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Article(id=1322916813910188785, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322916705915249052, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2025.11.399, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1763568000000, receivedDateStr=2025-11-20, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793337547042, onlineDateStr=2026-10-30, pubDate=1784044800000, pubDateStr=2026-07-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793337547042, onlineIssueDateStr=2026-10-30, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793337547042, creator=13701087609, updateTime=1793337547042, updator=13701087609, issue=Issue{id=1322916705915249052, tenantId=1146029695717560320, journalId=1281212831689347082, year='2026', volume='46', issue='7', pageStart='790', pageEnd='912', issueExtLink='null', onlineDate='null', pubDate='1784044800000', pubDateStr='2026-07-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793337521294, creator='13701087609', updateTime=1793337521294, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=816, endPage=824, ext={EN=ArticleExt(id=1322916814124098290, articleId=1322916813910188785, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=Non-Equal-Weighted Maximum Correntropy Filtering Algorithm for the GNSS/SINS Integrated Navigation System, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

In the maximum correntropy Kalman filter based on equal weighting (MCKF), since the error samples with different qualities are weighted equally, it affects the filtering accuracy of the GNSS/SINS integrated navigation system based on MCKF. To solve this problem, a non-equal-weighted maximum correntropy Kalman filter (NWMCKF) is proposed. Firstly, based on the equal weighting maximum correntropy criterion (MCC), a non-equal-weighted maximum correntropy criterion (NWMCC) is proposed to reduce the adverse effects caused by MCC applying the same weight to all error samples. Then, based on NWMCC, the algorithm equation of NWMCKF is derived, and the fixed-point iterative recursive algorithm flow of NWMCKF is given. Finally, experiments on the GNSS/SINS integrated navigation system based on KF, MCKF and NWMCKF are carried out for verification. The experimental results show that when the measurement noise is Gaussian noise, the filtering performance of NWMCKF is slightly better than that of MCKF; when the measurement noise is non-Gaussian noise, the filtering performance of NWMCKF is significantly better than that of MCKF.

, authors=Siming LIU1, Xueyuan LIN2, Yuxin QIAO1, *, authorsList=Siming LIU, Xueyuan LIN, Yuxin QIAO, authorCompany=null, correspAuthors=Yuxin QIAO, authorNote=null, correspAuthorsNote=null, copyrightStatement=All rights reserved. Unauthorized reproduction is prohibited., copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, fund=null), CN=ArticleExt(id=1322916815642436349, articleId=1322916813910188785, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=GNSS/SINS组合导航系统的非等加权最大相关熵滤波算法, columnId=1281909276545356011, journalTitle=大地测量与地球动力学, columnName=大地测量学, runingTitle=null, highlight=null, articleAbstract=

在基于等加权的最大相关熵卡尔曼滤波器(MCKF)中，由于质量不同的误差样本采用相同的加权，进而对基于MCKF的GNSS/SINS组合导航系统滤波精度产生影响。为解决这一问题，提出一种基于非等加权的最大相关熵卡尔曼滤波器(NWMCKF)。首先基于等加权的最大相关熵准则(MCC)，提出一种非等加权最大相关熵准则(NWMCC)，以降低MCC对所有误差样本施加相同权重所产生的不良影响；然后，以NWMCC为基础，推导了NWMCKF算法方程，并给出NWMCKF的定点迭代递归算法流程；最后，进行基于KF、MCKF和NWMCKF的GNSS/SINS组合导航系统实验验证。结果表明，测量噪声为高斯噪声时，NWMCKF的滤波性能略优于MCKF；测量噪声为非高斯噪声时，NWMCKF的滤波性能明显优于MCKF。

, authors=刘思明1, 林雪原2, 乔玉新1, *, authorsList=刘思明, 林雪原, 乔玉新, authorCompany=null, correspAuthors=乔玉新, authorNote=

刘思明，讲师，主要从事信息处理及导航技术研究，E-mail：

, correspAuthorsNote=
林雪原，博士，教授，主要从事组合导航及信息融合研究，E-mail：
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刘思明，讲师，主要从事信息处理及导航技术研究，E-mail：

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刘思明，讲师，主要从事信息处理及导航技术研究，E-mail：

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tableContent=null), ArticleFig(id=1322916820595909415, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916813910188785, language=CN, label=图5, caption=测量噪声为非高斯噪声时基于MCKF和NWMCKF的导航参数绝对误差变化曲线, figureFileSmall=pk23Z6v4Bl4YRWJWLPgaoA==, figureFileBig=HBxQ5AsA9/qBwf0RYE2KsA==, tableContent=null), ArticleFig(id=1322916822227493672, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916813910188785, language=EN, label=Tab. 1, caption=

Filter accuracy comparison between NWMCKF and MCKF when the measured noise is Gaussian noise

, figureFileSmall=null, figureFileBig=null, tableContent=
滤波器 经度误差/m 纬度误差/m 高度误差/m E向速度误差/m · s-1 N向速度误差/m · s-1 U向速度误差/m · s-1 横滚角误差/(°) 俯仰角误差/(°) 航向角误差/(°)
MCKF 1.328 1.317 1.413 0.115 0.114 0.080 1 0.044 6 0.061 7 0.443
NWMCKF 1.309 1.297 1.410 0.114 0.113 0.080 0 0.044 4 0.061 6 0.443
), ArticleFig(id=1322916822659506986, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916813910188785, language=CN, label=表1, caption=

测量噪声为高斯噪声时NWMCKF和MCKF滤波精度对比

, figureFileSmall=null, figureFileBig=null, tableContent=
滤波器 经度误差/m 纬度误差/m 高度误差/m E向速度误差/m · s-1 N向速度误差/m · s-1 U向速度误差/m · s-1 横滚角误差/(°) 俯仰角误差/(°) 航向角误差/(°)
MCKF 1.328 1.317 1.413 0.115 0.114 0.080 1 0.044 6 0.061 7 0.443
NWMCKF 1.309 1.297 1.410 0.114 0.113 0.080 0 0.044 4 0.061 6 0.443
), ArticleFig(id=1322916822747587371, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916813910188785, language=EN, label=Tab. 2, caption=

