Design of minimax robust filtering for an integrated GPS/INS system

The problem of navigation systems with uncertain noise is considered. A minimax robust filtering which can minimize the worst performance under noise uncertainties using the game theory is proposed. This new filter is applied to an integrated GPS/INS navigation system. A high dynamics aircraft trajectory is designed to test the new filter. The results show that minimax robust filtering performs better than standard Kalman filtering when noise parameters of an inertial measurement unit change their statistical properties. Received: 21 October 1997 / Accepted: 26 May 1999     

Adaptive Kalman Filtering for INS/GPS

After reviewing the two main approaches of adaptive Kalman filtering, namely, innovation-based adaptive estimation (IAE) and multiple-model-based adaptive estimation (MMAE), the detailed development of an innovation-based adaptive Kalman filter for an integrated inertial navigation system/global positioning system (INS/GPS) is given. The developed adaptive Kalman filter is based on the maximum likelihood criterion for the proper choice of the filter weight and hence the filter gain factors. Results from two kinematic field tests in which the INS/GPS was compared to highly precise reference data are presented. Results show that the adaptive Kalman filter outperforms the conventional Kalman filter by tuning either the system noise variance–covariance (V–C) matrix `Q' or the update measurement noise V–C matrix `R' or both of them. Received: 14 September 1998 / Accepted: 21 December 1998     

有色噪声作用下的抗差卡尔曼滤波

根据用GPS载波相位三差观测量进行动态定位或精密导航的需求,推导了动态噪声、观测噪声为有色噪声的抗差卡尔曼滤波公式。白噪声的抗差卡尔曼滤波是有色噪声的抗差卡尔曼滤波的特例,有色噪声的抗差卡尔曼滤波为白噪声的抗差卡尔曼滤波的推广。     

抗差卡尔曼滤波在GPS动态定位中的应用

基于Kalman滤波的GPS动态定位中，动态观测量及其相应的动态模型可能存在异常，若数据处理不考虑对这些异常的特别处理，则模糊度的估值及其所提供的动态信息将极不可靠，按抗差估计原理，文中构造了状态向量和观测值对模糊度的影响函数，并由此建立了动态GPS定位的抗差Kalman滤波解法，实际计算验证了该方法的实用性和可靠性。     

多源传感器动、静态滤波融合导航

首先给出联邦滤波各局部输出量之间的相关协方差矩阵，进而给出了基于各传感器独立观测信息的动、静态滤波解法，这种解法避免了重复使用载体状态方程信息的问题，保证了多传感器数据融合的最优性，而且很容易扩展到抗差滤波和自适应滤波融合。     

Biases and accuracy of, and an alternative to, discrete nonlinear filters

The biases and accuracy of the extended Kalman filter (EKF) and a second-order nonlinear filter (SONF) are discussed from the point of view of a frequentist; these are often derived by applying the relevant conditional quantities to the linear Kalman algorithm under the Bayesian framework. The EKF and the SONF are biased, although the SONF has been derived in the hope of improving first-order filters. Unfortunately the biases of the SONF may be magnified further, because the second-order terms of the relevant Bayesian conditional quantities have never been properly used to derive the SONF from the frequentist point of view. The variance–covariance matrix of the SONF given in the literature is proven to be incorrect up to the second-order approximation, and the correct one is derived. Finally, also from the point of view of a frequentist, an alternative, almost unbiased SONF is proposed, if the randomness of partials is neglected. Received: 12 July 1997 / Accepted: 5 October 1998     

后向平滑与抗差估计融合的SRCKF滤波

将后向平滑平方根容积卡尔曼滤波用于GPS动态单点定位数据处理,并探讨了粗差对后向平滑滤波的影响。借鉴经典卡尔曼滤波抗差估计思想,给出平方根容积卡尔曼滤波的抗差算法以抵抗量测粗差,而当判断不含粗差时使用后向平滑算法,在有效提高滤波精度的同时避免了抗差滤波对每个历元都需进行迭代运算。实测GPS动态数据验证了算法的有效性。     

