against impulsive noises, a new filter for non-linear systems is proposed in this work, namely the maximum correntropy unscented filter (MCUF). In MCUF, the UT is applied to obtain the prior estimates of the state and covariance matrix, and a robust statistical linearisation regression based on the maximum correntropy criterion is then used to obtain the posterior estimates of the state and covariance matrix. The satisfying performance of the new algorithm is confirmed by two illustrative examples.     

Square‐Root Cubature‐Quadrature Kalman Filter

In this paper, an on‐going work introducing square‐root extension of cubature‐quadrature based Kalman filter is reported. The proposed method is named square‐root cubature‐quadrature Kalman filter (SR‐CQKF). Unlike ordinary cubature‐quadrature Kalman filter (CQKF), the proposed method propagates and updates square‐root of the error covariance without performing Cholesky decomposition at each step. Moreover SR‐CQKF ensures positive semi‐definiteness of the state covariance matrix. With the help of two examples we show the superior performance of SR‐CQKF compared to EKF and square root cubature Kalman filter.     

无味变换与无味卡尔曼滤波

综述了非线性估计问题的由来、无味变换（UT,Unscented Transformation）的基本思路与基本算法、各种衍变形式、σ点集的设计原则、无味卡尔曼滤波（UKF,Unscented Kalman Filtering）的基本算法及其各种改进算法、UT的本质、UKF与几种免微分非线性滤波方法的比较、UT与UKF的相关应用、针对几种UKF算法的仿真实例,以及目前在UT与UKF的研究中尚存在的一些问题和对今后研究的展望等;提出了笔者的一些最新研究成果和见解。