Comparison of filter accuracy between NWMCKF and MCKF when the measured noise is color noise

, figureFileSmall=null, figureFileBig=null, tableContent=
滤波器 经度误差/m 纬度误差/m 高度误差/m E向速度误差/m · s-1 N向速度误差/m · s-1 U向速度误差/m · s-1 横滚角误差/(°) 俯仰角误差/(°) 航向角误差/(°)
KF 2.411 2.531 2.278 0.184 0.169 0.096 2 0.078 2 0.900 0 0.502
MCKF 1.968 2.013 1.836 0.142 0.132 0.081 9 0.061 1 0.072 2 0.433
NWMCKF 1.744 1.725 1.627 0.120 0.113 0.068 7 0.051 3 0.064 0 0.343
), ArticleFig(id=1322916823049577261, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916813910188785, language=CN, label=表2, caption=

测量噪声为非高斯噪声时NWMCKF和MCKF滤波精度对比

, figureFileSmall=null, figureFileBig=null, tableContent=
滤波器 经度误差/m 纬度误差/m 高度误差/m E向速度误差/m · s-1 N向速度误差/m · s-1 U向速度误差/m · s-1 横滚角误差/(°) 俯仰角误差/(°) 航向角误差/(°)
KF 2.411 2.531 2.278 0.184 0.169 0.096 2 0.078 2 0.900 0 0.502
MCKF 1.968 2.013 1.836 0.142 0.132 0.081 9 0.061 1 0.072 2 0.433
NWMCKF 1.744 1.725 1.627 0.120 0.113 0.068 7 0.051 3 0.064 0 0.343
), ArticleFig(id=1322916823519339310, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916813910188785, language=EN, label=Tab. 3, caption=

Comparison of filtering accuracy of NWMCKF algorithms with different a (ε=10-9)

, figureFileSmall=null, figureFileBig=null, tableContent=
a值 经度误差/m 纬度误差/m 高度误差/m E向速度误差/m · s-1 N向速度误差/m · s-1 U向速度误差/m · s-1 横滚角误差/(°) 俯仰角误差/(°) 航向角误差/(°)
a=2 2.191 2.351 2.248 0.167 0.164 0.088 4 0.093 0 0.115 0.333
a=4 2.199 2.256 2.163 0.177 0.172 0.089 2 0.094 6 0.116 0.313
a=6 2.227 2.259 2.159 0.180 0.174 0.089 6 0.094 9 0.116 0.311
a=8 2.242 2.263 2.158 0.181 0.175 0.089 7 0.095 0 0.116 0.308
a=10 2.251 2.267 2.158 0.181 0.176 0.089 8 0.095 0 0.116 0.307
a=12 2.257 2.269 2.159 0.182 0.176 0.089 9 0.095 1 0.116 0.307
), ArticleFig(id=1322916823930381103, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916813910188785, language=CN, label=表3, caption=

a不同时NWMCKF滤波精度对比(ε=10-9)

, figureFileSmall=null, figureFileBig=null, tableContent=
a值 经度误差/m 纬度误差/m 高度误差/m E向速度误差/m · s-1 N向速度误差/m · s-1 U向速度误差/m · s-1 横滚角误差/(°) 俯仰角误差/(°) 航向角误差/(°)
a=2 2.191 2.351 2.248 0.167 0.164 0.088 4 0.093 0 0.115 0.333
a=4 2.199 2.256 2.163 0.177 0.172 0.089 2 0.094 6 0.116 0.313
a=6 2.227 2.259 2.159 0.180 0.174 0.089 6 0.094 9 0.116 0.311
a=8 2.242 2.263 2.158 0.181 0.175 0.089 7 0.095 0 0.116 0.308
a=10 2.251 2.267 2.158 0.181 0.176 0.089 8 0.095 0 0.116 0.307
a=12 2.257 2.269 2.159 0.182 0.176 0.089 9 0.095 1 0.116 0.307
), ArticleFig(id=1322916824349811504, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916813910188785, language=EN, label=Tab. 4, caption=

Comparison of filtering accuracy of NWMCKF algorithms with different ε (a=2)

, figureFileSmall=null, figureFileBig=null, tableContent=
ε值 经度误差/m 纬度误差/m 高度误差/m E向速度误差/m · s-1 N向速度误差/m · s-1 U向速度误差/m · s-1 横滚角误差/(°) 俯仰角误差/(°) 航向角误差/(°) 总耗时量/s
ε=10-4 2.338 2.477 2.372 0.165 0.161 0.089 4 0.095 2 0.118 0.394 25.0
ε=10-5 2.338 2.477 2.372 0.165 0.161 0.089 4 0.095 2 0.118 0.394 25.0
ε=10-6 2.326 2.481 2.371 0.165 0.161 0.089 4 0.093 6 0.115 0.382 25.3
ε=10-7 2.192 2.333 2.230 0.166 0.164 0.088 4 0.092 7 0.115 0.326 27.8
ε=10-8 2.192 2.350 2.201 0.167 0.164 0.088 3 0.093 0 0.115 0.335 32.9
ε=10-9 2.191 2.351 2.248 0.167 0.164 0.088 4 0.093 0 0.115 0.333 32.8
), ArticleFig(id=1322916824706327345, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916813910188785, language=CN, label=表4, caption=

ε不同时NWMCKF滤波精度对比(a=2)

