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Article(id=1322566777196888488, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322566763112415546, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2024.12.577, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1734883200000, receivedDateStr=2024-12-23, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793254091786, onlineDateStr=2026-10-29, pubDate=1768406400000, pubDateStr=2026-01-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793254091786, onlineIssueDateStr=2026-10-29, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793254091786, creator=13701087609, updateTime=1793254091786, updator=13701087609, issue=Issue{id=1322566763112415546, tenantId=1146029695717560320, journalId=1281212831689347082, year='2026', volume='46', issue='1', pageStart='1', pageEnd='138', issueExtLink='null', onlineDate='null', pubDate='1768406400000', pubDateStr='2026-01-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793254088427, creator='13701087609', updateTime=1793254088427, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=26, endPage=33, ext={EN=ArticleExt(id=1322566777360466345, articleId=1322566777196888488, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=Satellite Orbit Determination Based on Extended Kalman Filter, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

Traditional satellite orbit determination methods primarily rely on post-processing on the ground, which results in significant time delays and fails to meet the real-time requirements of autonomous on-board tasks. To address this issue, this paper proposes an autonomous real-time orbit determination algorithm based on the extended Kalman filter (EKF). The algorithm integrates quality control of observational data, dynamic orbit integration, and parameter estimation techniques to achieve real-time orbit determination for low-Earth orbit (LEO) satellites. The accuracy of the algorithm was assessed by comparing the real-time and predicted orbits with the ground-based post-processed precise reference orbits. Results show that, after accumulating 72 hours of observational data, the three-dimensional position error of real-time orbit determination was 15.18 cm, and the three-dimensional position error of orbit prediction within 2 hours was better than 18.75 cm, meeting the requirements for autonomous mission planning. Moreover, validated on an embedded platform, the algorithm has a single calculation cycle of less than 35 seconds and a memory footprint of less than 4 MB, complying with the onboard resource requirements. This study not only provides a theoretical and technical foundation for the design of autonomous navigation systems for medium and high Earth orbit satellites but also holds engineering value for enhancing the real-time response capabilities of satellites in orbit.

, authors=Zhongyuan ZHANG1, Yulong FEI1, *, Xiangyang LU1, authorsList=Zhongyuan ZHANG, Yulong FEI, Xiangyang LU, authorCompany=null, correspAuthors=Yulong FEI, 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=1322566779084325304, articleId=1322566777196888488, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=基于扩展卡尔曼滤波的卫星定轨研究, columnId=1281909276545356011, journalTitle=大地测量与地球动力学, columnName=大地测量学, runingTitle=null, highlight=null, articleAbstract=

传统卫星轨道解算方法主要依赖地面事后处理，存在较大时延，难以满足星上自主任务的实时性需求。针对这一问题，提出一种基于扩展卡尔曼滤波的自主实时定轨算法。该算法融合了观测数据质量控制、动力学轨道积分以及参数估计技术，能够实现低轨卫星在轨实时解算。通过将实时轨道和预报轨道分别与地面事后精密参考轨道进行对比，对算法的准确性进行评估。结果表明，在累积72 h观测数据后，实时定轨的三维位置误差为15.18 cm, 而轨道预报在2 h内的三维位置误差优于18.75 cm, 能够满足自主任务规划的需求。此外，经嵌入式平台验证，该算法单次解算周期小于35 s, 内存占用低于4 MB, 符合星载资源的要求。该研究不仅为中高轨卫星自主导航系统的设计提供了理论与技术支撑，而且对提升卫星在轨实时响应能力具有工程应用价值。

, authors=张中原1, 费玉龙1, *, 卢向阳1, authorsList=张中原, 费玉龙, 卢向阳, authorCompany=null, correspAuthors=费玉龙, authorNote=

张中原, 助理研究员, 主要从事合成孔径雷达成像算法和控制研究, E-mail:

, correspAuthorsNote=
费玉龙, 副研究员, 主要从事合成孔径雷达成像算法和控制研究, E-mail:
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张中原, 助理研究员, 主要从事合成孔径雷达成像算法和控制研究, E-mail:

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张中原, 助理研究员, 主要从事合成孔径雷达成像算法和控制研究, E-mail:

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Hardware platform for real-time precise orbit determination

, figureFileSmall=null, figureFileBig=null, tableContent=
硬件环境主要参数
DSP芯片TMS320C6678
晶振频率1 GHz
看门狗频率5 s
FPGA芯片JFM4VSX55RT-CCGA1140
FLASH芯片256 M×3 32 bits
DDR3RAM1 G 64 bits
总线通信CAN
数据通信EMIF
数传接口LVDS
调试串口UART
), ArticleFig(id=1322566783664505316, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566777196888488, language=CN, label=表1, caption=

实时精密定轨硬件平台

, figureFileSmall=null, figureFileBig=null, tableContent=
硬件环境主要参数
DSP芯片TMS320C6678
晶振频率1 GHz
看门狗频率5 s
FPGA芯片JFM4VSX55RT-CCGA1140
FLASH芯片256 M×3 32 bits
DDR3RAM1 G 64 bits
总线通信CAN
数据通信EMIF
数传接口LVDS
调试串口UART
), ArticleFig(id=1322566783752585701, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566777196888488, language=EN, label=Tab. 2, caption=

Real-time precise orbit determination processing strategy

, figureFileSmall=null, figureFileBig=null, tableContent=
相关内容处理策略
采样起始2018-02-01 00:00:00
采样间隔30 s
采样截止2018-02-08 00:00:00
观测量双频伪距和载波相位
GNSS卫星选择BDS
观测频率选择B1I、B3I
截止高度角15°
周跳检测TurboEdit
天线相位中心改正地面校正值igs文件
卫星天线相位缠绕模型改正
卫星钟差地面站计算改正数
对流层延迟无
电离层延迟无
地球重力场EIGEN_GL04C(120×120)
N体引力JPL DE405行星星历
固体潮IERS Conventions 2010
相对论效应IERS Conventions 2010
太阳光压Box-wing模型
经验力模型ACR周期性经验力
模糊度参数浮点解、采用常数模型估计
参数估计方法EKF
), ArticleFig(id=1322566783819694566, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566777196888488, language=CN, label=表2, caption=

实时精密定轨处理策略

, figureFileSmall=null, figureFileBig=null, tableContent=
相关内容处理策略
采样起始2018-02-01 00:00:00
采样间隔30 s
采样截止2018-02-08 00:00:00
观测量双频伪距和载波相位
GNSS卫星选择BDS
观测频率选择B1I、B3I
截止高度角15°
周跳检测TurboEdit
天线相位中心改正地面校正值igs文件
卫星天线相位缠绕模型改正
卫星钟差地面站计算改正数
对流层延迟无
电离层延迟无
地球重力场EIGEN_GL04C(120×120)
N体引力JPL DE405行星星历
固体潮IERS Conventions 2010
相对论效应IERS Conventions 2010
太阳光压Box-wing模型
经验力模型ACR周期性经验力
模糊度参数浮点解、采用常数模型估计
参数估计方法EKF
), ArticleFig(id=1322566783916163559, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566777196888488, language=EN, label=Tab. 3, caption=

Statistical results of daily orbit determination error RMS

, figureFileSmall=null, figureFileBig=null, tableContent=
定轨时间/dRMS/cm
ACR3D
119.8425.3726.1541.49
29.7110.689.8817.49
38.098.689.4715.18
47.178.188.7913.99
57.208.178.7713.98
67.168.198.8013.99
77.108.108.6413.81
), ArticleFig(id=1322566784000049640, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566777196888488, language=CN, label=表3, caption=

每日定轨误差RMS统计结果

, figureFileSmall=null, figureFileBig=null, tableContent=
定轨时间/dRMS/cm
ACR3D
119.8425.3726.1541.49
29.7110.689.8817.49
38.098.689.4715.18
47.178.188.7913.99
57.208.178.7713.98
67.168.198.8013.99
77.108.108.6413.81
), ArticleFig(id=1322566785656799721, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566777196888488, language=EN, label=Tab. 4, caption=