基于抗差卡尔曼滤波的GM(1,1)模型在变形预计中的应用

针对传统灰色模型建模过程中易受观测数据随机噪声干扰的影响,利用抗差卡尔曼滤波理论能够有效地估计含有噪声的观测值的优点,构建了基于抗差卡尔曼滤波的GM(1,1)模型。结合实例,验证了该模型在一定程度上可以提高变形监测预测精度,更好地反映观测对象的变形趋势。     

基于EKF和自适应抗差滤波的星载GPS定轨方法

针对在星载GPS卫星定轨中由于卫星动力学模型误差和不可避免的观测异常严重影响定轨精度的问题,通过采用适当的自适应控制因子和应用抗差估计原理,构造自适应抗差扩展卡尔曼滤波(RAEKF)来实现星载GPS卫星定轨。实测计算表明,自适应抗差扩展卡尔曼滤波对观测误差和状态扰动有一定的抵制能力,与一般扩展卡尔曼滤波相比提高了精度,证明其理论的可行性。     

精密单点定位不等精度观测值的RKF研究

推导了精密单点定位含有粗差观测数据的M-LS滤波原理,对等价权阵采用三段降权函数实现抗差。从新息和残差的协方差关系出发,利用对粗差敏感的残差标准差作为抗差因子。通过迭代减弱卫星间载波残差及其抗差因子的相关性。针对载波和伪距观测值不等观测精度和不相关性,采用双抗差因子实现静态抗差卡尔曼滤波（robust Kalman filtering,RKF）。采用标准卡尔曼滤波、基于新息RKF、基于残差的增益矩阵双抗差因子RKF、基于残差的等价权阵双抗差因子RKF等4种模型,分别对一组实测数据解算分析。结果表明,基于新息RKF对精度较高的载波粗差不敏感;基于残差的增益矩阵RKF对载波较小的粗差抗差效果较差,且发生粗差历元时刻的状态参数与真值偏差较大;而基于残差构造的等价权阵双抗差因子RKF可以非常精确和高效地实现抗差,单个卫星粗差对测站位置参数影响小于1 mm。     

抗差估计在多站多星卡尔曼滤波精密定轨中的应用

针对在多站多星卡尔曼滤波定轨中,粗差探测判断标准选择不合适时会影响卡尔曼滤波解精度的问题,采用放宽粗差探测阈值并在滤波阶段采用抗差估计的方法来控制滤波的精度。利用IGS站和BD站的观测数据,分别计算了GPS、BD卫星单天弧度的定轨结果,并与IGS精密星历、武大精密星历作比较。结果表明,利用抗差卡尔曼滤波方法轨道精度得到了提高。     

Robust adaptive filter for shipborne kinematic positioning and velocity determination during the Baltic Sea experiment

Information on trajectory and attitude is essential for analyzing gravimetric data collected on kinematic platforms. Usually, a Kalman filter is used to obtain high-accuracy positional and velocity information. However, this can be affected by measurement outliers and by state disturbances that occur frequently under a fast-changing environment. To overcome these problems, a robust adaptive Kalman filtering algorithm is applied for state estimates, which introduces an equivalent weight to resist measurement outliers and an optimal adaptive factor to balance the contributions of the kinematic model information and the measurements. In addition to the conventional robust estimator, an improved Current Statistical (CS) model is proposed. The improved CS model adopts a variance adaptive learning algorithm, and it can perform self-adaptation of acceleration variance with the innovation information; thus, it can overcome the shortcoming of lower tracking accuracy and avoid setting the maximum acceleration. Following a gravimetry campaign on the Baltic Sea, it is shown in theory and in practice that the robust adaptive Kalman filter is not only simple in its calculation but also more reliable in controlling the colored observation noise and kinematic state disturbance compared with the classical Kalman filter. The improved CS model performs best, especially when analyzing the positioning errors at the turns due to the target maneuvering. Compared to the CS model, the RMS values of the positional estimates derived from the improved CS model decrease by almost 30% in the horizontal direction, and no significant improvement in the vertical direction is found.     