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ε值 经度误差/m 纬度误差/m 高度误差/m E向速度误差/m · s-1 N向速度误差/m · s-1 U向速度误差/m · s-1 横滚角误差/(°) 俯仰角误差/(°) 航向角误差/(°) 总耗时量/s
ε=10-4 2.338 2.477 2.372 0.165 0.161 0.089 4 0.095 2 0.118 0.394 25.0
ε=10-5 2.338 2.477 2.372 0.165 0.161 0.089 4 0.095 2 0.118 0.394 25.0
ε=10-6 2.326 2.481 2.371 0.165 0.161 0.089 4 0.093 6 0.115 0.382 25.3
ε=10-7 2.192 2.333 2.230 0.166 0.164 0.088 4 0.092 7 0.115 0.326 27.8
ε=10-8 2.192 2.350 2.201 0.167 0.164 0.088 3 0.093 0 0.115 0.335 32.9
ε=10-9 2.191 2.351 2.248 0.167 0.164 0.088 4 0.093 0 0.115 0.333 32.8
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GNSS/SINS组合导航系统的非等加权最大相关熵滤波算法
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刘思明 1 , 林雪原 2 , 乔玉新 1, *
大地测量与地球动力学 | 大地测量学 2026,46(7): 816-824
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大地测量与地球动力学 |大地测量学 2026 , 46 (7) : 816 -824
GNSS/SINS组合导航系统的非等加权最大相关熵滤波算法
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刘思明1 , 林雪原2, 乔玉新1, *
作者信息
  • 1 烟台南山学院智能科学与工程学院，烟台，265713
  • 2 山东外事职业大学人工智能学院，威海，264500
通讯作者:
林雪原，博士，教授，主要从事组合导航及信息融合研究，E-mail：
作者简介:

刘思明，讲师，主要从事信息处理及导航技术研究，E-mail：

Non-Equal-Weighted Maximum Correntropy Filtering Algorithm for the GNSS/SINS Integrated Navigation System
Siming LIU1 , Xueyuan LIN2, Yuxin QIAO1, *
Affiliations
  • 1 College of Intelligent Science and Engineering, Yantai Nanshan University, Yantai 265713, China
  • 2 School of Artificial Intelligence, Shandong Vocational University of Foreign Affairs, Weihai 264500, China
出版时间: 2026-07-15 doi: 10.14075/j.jgg.2025.11.399
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摘要
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在基于等加权的最大相关熵卡尔曼滤波器(MCKF)中，由于质量不同的误差样本采用相同的加权，进而对基于MCKF的GNSS/SINS组合导航系统滤波精度产生影响。为解决这一问题，提出一种基于非等加权的最大相关熵卡尔曼滤波器(NWMCKF)。首先基于等加权的最大相关熵准则(MCC)，提出一种非等加权最大相关熵准则(NWMCC)，以降低MCC对所有误差样本施加相同权重所产生的不良影响；然后，以NWMCC为基础，推导了NWMCKF算法方程，并给出NWMCKF的定点迭代递归算法流程；最后，进行基于KF、MCKF和NWMCKF的GNSS/SINS组合导航系统实验验证。结果表明，测量噪声为高斯噪声时，NWMCKF的滤波性能略优于MCKF；测量噪声为非高斯噪声时，NWMCKF的滤波性能明显优于MCKF。

非等加权最大相关熵准则  /  非等加权最大相关熵卡尔曼滤波器  /  组合导航  /  定点迭代递归算法

In the maximum correntropy Kalman filter based on equal weighting (MCKF), since the error samples with different qualities are weighted equally, it affects the filtering accuracy of the GNSS/SINS integrated navigation system based on MCKF. To solve this problem, a non-equal-weighted maximum correntropy Kalman filter (NWMCKF) is proposed. Firstly, based on the equal weighting maximum correntropy criterion (MCC), a non-equal-weighted maximum correntropy criterion (NWMCC) is proposed to reduce the adverse effects caused by MCC applying the same weight to all error samples. Then, based on NWMCC, the algorithm equation of NWMCKF is derived, and the fixed-point iterative recursive algorithm flow of NWMCKF is given. Finally, experiments on the GNSS/SINS integrated navigation system based on KF, MCKF and NWMCKF are carried out for verification. The experimental results show that when the measurement noise is Gaussian noise, the filtering performance of NWMCKF is slightly better than that of MCKF; when the measurement noise is non-Gaussian noise, the filtering performance of NWMCKF is significantly better than that of MCKF.