Statistical results of daily orbit forecast error RMS

, figureFileSmall=null, figureFileBig=null, tableContent=
数据累积时间/hRMS/cm
ΔAΔCΔRΔ3D
647.1342.1578.49100.79
306.4923.8316.9329.94
5413.385.7711.8018.75
), ArticleFig(id=1322566785744880106, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566777196888488, language=CN, label=表4, caption=

每日预报误差RMS统计结果

, figureFileSmall=null, figureFileBig=null, tableContent=
数据累积时间/hRMS/cm
ΔAΔCΔRΔ3D
647.1342.1578.49100.79
306.4923.8316.9329.94
5413.385.7711.8018.75
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基于扩展卡尔曼滤波的卫星定轨研究
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张中原 1 , 费玉龙 1, * , 卢向阳 1
大地测量与地球动力学 | 大地测量学 2026,46(1): 26-33
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大地测量与地球动力学 |大地测量学 2026 , 46 (1) : 26 -33
基于扩展卡尔曼滤波的卫星定轨研究
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张中原, 助理研究员, 主要从事合成孔径雷达成像算法和控制研究, E-mail:

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张中原1 , 费玉龙1, * , 卢向阳1
作者信息
  • 1 中国科学院空天信息创新研究院, 北京, 100094
通讯作者:
费玉龙, 副研究员, 主要从事合成孔径雷达成像算法和控制研究, E-mail:
作者简介:

张中原, 助理研究员, 主要从事合成孔径雷达成像算法和控制研究, E-mail:

Satellite Orbit Determination Based on Extended Kalman Filter
Zhongyuan ZHANG1 , Yulong FEI1, * , Xiangyang LU1
Affiliations
  • 1 Aerospace Information Research Institute, CAS, Beijing 100094, China
出版时间: 2026-01-15 doi: 10.14075/j.jgg.2024.12.577
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摘要
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传统卫星轨道解算方法主要依赖地面事后处理，存在较大时延，难以满足星上自主任务的实时性需求。针对这一问题，提出一种基于扩展卡尔曼滤波的自主实时定轨算法。该算法融合了观测数据质量控制、动力学轨道积分以及参数估计技术，能够实现低轨卫星在轨实时解算。通过将实时轨道和预报轨道分别与地面事后精密参考轨道进行对比，对算法的准确性进行评估。结果表明，在累积72 h观测数据后，实时定轨的三维位置误差为15.18 cm, 而轨道预报在2 h内的三维位置误差优于18.75 cm, 能够满足自主任务规划的需求。此外，经嵌入式平台验证，该算法单次解算周期小于35 s, 内存占用低于4 MB, 符合星载资源的要求。该研究不仅为中高轨卫星自主导航系统的设计提供了理论与技术支撑，而且对提升卫星在轨实时响应能力具有工程应用价值。

实时定轨  /  轨道预报  /  低轨卫星  /  扩展卡尔曼滤波  /  嵌入式平台

Traditional satellite orbit determination methods primarily rely on post-processing on the ground, which results in significant time delays and fails to meet the real-time requirements of autonomous on-board tasks. To address this issue, this paper proposes an autonomous real-time orbit determination algorithm based on the extended Kalman filter (EKF). The algorithm integrates quality control of observational data, dynamic orbit integration, and parameter estimation techniques to achieve real-time orbit determination for low-Earth orbit (LEO) satellites. The accuracy of the algorithm was assessed by comparing the real-time and predicted orbits with the ground-based post-processed precise reference orbits. Results show that, after accumulating 72 hours of observational data, the three-dimensional position error of real-time orbit determination was 15.18 cm, and the three-dimensional position error of orbit prediction within 2 hours was better than 18.75 cm, meeting the requirements for autonomous mission planning. Moreover, validated on an embedded platform, the algorithm has a single calculation cycle of less than 35 seconds and a memory footprint of less than 4 MB, complying with the onboard resource requirements. This study not only provides a theoretical and technical foundation for the design of autonomous navigation systems for medium and high Earth orbit satellites but also holds engineering value for enhancing the real-time response capabilities of satellites in orbit.