GPS navigation processing using the quaternion-based divided difference filter

Divided difference filter (DDF) with quaternion-based dynamic process modeling is applied to global positioning system (GPS) navigation. Using techniques similar to those of the unscented Kalman filter (UKF), the DDF uses divided difference approximations of derivatives based on Stirling’s interpolation formula which results in a similar mean but different posterior covariance compared to the extended Kalman filter (EKF) solutions. The second-order divided difference is obtained from the mean and covariance in second-order polynomial approximation. The quaternion-based dynamic model is adopted for avoiding the singularity problems encountered in the Euler angle method and enhancing the computational efficiency. The proposed method is applied to GPS navigation to increase the navigation estimation accuracy at high-dynamic regions while preserving (without sacrificing) the precision at low-dynamic regions. For the illustrated example, the second-order DDF can deliver about 41–82% accuracy improvement as compared to the EKF. Some properties and performance are assessed and compared to those of the EKF and UKF approaches.     

GPS/INS组合导航系统抗差滤波器设计

常规Ｋａｌｍａｎ滤波器已经广泛用于ＧＰＳ／ＩＮＳ组合导航系统,其中假设系统动态模型和噪声统计特性是精确已知的。事实上,这种假设是不符合实际情况的。在组合导航系统中,惯性测量器件的质量不稳定,ＧＰＳ测量误差受外界环境的影响,因而对组合导航系统进行抗差设计是十分必要的。本文利用对策论设计了能使不确定噪声下性能最好的极小极大抗差滤波器,并将其应用到ＧＰＳ／ＩＮＳ组合导航系统中。考虑一个ＩＯ状态的ＧＰＳ／     

Shaping filters and the upward continuation problem

An attempt is made to bridge the gap between closed-form harmonic upward continuation (HUC) of analytic covariance functions of the disturbing potential of the anomalous local gravity field and the numerical shaping filter construction when the local gravity vector is modelled in the framework of Kalman filtering. Some fundamental concepts of the local gravity field, interpreted as a stochastic process that is stationary in the plane and harmonic in the upper half space, are reviewed. The shaping-filter modelling technique for the local gravity vector is introduced. To determine the relation between the disturbing potential covariance function and the gravity vector covariance matrix, the role of the so-called admissible pair is established. It is shown that rescaling an admissible pair leads to an analogue rescaling of the shaping filter matrices derived hereof; no cumbersome numerical recalculations are necessary. The class of covariance functions whose corresponding shaping filters possess a closed-form HUC are identified as models whose HUC can be interpreted as a rescaling. Received: 17 December 1997 / Accepted: 7 September 1998     

基于抗差EKF的GNSS/INS紧组合算法研究

提出了GNSS/INS紧组合导航的抗差EKF算法,采用21状态GNSS/INS紧组合状态方程,根据多余观测分量及预测残差统计构造抗差等价增益矩阵,建立抗差EKF算法,通过迭代给出GNSS/INS组合导航的抗差解,并开发GNSS/INS紧组合导航模拟平台,通过对观测值加入单粗差、多粗差及缓慢增长三类误差,测试本文算法对不同粗差的抑制能力。分析表明,抗差EKF可以将三类粗差抑制在相应观测值的残差中,达到削弱其对状态参数估计的影响。本文算例证明,抗差EKF算法可将导航解的误差精度从dm级提高为cm级甚至mm级,导航精度及可靠性得到明显提高。     

基于修正嵌入式容积卡尔曼滤波的相位展开算法

针对干涉图展开问题,将Levenberg-Marquardt方法修正后的嵌入式容积卡尔曼滤波器应用于缠绕相位图像的展开中,结合基于修正矩阵束模型的局部相位梯度估计算法以及量化跟踪策略,提出一种基于修正嵌入式容积卡尔曼滤波的相位展开算法。该算法不仅可以精确和快速地展开缠绕像元,还可以在展开缠绕像元的同时抑制相位噪声,降低前置预滤波器的难度与复杂度,甚至可以在处理受噪声污染不严重的干涉图时免去前置预滤波处理步骤。试验结果表明本文算法具有较高的效率和良好的稳键性。     

动态系统的抗差Kaliman滤波

离散历元的动态观测量及其相应的动态模型可能存在异常，若数据处理模型不考虑对这些异常的特别处理，则动态模型参数估值及其所提供的动态信息将极不可靠。基于贝叶斯统计和抗差估计原理，我们构造了一种抗差滤波算法。该算法考虑观测分布和参数验前分布均为污染分布。并利用一个实测网验算该算法和模型的可靠性。