non-equal-weighted maximum correlation entropy criterion(NWMCC)  /  non-equal-weighted maximum correlation entropy Kalman filter(NWMCKF)  /  integrated navigation  /  fixed-point iterative recursive algorithm
刘思明, 林雪原, 乔玉新. GNSS/SINS组合导航系统的非等加权最大相关熵滤波算法. 大地测量与地球动力学, 2026 , 46 (7) : 816 -824 . DOI: 10.14075/j.jgg.2025.11.399
Siming LIU, Xueyuan LIN, Yuxin QIAO. Non-Equal-Weighted Maximum Correntropy Filtering Algorithm for the GNSS/SINS Integrated Navigation System[J]. Journal of Geodesy and Geodynamics, 2026 , 46 (7) : 816 -824 . DOI: 10.14075/j.jgg.2025.11.399
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状态估计技术在全球卫星导航系统/捷联惯性导航系统(GNSS/SINS)组合导航系统领域发挥着重要作用 [1]，常规卡尔曼滤波器(KF)是常用的状态估计方法，在高斯假设且最小均方误差(MMSE)准则下可为系统提供最优递归解[2]。这是因为MMSE准则利用了误差信号的二阶统计量(如方差和协方差)。但在实际情况下，测量噪声通常是典型的非高斯噪声，此时KF失去最优性并导致滤波精度下降，因此需要采用其他优化准则[3-4]对KF进行优化。相对于MMSE准则，信息理论学习中的优化准则可从样本数据中获得更高阶的统计信息，在信息处理应用中能显著提高性能[5]。其中，相关熵作为一种广义的相似性度量，在信息处理中可通过最大化模型输出和预期响应之间的相关熵构成最大相关熵准则(MCC)。近年来，MCC已被成功应用于鲁棒滤波领域[6]，并由此衍生出基于MCC的卡尔曼滤波器(MCKF)[7]。
MCKF在GNSS/SINS组合导航系统中得到许多研究与应用[8-10]，其算法是建立于具有相等权重的MCC基础之上，然而实际情况下各误差样本质量不一，相同的权重必然导致滤波精度下降，质量不同的误差样本应该有不同的权重以提升滤波精度。为此，本文在MCC的基础上提出了具有非等加权的MCC准则(NWMCC)及基于非等加权的最大熵卡尔曼滤波算法(NWMCKF)。
1 GNSS/SINS组合导航系统模型
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本文以基于位置与速度的GNSS/SINS松组合导航系统为研究对象，选用东北天(ENU)地理坐标系为导航坐标系，其状态方程为：
$\boldsymbol{X}_k=\boldsymbol{\mathit{\pmb{\Phi}}}_{k\; \mid \;k-1} \boldsymbol{X}_{k-1}+\boldsymbol{\mathit{\pmb{\Gamma}}}_k \boldsymbol{W}_k$
式中，下标k代表第k时刻；Xk∈Rn×1代表状态向量；Φk | k－1 ∈ Rn×n为状态转移矩阵；Γk∈Rn×q为系统噪声分配矩阵；Wk∈Rq×1为系统噪声向量, 其协方差矩阵为Qk∈Rq×q。Xk中的变量包括：
$\boldsymbol{X}_k=\left[\begin{array}{lllll}\boldsymbol{\varphi} & \delta \boldsymbol{v} & \delta \boldsymbol{p} & \boldsymbol{\mathit{\pmb{\varepsilon}}}^{\mathrm{b}} & \boldsymbol{\mathit{\pmb{\nabla}}}^{\mathrm{b}}\end{array}\right]^{\mathrm{T}}$
其中, φ = [ φE    φN    φU ] 为SINS的失准角信息; δv = [ δvE    δvN    δvU ] 为SINS速度的误差信息; δp = [ δλ    δL    δh ] 为SINS位置的误差信息；εb、$\boldsymbol{\nabla}^{\mathrm{b}}$分别为陀螺、加速度计零偏。
测量方程为：
$\boldsymbol{Z}_k=\boldsymbol{H}_k \boldsymbol{X}_k+\boldsymbol{V}_k$
式中, Zk∈ Rm×1为测量向量，可表示GNSS与SINS各自输出的位置及速度之差；Hk∈ Rm×n为测量矩阵；Vk∈ Rm×1为测量噪声向量, 其协方差矩阵为Rk∈ Rm×m。
2 基于非等加权的最大熵滤波算法
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2.1 非等加权相关熵
相关熵[7]是随机变量之间的广义相似性度量，定义为：
$V(X, Y)=E[\kappa(X, Y)]=\int \kappa(x, y) \cdot \mathrm{d} F_{X Y}(x, y)$
式中，X, Y∈R为具有联合分布函数FXY(x, y)的2个随机变量；E[  ·  ]代表期望运算；κ( · ,  · )代表核函数。
本文选用高斯核函数：
$\kappa(x, y)=G_\sigma(e)=\exp \left(-\frac{e^2}{2 \sigma^2}\right)$
式中，e=x－y，σ>0代表核带宽。
在实际中，通常只能获得1组有限的数据，而不知道这些数据的联合分布情况。为此，使用样本值来估算相关熵[11]：
$\hat{V}(X, Y)=\frac{1}{N} \sum\limits_{i=1}^N G_\sigma[e(i)]$
式中，e(i)=x(i)－y(i)，且$\{x(i), y(i)\}_{i=1}^N$为从FXY(x, y)中提取的N个样本。
根据式(6)可以看出，所有样本对相关熵$\hat{V}(X, Y)$的权重是相同的。但在实际情况下，不同样本对状态估计的作用是不同的，即不同的样本应具有不同的权重。因此基于非等加权的相关熵表示为[12]：
$\hat{V}(X, Y)=\sum\limits_{i=1}^N \eta_i^a G_\sigma[e(i)]$
式中，a∈[1, +∞)为权重指数，一般设为a=2[13]；ηi为权重，且满足如下关系式：
$\sum\limits_{i=1}^N \eta_i=1, \eta_i \in[0, 1]$
2.2 NWMCKF的推导
与常规卡尔曼滤波器(KF)一样，NWMCKF的先验估计可通过一步预测的方式得到：
$\hat{\boldsymbol{X}}_{k\; \mid \;k-1}=\boldsymbol{\mathit{\pmb{\Phi}}}_{k\; \mid \;k-1} \hat{\boldsymbol{X}}_{k-1}$
$\boldsymbol{P}_{k\; \mid \;k-1}=\boldsymbol{\mathit{\pmb{\Phi}}}_{k\; \mid \;k-1} \boldsymbol{P}_{k-1} \boldsymbol{\mathit{\pmb{\Phi}}}_{k\; \mid \;k-1}^{\mathrm{T}}+\boldsymbol{\mathit{\pmb{\Gamma}}}_{k-1} \boldsymbol{Q}_{k-1} \boldsymbol{\mathit{\pmb{\Gamma}}}_{k-1}^{\mathrm{T}}$