real-time orbit determination  /  orbit prediction  /  low Earth orbit (LEO) satellites  /  extended Kalman filtering (EKF)  /  embedded platform
张中原, 费玉龙, 卢向阳. 基于扩展卡尔曼滤波的卫星定轨研究. 大地测量与地球动力学, 2026 , 46 (1) : 26 -33 . DOI: 10.14075/j.jgg.2024.12.577
Zhongyuan ZHANG, Yulong FEI, Xiangyang LU. Satellite Orbit Determination Based on Extended Kalman Filter[J]. Journal of Geodesy and Geodynamics, 2026 , 46 (1) : 26 -33 . DOI: 10.14075/j.jgg.2024.12.577
正文
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近期，国际形势急剧变化，各类产业技术飞速发展。在军事、环境、通信等诸多关键领域，卫星技术面临着愈发严苛的挑战与更高层次的要求，其中实时轨道解算和轨道预报成为关键环节，这对于实现地面目标的精准定位至关重要。目前，获取在轨卫星实时精密轨道信息的主要手段是依靠地面站与全球导航卫星系统(GNSS)协同配合，通过观测和计算来实现[1]。然而，这一传统方式存在诸多弊端：一方面，星地传输资源的限制以及卫星过境时间的约束，使得计算出的精密轨道相对起始观测时刻存在较大的时间滞后，难以满足实时性的需求；另一方面，分段计算的原则导致轨道在不同段落之间出现不连续的情况，在边界处存在较大的跳变，这在很大程度上降低了轨道精度；此外，卫星在轨运行期间周期性地从地面站获取精密轨道信息，这一过程不可避免地增加了星地之间的传输资源占用，影响整个系统的运行效率。因此，实现卫星在轨自主实时精密轨道解算，不仅能够有效缩短轨道滞后时间，提升轨道精度，还能显著减少星地传输资源的占用，进而加速在轨卫星自主参数计算、成像、处理一体化进程。
在相关研究领域，众多学者已经开展了大量卓有成效的工作。例如，Hauschild等[2]利用伽利略服务组合中的高精度服务，对Sentinel-6A卫星进行定轨分析，相比于仅使用GPS、GPS+伽利略和仅使用伽利略的情况，三维位置误差分别降低40%、10%和5%，成功实现7~9 cm的轨道确定误差；Yoshinari等[3]采用日本宇宙航空研究开发机构(JAXA)提供的精密GNSS轨道和时钟估计系统实时数据，从而获得更精确的钟差信息，最终将定轨精度提升至9.1 cm；Kang等[4]则聚焦于GRACE-FO两颗编队飞行的卫星轨道处理，仅凭借星载GPS双差观测量进行定轨分析，最终得到的SLR残差均方根(RMS)误差为8 cm，三维轨道差异的RMS为13 cm；Bhattacharjee等[5]结合法方程和Cholesky分解，在GNSS惯性导航系统中成功生成高精度的大轨道弧，并借助LAGEOS-1卫星数据，生成cm级小弧(每周弧)和dm级大弧(每月弧)；Dai等[6]引入平方根信息滤波(square root information filter, SRIF)实时定轨策略，针对倾斜地球同步轨道(inclined GSO, IGSO)和中圆地球轨道(MEO)卫星进行研究，结果显示, SRIF解决方案与最终产品之间的轨道差异三维RMS平均值分别为29.1 cm和22.5 cm，同时有力证明了生成的轨道在所有时间都是连续的；张万威等[7]构建伪模糊度随机模型，在优化过程中精准确定KF待估滤波状态量，使定轨三维位置误差达到4.62~7.56 cm；Shi等[8]利用精密单点定位(PPP)B2b增强实时精密定轨技术，对天津大学一号(TJU-01)卫星进行定轨分析，在径向、沿向和横向方向上，轨道误差RMS分别达到10 cm、13 cm和9 cm；Chen等[9]提出一种改进的鲁棒自适应分层滤波算法，该算法在减少拟合后残差和提高轨道预测精度方面表现出色，展现出卓越的鲁棒性，三轴轨道确定内部一致性精度可达到cm级。
尽管在轨卫星实时精密轨道解算已经达到cm级精度，但目前还没有应用于实际的卫星系统中，只停留在理论层面。为此，本文搭建星载精密定轨处理器平台，通过对观测量进行质量控制，建立卫星轨道积分，并采用扩展卡尔曼滤波(extended Kalman filter, EKF)作为参数估计方法，以确定初始时刻卫星轨道状态和动力学模型参数的最佳估值，实现卫星在轨实时精密定轨。此外，还采用实测数据进行实时精密定轨实验并进行结果分析，以评估算法的精度。