式中，$\hat{\boldsymbol{X}}_{k-1}$和Pk－1为状态Xk－1的估计值及估计误差协方差矩阵；$\hat{\boldsymbol{X}}_{k\; \mid \;k-1}$和Pk  |  k－1为状态Xk的一步预测值和一步预测协方差矩阵。
基于式(9)和式(3)有：
$\left[\begin{array}{c}\hat{\boldsymbol{X}}_{k\; \mid \;k-1} \\\boldsymbol{Z}_k\end{array}\right]=\left[\begin{array}{c}\boldsymbol{I} \\\boldsymbol{H}_k\end{array}\right] \boldsymbol{X}_k+\boldsymbol{F}_k$
式中, I为n×n的单位矩阵。Fk为：
$\boldsymbol{F}_k=\left[\begin{array}{c}\hat{\boldsymbol{X}}_{k\; \mid \;k-1}-\boldsymbol{X}_k \\\boldsymbol{V}_k\end{array}\right]$
且
$\begin{array}{*{20}{c}}{E\left[ {{{\mathit{\pmb{F}}}_k}{\mathit{\pmb{F}}}_k^{\rm{T}}} \right] = \left[ {\begin{array}{*{20}{c}}{{{\mathit{\pmb{P}}}_{k{\rm{\;|\;}}k - 1}}}&\bf{0}\\\bf{0}&{{{\mathit{\pmb{R}}}_k}}\end{array}} \right] = }\\{\left[ {\begin{array}{*{20}{c}}{{{\mathit{\pmb{B}}}_{P,k{\rm{\;|}\;}k - 1}}{\mathit{\pmb{B}}}_{P,k{\rm{\;|\;}}k - 1}^{\rm{T}}}&\bf{0}\\\bf{0}&{{{\mathit{\pmb{B}}}_{V,k}}{\mathit{\pmb{B}}}_{V,k}^{\rm{T}}}\end{array}} \right] = {{\mathit{\pmb{B}}}_k}{\mathit{\pmb{B}}}_k^{\rm{T}}}\end{array}$
式中，Bk可通过 $E\left[\boldsymbol{F}_k \boldsymbol{F}_k^{\mathrm{T}}\right]$的Cholesky分解得到。
用 ${\mathit{\pmb{B}}}_k^{ - 1}$左乘以式(11)有：
$\boldsymbol{D}_k=\boldsymbol{M}_k \boldsymbol{X}_k+\boldsymbol{E}_k$
或
$\boldsymbol{E}_k=\boldsymbol{D}_k-\boldsymbol{M}_k \boldsymbol{X}_k$
式中，${{\mathit{\pmb{D}}}_k} = {\mathit{\pmb{B}}}_k^{ - 1}\left[{\begin{array}{*{20}{c}}{{{\widehat {\mathit{\pmb{X}}}}_{k\;\mid \;k - 1}}}\\{{{\mathit{\pmb{Z}}}_k}}\end{array}} \right]$；${{\mathit{\pmb{M}}}_k} = {\mathit{\pmb{B}}}_k^{ - 1}\left[{\begin{array}{*{20}{c}}{\mathit{\pmb{I}}}\\{{{\mathit{\pmb{H}}}_k}}\end{array}} \right]$；${{\mathit{\pmb{E}}}_k} = {\mathit{\pmb{B}}}_k^{ - 1}{{\mathit{\pmb{F}}}_k}$。可以看出 $E\left[\boldsymbol{E}_k \boldsymbol{E}_k^{\mathrm{T}}\right]=\boldsymbol{I}$。
通过上述系统性重构，得到估计误差 Ek。可以看出， Ek是衡量状态估计值与先验估计值之间差异的指标，也是实际测量值与预测测量值之间差异的衡量指标。对其进行优化，可以得到最优估计值。
基于式(7)，构造目标函数为：
$J_L\left[\boldsymbol{X}_k, \eta_{i, k}\right]=\sum\limits_{i=1}^L \eta_{i, k}^a G_\sigma\left(e_{i, k}\right)$
式中，L=n+m；ηi, k∈[0, 1]为第i维在时刻k的权重；ei, k=di, k－ Mi, k Xk，ei, k和di, k分别为Ek和Dk的第i个元素，而 Mi, k为Mk的第i行。
基于式(16)， Xk的最优解为：
$\hat{\boldsymbol{X}}_k=\underset{\boldsymbol{X}_k}{\operatorname{argmax}} J_L\left(\boldsymbol{X}_k, \eta_{i, k}\right)$
式中, $\sum\limits_{i=1}^L \eta_{\mathrm{i}, \mathrm{k}}=1$。利用拉格朗日乘法求解：
$l_L=\sum\limits_{i=1}^L \eta_{i, k}^a G_\sigma\left(e_{i, k}\right)+\lambda_k\left(1-\sum\limits_{i=1}^L \eta_{i, k}\right)$
可得：
$\frac{\partial l_L}{\partial \lambda_k}=1-\sum\limits_{i=1}^L \eta_{i, k}=0$
$\frac{\partial l_L}{\partial \eta_{i, k}}=a \cdot \eta_{i, k}^{a-1} G_\sigma\left(e_{i, k}\right)-\lambda_k=0$
根据式(20)，ηi, k可写为：
$\eta_{i, k}=\left[\frac{\lambda_k}{a\; \cdot \;G_\sigma\left(e_{i, k}\right)}\right]^{\frac{1}{a-1}}$
将式(21)代入式(19)得：
$\sum\limits_{i=1}^L \eta_{i, k}=\sum\limits_{i=1}^L\left[\frac{\lambda_k}{a\; \cdot \;G_\sigma\left(e_{i, k}\right)}\right]^{\frac{1}{a-1}}=1$
因此，
$\left(\frac{\lambda_k}{a}\right)^{\frac{1}{a-1}}=\frac{1}{\sum\limits_{i=1}^L\left[\frac{1}{G_\sigma\left(e_{i, k}\right)}\right]^{\frac{1}{a-1}}}$
将式(23)代入式(21)可得：
$\begin{array}{*{20}{c}}{{\eta _{i, k}} = \frac{1}{{{{\left[ {{G_\sigma }\left( {{e_{i, k}}} \right)} \right]}^{\frac{1}{{a - 1}}}}\sum\limits_{j = 1}^L {{{\left[ {\frac{1}{{{G_\sigma }\left( {{e_{j, k}}} \right)}}} \right]}^{\frac{1}{{a - 1}}}}} }} = }\\{\frac{1}{{\sum\limits_{j = 1}^L {{{\left[ {\frac{{{G_\sigma }\left( {{e_{i, k}}} \right)}}{{{G_\sigma }\left( {{e_{j, k}}} \right)}}} \right]}^{\frac{1}{{a - 1}}}}} }}}\end{array}$