1 实时精密定轨算法
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在轨卫星实时精密定轨是对轨道的n维状态向量进行实时解算，包含轨道六根数或者卫星位置、速度信息(记为x)，状态向量满足的微分方程可表示为：
$\left\{\begin{array}{l}\dot{\boldsymbol{x}}=F(\boldsymbol{x}, t) \\\left.\boldsymbol{x}\right|_{t_{0}}=\boldsymbol{x}_{0}\end{array}\right.$
式中，F为关于状态向量和时间的n维非线性函数；$\boldsymbol{x}_{0}$为轨道初始状态。式(1)的解为卫星在t时刻的参考轨道$\boldsymbol{x}^{*}$。
令$\boldsymbol{X}(t)=\boldsymbol{x}(t)-\boldsymbol{x}^{*}(t), A(t)= \left.\frac{\partial F(\boldsymbol{x}, t)}{\partial \boldsymbol{x}}\right|_{*}$，则式(1)可表示为：
$\begin{equation*}\dot{\boldsymbol{X}}(t)=A(t) \boldsymbol{X}(t) \end{equation*}$
式中，$\boldsymbol{x}(t)$为轨道真实状态；$\boldsymbol{x}^{*}(t)$为轨道参考状态；$\boldsymbol{X}(t)$为轨道修正量。
式(2)的解的形式为：
$\begin{equation*}\boldsymbol{X}(t)=\boldsymbol{\varPhi}\left(t, t_{0}\right) \boldsymbol{X}_{0}\end{equation*}$
并且有
$\left\{\begin{array}{l}\boldsymbol{\varPhi}\left(\dot{t}, t_{0}\right)=A(t) \boldsymbol{\varPhi}\left(t, t_{0}\right) \\\boldsymbol{\varPhi}\left(t_{0}, t_{0}\right)=\boldsymbol{I}\end{array}\right.$
式中，$\boldsymbol{\varPhi}\left(t, t_{0}\right)$为状态转移矩阵；$\boldsymbol{I}$为单位矩阵。因此，运动方程的求解就变成了状态转移矩阵的求解问题，式(4)可通过数值积分法进行求解[10]。
GNSS卫星对在轨卫星的相对位置进行观测得到的m维向量称为观测量(记作y)，其与观测量的真值$\boldsymbol{G}(\boldsymbol{x}, t)$以及观测噪声V之间的关系为：
$\begin{equation*}\boldsymbol{y}=\boldsymbol{G}(x, t)+\boldsymbol{V}\end{equation*}$
类似的有：
$\begin{equation*}\boldsymbol{Y}(t)=H(t) \boldsymbol{X}(t)+\boldsymbol{V}\end{equation*}$
式中，$\boldsymbol{Y}(t)$为观测量改正值; $H(t)$为参观测量对该时刻历元参数的偏导数。联合式(3)可得式(6)的解为：
$\begin{equation*}\boldsymbol{Y}(t)=H(t) \boldsymbol{\varPhi}\left(t, t_{0}\right) \boldsymbol{X}_{0}+\boldsymbol{V}\end{equation*}$
将式(7) 写为更为常见的形式：
$\begin{equation*}\boldsymbol{L}=\boldsymbol{B} \boldsymbol{X}_{0}+\boldsymbol{V} \end{equation*}$
其中,