根据式(24)，在核带宽σ不变的前提下，当误差ei, k增大时，Gσ(ei, k)减小，ηi, k变大，进而导致大误差会赋予更大的权重，这与传统鲁棒估计的常理不一致。为此对式(24)重新调整如下：
$\tilde{\eta}_{i, k}=\frac{\left(\eta_{i, k}\right)^{-1}}{\sum\limits_{j=1}^L\left(\eta_{j, k}\right)^{-1}}$
此时，当误差ei, k增大时，${{\tilde \eta }_{i, k}}$变小，并以式(25)中的${{\tilde \eta }_{i, k}}$取代式(16)中的${{\tilde \eta }_{i, k}}$，即对大误差赋予更小的权重。
通过如下方程式：
$\frac{\partial l_L}{\partial \boldsymbol{X}_k}=\sum\limits_{i=1}^L\left[\tilde{\eta}_{i, k}^a G_\sigma\left(e_{i, k}\right) \boldsymbol{M}_{i, k}^{\mathrm{T}}\left(d_{i, k}-\boldsymbol{M}_{i, k} \boldsymbol{X}_k\right)\right]=0$
可计算 Xk为：
$\begin{aligned}\boldsymbol{X}_k= & \left\{\sum\limits_{i=1}^L\left[\tilde{\eta}_{i, k}^a G_\sigma\left(e_{i, k}\right) \boldsymbol{M}_{i, k}^{\mathrm{T}} \boldsymbol{M}_{i, k}\right]\right\}^{-1} \times \\& \left\{\sum\limits_{i=1}^L\left[\tilde{\eta}_{i, k}^a G_\sigma\left(e_{i, k}\right) \boldsymbol{M}_{i, k}^{\mathrm{T}} d_{i, k}\right]\right\}\end{aligned}$
根据式(15)可得，ei, k=di, k－ Mi, k Xk，式(27)的最优解具有如下的迭代形式：
$\boldsymbol{X}_k=f\left[\boldsymbol{X}_k\right]$
其中, $f\left[ {{\mathit{\pmb{X}}_k}} \right] = {\left\{ {\sum\limits_{i = 1}^L {\left[ {\tilde \eta _{i,k}^a{G_\sigma }\left( {{d_{i,k}} - {\mathit{\pmb{M}}_{i,k}}{\mathit{\pmb{X}}_k}} \right)\mathit{\pmb{M}}_{i,k}^{\rm{T}}} \right.} {\mathit{\pmb{M}}_{i,k}}} \right\}^{ - 1}} \times \left\{ {\sum\limits_{i = 1}^L {\left[ {\tilde \eta _{i,k}^a{G_\sigma }\left( {{d_{i,k}} - {\mathit{\pmb{M}}_{i,k}}{\mathit{\pmb{X}}_k}} \right)\mathit{\pmb{M}}_{i,k}^{\rm{T}}{d_{i,k}}} \right]} } \right\}$。为此，采用定点迭代算法求解状态向量 Xk的估计值$\hat{\boldsymbol{X}}_k$：
$\hat{\boldsymbol{X}}_k^{(t)}=f\left[\hat{\boldsymbol{X}}_k^{(t-1)}\right]$
式中, $\hat{\boldsymbol{X}}_k^{(t)}$为定点迭代算法在第t次迭代中的解。
将式(27)进一步表示为矩阵形式：
$\boldsymbol{X}_k=\left(\boldsymbol{M}_k^{\mathrm{T}} \boldsymbol{C}_k \boldsymbol{M}_k\right)^{-1}\left(\boldsymbol{M}_k^{\mathrm{T}} \boldsymbol{C}_k \boldsymbol{D}_k\right)$
其中
$\boldsymbol{C}_k=\left[\begin{array}{cc}\boldsymbol{C}_{X, k} & \mathbf{0} \\\mathbf{0} & \boldsymbol{C}_{Z, k}\end{array}\right]$
$\boldsymbol{C}_{X, k}=\operatorname{diag}\left(\tilde{\eta}_{1, k}^a G_\sigma\left(e_{1, k}\right), \cdots, \tilde{\eta}_{n, k}^a G_\sigma\left(e_{n, k}\right)\right)$
$\boldsymbol{C}_{Z, k}=\operatorname{diag}\left(\tilde{\eta}_{n+1, k}^a G_\sigma\left(e_{n+1, k}\right), \cdots, \tilde{\eta}_{n+m, k}^a G_\sigma\left(e_{n+m, k}\right)\right)$
根据矩阵逆引理知识[13]，式(30)可转换成类似于KF的形式：
$\hat{\boldsymbol{X}}_{k\; \mid \;k}=\hat{\boldsymbol{X}}_{k\; \mid \;k-1}+\overline{\boldsymbol{K}}_k\left(\boldsymbol{Z}_k-\boldsymbol{H}_k \hat{\boldsymbol{X}}_{k\; \mid \;k-1}\right)$
其中,
$\left\{\begin{array}{c}\overline{\boldsymbol{K}}_k=\overline{\boldsymbol{P}}_{k\; \mid \;k-1} \boldsymbol{H}_k^{\mathrm{T}}\left(\boldsymbol{H}_k \overline{\boldsymbol{P}}_{k\; \mid \;k-1} \boldsymbol{H}_k^{\mathrm{T}}+\overline{\boldsymbol{R}}_k\right)^{-1} \\\overline{\boldsymbol{P}}_{k\; \mid \;k-1}=\boldsymbol{B}_{P, k\; \mid \;k-1} \boldsymbol{C}_{X, k}^{-1} \boldsymbol{B}_{P, k\; \mid \;k-1}^{\mathrm{T}} \\\overline{\boldsymbol{R}}_k=\boldsymbol{B}_{V, k\; \mid \;k-1} \boldsymbol{C}_{Z, k}^{-1} \boldsymbol{B}_{V, k\; \mid \;k-1}^{\mathrm{T}}\end{array}\right.$
后验协方差矩阵，即估值$\hat{\boldsymbol{X}}_k$的均方误差矩阵 Pk为[1]：
$\begin{gathered}\boldsymbol{P}_k=\left(\boldsymbol{I}-\overline{\boldsymbol{K}}_k \boldsymbol{H}_k\right) \overline{\boldsymbol{P}}_{k\; \mid \;k-1}\left(\boldsymbol{I}-\overline{\boldsymbol{K}}_k \boldsymbol{H}_k\right)^{\mathrm{T}}+ \\\overline{\boldsymbol{K}}_k \boldsymbol{R}_k \overline{\boldsymbol{K}}_k^{\mathrm{T}}\end{gathered}$
综上所述，本文提出的NWMCKF算法总结如下：
1) 初始化。选择合适的权重指数a、核带宽σ和较小的迭代停止阈值ε；设置初始估计$\hat{\boldsymbol{X}}_0$和初始协方差矩阵 P0；令k=0及t=0。
2) 假设k=1、2、…，根据式(9)和式(10)计算$\hat{\boldsymbol{X}}_{k\; \mid \;k-1}$和 Pk | k－1，通过Cholesky分解计算 Bp,k | k－1和BV,k；令$\widehat {\mathit{\pmb{X}}}_{\left. k \right|k}^{(0)} = {\widehat {\mathit{\pmb{X}}}_{k\mid k - 1}}$，其中$\widehat {\mathit{\pmb{X}}}_{k{\rm{|}}k - 1}^{\left(t \right)}$代表第t次定点迭代的状态估计；令t=t+1，计算$\widehat {\mathit{\pmb{X}}}_{k{\rm{|}}k - 1}^{\left(t \right)}$如下：
$\widehat {\mathit{\pmb{X}}}_{k{\rm{|}}k}^{\left( t \right)} = {\widehat {\mathit{\pmb{X}}}_{k\;\mid \;k - 1}} + {{\mathit{\pmb{K}}}_k}\left( {{{\mathit{\pmb{Z}}}_k} - {{\mathit{\pmb{H}}}_k}{{\widehat {\mathit{\pmb{X}}}}_{k\;\mid \;k - 1}}} \right)$
$\widetilde{\boldsymbol{K}}_k=\widetilde{\boldsymbol{P}}_{k\; \mid \;k-1} \boldsymbol{H}_k^{\mathrm{T}}\left(\boldsymbol{H}_k \widetilde{\boldsymbol{P}}_{k\; \mid \;k-1} \boldsymbol{H}_k^{\mathrm{T}}+\widetilde{\boldsymbol{R}}_k\right)$
$\widetilde{\boldsymbol{P}}_{k\; \mid \;k-1}=\boldsymbol{B}_{P, k\; \mid \;k-1} \widetilde{\boldsymbol{C}}_{X, k}^{-1} \boldsymbol{B}_{P, k\; \mid \;k-1}^{\mathrm{T}}$
$\widetilde{\boldsymbol{R}}_k=\boldsymbol{B}_{V, k\; \mid \;k-1} \widetilde{\boldsymbol{C}}_{\boldsymbol{Z}, k}^{-1} \boldsymbol{B}_{V, k\; \mid \;k-1}^{\mathrm{T}}$
$\widetilde{\boldsymbol{C}}_{X, k}=\operatorname{diag}\left(\tilde{\eta}_{1, k}^a G_\sigma\left(\tilde{e}_{1, k}\right), \cdots, \tilde{\eta}_{n, k}^a G_\sigma\left(\tilde{e}_{n, k}\right)\right)$
$\begin{gathered}\tilde{\boldsymbol{C}}_{\boldsymbol{Z}, k}=\operatorname{diag}\left(\tilde{\eta}_{n+1, k}^a G_\sigma\left(\tilde{e}_{n+1, k}\right), \cdots, \tilde{\eta}_{n+m, k}^a\right. \\\left.G_\sigma\left(\tilde{e}_{n+m, k}\right)\right)\end{gathered}$
$ {{\tilde e}_{i, k}} = {d_{i, k}} - {{\mathit{\pmb{M}}}_{i, k}}\widehat {\mathit{\pmb{X}}}_{k{\rm{\;|\;}}k}^{(t - 1)} $
直到
$\frac{{\left\| \;{\widehat {\mathit{\pmb{X}}}_{k{\rm{|}}k}^{(t)} - \widehat {\mathit{\pmb{X}}}_{k{\rm{\;|\;}}k}^{(t - 1)}}\; \right\|}}{{\left\| \;{\widehat {\mathit{\pmb{X}}}_{k{\rm{\;|\;}}k}^{(t)}}\; \right\|}}\varepsilon $
令${\widehat {\mathit{\pmb{X}}}_{k{\rm{\;|\;}}k}} = \widehat {\mathit{\pmb{X}}}_{k{\rm{\;|\;}}k}^{\left(t \right)}$，更新后验协方差矩阵为：
$\begin{gathered}\widetilde{\boldsymbol{P}}_k=\left(\boldsymbol{I}-\widetilde{\boldsymbol{K}}_k \boldsymbol{H}_k\right) \widetilde{\boldsymbol{P}}_{k\; \mid \;k-1}\left(\boldsymbol{I}-\widetilde{\boldsymbol{K}}_k \boldsymbol{H}_k\right)^{\mathrm{T}}+ \\\widetilde{\boldsymbol{K}}_k \boldsymbol{R}_k \widetilde{\boldsymbol{K}}_k^{\mathrm{T}}\end{gathered}$
2.3 NWMCKF的鲁棒性分析
对于MCKF算法，其目标函数为：
$J_L\left[\boldsymbol{X}_k, \eta_{i, k}\right]=\frac{1}{L} \sum\limits_{i=1}^L G_\sigma\left(e_{i, k}\right)$
当式(41)和式(42)中的 $\eta_{i, k}^a=1 / L$时，NWMCKF将会等效于MCKF。
对于非高斯噪声，MCKF算法具有较强的鲁棒性[7-11]。MCKF算法与NWMCKF算法的唯一不同之处在于，对应于式(30)中的 Ck，MCKF算法的对应表达式为：
$\overline{\boldsymbol{C}}_k=\left[\begin{array}{cc}\overline{\boldsymbol{C}}_{X, k} & \mathbf{0} \\\mathbf{0} & \overline{\boldsymbol{C}}_{Z, k}\end{array}\right]$