$\boldsymbol{L}=\left[\begin{array}{c}\boldsymbol{Y}_{1} \\\vdots \\\boldsymbol{Y}_{l}\end{array}\right], \boldsymbol{B}=\left[\begin{array}{c}H_{1} \boldsymbol{\varPhi}\left(t_{1}, t_{0}\right) \\\vdots \\H_{l} \boldsymbol{\varPhi}\left(t_{l}, t_{0}\right)\end{array}\right], \boldsymbol{V}=\left[\begin{array}{c}\boldsymbol{V}_{1} \\\vdots \\\boldsymbol{V}_{l}\end{array}\right]$
综上所述，在轨卫星实时精密定轨就转化为状态估计问题：利用带有随机观测误差$\boldsymbol{V}_{i}(i=1$，$2, \cdots, l)$的测量序列$\boldsymbol{Y}_{i}(i=1, 2, \cdots, l)$和已知的初始状态$\boldsymbol{x}_{0}$，确定在某种意义下初始状态$\boldsymbol{x}_{0}$的改正量$\boldsymbol{X}_{0}$的最优估计值$\hat{\boldsymbol{X}}_{0}$。
由上述精密定轨算法可以看出，轨道积分、参数估计是精密定轨算法的核心步骤，而利用质量控制剔除粗差，保证观测数据无偶发跳变影响同样重要。
1.1 质量控制
数据质量控制是高精度GNSS应用的关键技术之一[11]，本文仅说明较为复杂的周跳探测。
观测方程可表示为[12]：
$\left\{\begin{align*}P_{\mathrm{g}, i}^{\mathrm{s}}= & \rho_{\mathrm{g}}^{\mathrm{s}}+c\left(t_{\mathrm{g}}-t^{\mathrm{s}}\right)+I_{\mathrm{g}, i}^{\mathrm{s}}+T_{\mathrm{g}}^{\mathrm{s}}+ \\& \rho_{\text {delay }, \mathrm{g}, i}-\rho_{\text {delay }, i}^{\mathrm{s}}+e_{\mathrm{g}, i}^{\mathrm{s}} \\L_{\mathrm{g}, i}^{\mathrm{s}}= & \rho_{\mathrm{g}}^{\mathrm{s}}+c\left(t_{\mathrm{g}}-t^{\mathrm{s}}\right)-I_{\mathrm{g}, i}^{\mathrm{s}}+T_{\mathrm{g}}^{\mathrm{s}}+ \\& \varphi_{\text {delay }, \mathrm{g}, i}-\varphi_{\text {delay }, i}^{\mathrm{s}}+\lambda_{i} N_{\mathrm{g}, i}^{\mathrm{s}}+\varepsilon_{\mathrm{g}, i}^{\mathrm{s}}\end{align*}\right.$
式中, 下标i为频率; 下标g为GNSS卫星; 上标s为被观测在轨卫星, P和L分别为伪距和载波相位观测量; ρ为GNSS卫星和在轨卫星的几何真实距离; c为光速; t为卫星钟差; I为电离层延迟; T为对流层延迟; ρdelay和φdelay分别为卫星伪距和载波相位信号延迟；e和ε分别为伪距和载波相位观测噪声；λ为载波波长；N为整周模糊度。
Blewitt[13]利用双频MW(Melbourne-Wübbena)和GF(geometry-free)组合观测量的特性，提出了探测和修复周跳及剔除野值的TurboEdit算法。本文在此基础上进行改进，从根本上排除电离层延迟影响，从而以更高的精度解算出周跳值。具体流程如下：
1) MW组合。利用载波相位观测量的宽巷组合和伪距观测量的窄巷组合相减可得MW组合观测模型为[14]：
$\begin{equation*}L_{\mathrm{MW}}=\frac{f_{1} L_{1}-f_{2} L_{2}}{f_{1}-f_{2}}-\frac{f_{1} P_{1}+f_{2} P_{2}}{f_{1}+f_{2}}=\lambda_{\mathrm{MW}} N_{\mathrm{MW}} \end{equation*}$
式中，LMW为组合观测量；Li=λiφi(i=1, 2)为不同频率的载波相位观测量，λi和φi分别为不同频率的波长和相位观测量；fi为相位对应频率；Pi为伪距观测量；λMW为组合波长；NMW为组合模糊度。
无周跳时，NMW近似为随机分布的不变常数[15]，而一旦本次历元观测量符合式(12)，则认为本次观测存在周跳，需要剔除本次观测：