$\overline{\boldsymbol{C}}_{X, k}=\frac{1}{L} \operatorname{diag}\left(G_\sigma\left(e_{1, k}\right), \quad \cdots \quad, G_\sigma\left(e_{n, k}\right)\right)$
$\overline{\boldsymbol{C}}_{Z, k}=\frac{1}{L} \operatorname{diag}\left(G_\sigma\left(e_{n+1, k}\right), \quad \cdots \quad, G_\sigma\left(e_{n+m, k}\right)\right)$
根据式(25)的分析，当误差ei,k增大时，${{\tilde \eta }_{i, k}}$变小且${{\tilde \eta }_{i, k}}\frac{1}{L}$。如当测量值受到严重干扰时，则CZ,k中的$\tilde{\eta}_{i, k}^a G_\sigma\left(e_{i, k}\right)(n+1 \leqslant i \leqslant n+m)$减小幅度变大且$\tilde{\eta}_{i, k}^a G_\sigma\left(e_{i, k}\right)＜\frac{1}{L} G_\sigma\left(e_{i, k}\right)$，导致式(35)中${\rm{\bar R}}$k的对应元素变大，则式(35)中增益矩阵${\rm{\bar K}}$k中的对应元素减小，进而实现强鲁棒性。
上述分析也说明，相对于MCKF算法，NWMCKF算法的鲁棒性更强一些。
3 实验与分析
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3.1 实验条件
利用实验室搭建的GNSS/SINS组合导航系统仿真平台对本文算法进行验证。其中，陀螺零偏为4°/h，角度随机游走误差为$0.4^\circ \sqrt {\rm{h}} $；加速度计零偏为50 μg，随机游走误差为$5 \mu g / \sqrt{\mathrm{Hz}}$；SINS采样频率为50 Hz；GNSS三维位置测量误差均为8 m，三维速度测量误差均为0.2 m/s，采样周期为1 s；实验时长为3 600 s。
以小型飞行器为研究对象，设计飞行轨迹以实时获取载体真实的导航参数，进而对算法的估计性能进行精确评估。飞行轨迹见图 1，因篇幅限制图 2仅给出部分导航参数的真值。
3.2 实验1：测量噪声为高斯噪声时
根据式(2)对状态向量 Xk的定义，不同状态变量的单位不同。为此，基于水平位置误差估计值选择迭代停止条件，并基于式(44)，取ε=10－9；对于NWMCKF，取a=2；对于MCKF和NWMCKF，其核带宽σ的取值均基于自适应核带宽方法[9, 14]。
当测量噪声为高斯噪声时，进行10次蒙特卡洛实验，基于MCKF和NWMCKF的各导航参数相对误差变化曲线见图 3，各导航参数误差RMSE如表 1所示。
由图 3和表 1可知，相对于MCKF，NWMCKF可提高约1.14%的位置精度，提高约0.78%的速度精度，平均提高约0.12%的姿态精度。总体而言，该精度的提高不明显，二者精度近似相同。
3.3 实验2：测量噪声为非高斯噪声时
SINS不受外界环境的干扰，且具有较强的自主性，为此可近似认为系统噪声为高斯噪声。此处仅考虑测量噪声为非高斯噪声的情况，测量噪声选用的模型为：
$V_k \sim 0.9 \cdot N\left(0, R_k\right)+0.1 \cdot N\left(0, 20 R_k\right)$
式中，Rk为标准方差，其取值取决于GNSS误差参数。
相关实验参数的设置与 § 3.2相同。进行10次蒙特卡洛实验，图 4为基于KF和MCKF的各导航参数绝对误差变化曲线，图 5为基于MCKF和NWMCKF的各导航参数绝对误差变化曲线，各导航参数误差RMSE如表 2所示。
从表 2可以看出，相对于KF，MCKF可分别提高位置精度、速度精度及姿态精度约19.5%、21.7%和18.9%；相对于KF，NWMCKF可分别提高位置精度、速度精度和姿态精度约29.5%、33.5%和31.2%，即相对于MCKF，NWMCKF具有更好的鲁棒性。这是由于NWMCKF通过基于非等加权的最大相关熵准则对MCKF进行优化，消除了普通相关熵对所有样本给予相同权重的缺陷。
由NWMCKF算法流程可以看出，参数a和ε对滤波精度均有影响。为此，进一步研究加权指数a对滤波精度的影响。以某次实验为例，在不同a值下(令ε不变，且ε=10－9)，表 3给出基于NWMCKF的各导航参数的RMSE。可以看出，a的变化对各导航参数的影响规律不一致，总体而言，a=2~4时，NWMCKF的滤波效果较好。由此可知，a值的选择也是影响滤波精度的因素之一。
表 4为在不同ε值下各导航参数的RMSE及滤波总耗时量(取α=2)。可以看出，较小的ε值会导致较小的RMSE，但计算时间会变长；当ε≥10－6时，各导航参数的RMSE较大，其原因为式(2)中水平误差δλ和δL的单位为rad而非m；同时，当ε≤10－7时，ε值对滤波精度的影响并不显著。
4 结语
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为改善基于MCKF的GNSS/SINS组合导航系统滤波精度，本文提出了基于非等加权的最大相关熵卡尔曼滤波算法(NWMCKF)。针对MCC准则中因权重相同而导致不同样本对状态估计影响不确定的问题，本文根据不同样本情况引入非等加权，进而构建了NWMCC准则，并在此基础上提出了NWMCKF算法。
实验结果表明，当测量噪声为高斯噪声时，NWMCKF与MCKF的滤波精度基本相同；而当测量噪声为非高斯噪声时，NWMCKF的性能明显优于MCKF。同时，本文也研究了指数权重及迭代阈值对滤波精度的影响。本文仅探讨了NWMCKF在GNSS/SINS线性组合导航系统的应用，下一步将集中针对NWMCKF在GNSS/SINS非线性组合导航系统的应用展开研究。
基金
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  • 国家自然科学基金(62171402)
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2026年第46卷第7期
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doi: 10.14075/j.jgg.2025.11.399
  • 接收时间：2025-11-20
  • 首发时间：2026-10-30
  • 出版时间：2026-07-15
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  • 收稿日期：2025-11-20
基金
国家自然科学基金(62171402)
作者信息
    1 烟台南山学院智能科学与工程学院，烟台，265713
    2 山东外事职业大学人工智能学院，威海，264500

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林雪原，博士，教授，主要从事组合导航及信息融合研究，E-mail：
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鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
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