$\left\{\begin{array}{l}\left|\bar{N}_{\mathrm{MW}}(i)-\bar{N}_{\mathrm{MW}}(i-1)\right| \geqslant 4 \sigma_{\mathrm{MW}}(i-1)\\\left|\bar{N}_{\mathrm{MW}}(i+1)-\bar{N}_{\mathrm{MW}}(i)\right| \leqslant 1\end{array}\right.$
式中，$\bar{N}_{\mathrm{MW}}$（i）和$\sigma_{\mathrm{MW}}^{2}$（i）分别为第i次观测组合模糊度的均值和方差$\bar{N}_{\mathrm{MW}}(i)$受噪声影响较为严重，很难准确判断周数较小的跳变，因此加入改进GF组合进行判断。
2) 改进GF组合。根据式(10)原始观测方程得到GF组合观测模型为[16]：
$$$\left\{\begin{align*}L_{\varDelta}= & L_{1}-L_{2}=-I_{\mathrm{g}, 1}^{\mathrm{s}}+I_{\mathrm{g}, 2}^{\mathrm{s}}+ \\& \lambda_{1} N_{\mathrm{g}, 1}^{\mathrm{s}}-\lambda_{2} N_{\mathrm{g}, 2}^{\mathrm{s}}+m \\m= & \varphi_{\text {delay }, \mathrm{g}, 1}-\varphi_{\text {delay }, 1}^{\mathrm{s}}-\varphi_{\text {delay }, \mathrm{g}, 2} \\& +\varphi_{\text {delay }, 2}^{\mathrm{s}}+\varepsilon_{\mathrm{r}, 1}^{\mathrm{s}} \mp \varepsilon_{\mathrm{r}, 2}^{\mathrm{s}}\end{align*}\right.$
式中，m代表在轨卫星和GNSS卫星端信号延迟和观测噪声在不同频率上的差值，其值相比其他值非常小，可以忽略不计。在轨卫星搭载GNSS接收机直接接收GNSS卫星对本卫星的观测量，中高轨卫星信号传输不穿越电离层，低轨卫星常见高度大于700 km，信号传输仅穿越上电离层，因此可以忽略电离层延迟。最终改进GF组合方程中只有整周模糊度项，适合较小周跳的检测和修复。式(13)可进一步改写为：
$\begin{equation*}\varDelta \varphi=\varDelta N_{\mathrm{g}, 1}^{\mathrm{s}}-\frac{\lambda_{2}}{\lambda_{1}} \varDelta N_{\mathrm{g}, 2}^{\mathrm{s}} \end{equation*}$
当Δφ在0附近波动时, 可认为本次历元无周跳。因此, 综合MW和GF组合可以对大小周跳进行准确探测[17]。
1.2 轨道积分
数值方法分为单步法和多步法，单步法通常为RK算法，多步法通常为Adams或者Adams-Cowell算法。本文采用单步法和多步法结合的方法进行轨道积分计算，两者同时使用可快速准确地求解轨道积分中的状态转移矩阵[18]。
1.2.1 单步法
将式(1)卫星运动方程按泰勒展开改写为s阶RK方程，其形式如下[19]：
$\left\{\begin{array}{l}\boldsymbol{x}_{i+1}=\boldsymbol{x}_{i}+c_{1} K_{1}+c_{2} K_{2}+\cdots+c_{s} K_{s} \\K_{1}=h f\left(t_{i}, \boldsymbol{x}_{i}\right) \\\cdots \\K_{s}=h f\left(t_{i}+a_{s} h, \boldsymbol{x}_{i}+h \sum\limits_{j=1}^{s-1} b_{s j} K_{j}\right), \\s=2, 3, \cdots\end{array}\right.$
由于RK方法可以自起步积分[20]，因此先利用RK方法得到足够的积分起算点，一般工程上使用4阶计算。
1.2.2 多步法
同阶中Adams-Cowell算法精度高于Adams算法，但Adams-Cowell算法不能处理与卫星速度相关的大气阻力模型[21]，因此只采用Adams算法作为多步法进行数值积分。
考虑到计算$\boldsymbol{x}_{i+1}$之前已得出一系列节点上的斜率值，因此利用这些已知量减少运算量，即引用之前几个点和该点处的导数值组成线性组合，可得Adams显式公式(式(16))和隐式公式(式(18))如下[22]:
$\begin{gather*}\left\{\begin{array}{l}\boldsymbol{x}_{i+1}=\boldsymbol{x}_{i}+h \sum\limits_{j=0}^{k} \beta_{k j} f_{i-j} \\\beta_{k j}=(-1)^{j} \sum\limits_{l=j}^{k}\left(\frac{l}{j}\right) a_{l} \\a_{l}=(-1)^{l} \int_{0}^{1}\left(\frac{-t}{l}\right) \mathrm{d} t\end{array}\right. \end{gather*}$
$\left(\frac{l}{j}\right)=\frac{l!}{j!(l-j)!}$
$\left\{\begin{array}{l}\boldsymbol{x}_{i+1}=\boldsymbol{x}_{i}+h \sum\limits_{j=0}^{k} \beta_{k j}^{*} f_{i-j+1} \\\beta_{k j}^{*}=(-1)^{j} \sum\limits_{l=j}^{k}\left(\frac{l}{j}\right) a_{l}^{*} \\a_{l}^{*}=(-1)^{l} \int_{-1}^{0}\left(\frac{-t}{l}\right) \mathrm{d} t\end{array}\right. $
显式公式是一个外推的过程，便于计算但稳定性较差；隐式公式精度较高，但需要迭代的次数更多，计算较复杂。因此利用预估-校正算法，通过显式公式预报、利用预报值计算右函数、代入隐式公式校正等步骤进行卫星轨道计算，将显示公式和隐式公式结合使用[23]。
Adams-PECE方法计算速度快且稳定性较好。为保证积分精度和收敛速度的平衡，单步法和多步法阶数保持一致，均取4阶[24]。
1.3 参数估计
轨道积分可建立卫星初始状态和任意时刻之间的联系，进而可以计算得到卫星的状态量，实现实时精密定轨。但是在卫星实际在轨运行中，初始状态一般是难以确定的，描述卫星运动模型的参数也有一定的误差，使得最终计算得到的卫星状态偏离真实轨道。为了提高精度，需要多颗GNSS卫星对在轨卫星进行实时观测，经过参数估计确定最优改正量，进而计算卫星状态量的最优估值。
目前常用的参数估计方法有最小二乘法和卡尔曼滤波[25]。最小二乘法针对一段时间内的卫星状态进行综合估计，需要累积时间足够长的观测量后才能拟合目标曲线，一旦某段时间GNSS卫星通信中断，实时定轨精度将会大打折扣，同时其对硬件平台有较高的存储性能要求，导致硬件体积变大、重量上升、散热增加等一系列问题，不适用于在轨卫星实时精密定轨；卡尔曼滤波使用上一时刻的最优结果预测这一时刻的预测值，同时使用这一时刻的观测量进行修正，即可得到这一时刻的最优结果，是一种迭代算法，因此降低了同一时间的计算量，从而降低卫星平台硬件功耗。本文采用EKF作为参数估计策略，并对协方差矩阵进行针对性处理，以有效抑制常规卡尔曼滤波中因累积迭代造成的增益矩阵发散问题，同时缩小病态矩阵各状态量之间的精度误差，提升参数估计的准确性与稳定性[26]。
由轨道积分可得卫星的运动方程，结合式(8)观测方程可得卫星系统方程组如下：
$\left\{\begin{array}{l}\boldsymbol{X}_{i}=\boldsymbol{\varPhi}_{i / i-1} \boldsymbol{X}_{i-1}+\boldsymbol{\varGamma}_{i / i-1} \boldsymbol{W}_{i-1} \\\boldsymbol{Y}_{i}=\boldsymbol{H}_{i} \boldsymbol{X}_{i}+\boldsymbol{V}_{i}\end{array}\right.$
式中，$\boldsymbol{X}_{i}$为卫星n维状态向量；$\boldsymbol{\varPhi}_{i / i-1}$为轨道积分得到的2个时刻之间卫星状态向量的转移矩阵；$\boldsymbol{\varGamma}_{i / i-1}$为噪声权重参数；$\boldsymbol{W}_{i}$为过程噪声，符合正态分布，均值为0，方差为$\boldsymbol{Q}_{i} ; \boldsymbol{Y}_{i}$为$m$维观测向量；$\boldsymbol{H}_{i}$为测量矩阵；$\boldsymbol{V}_{i}$为测量噪声，符合正态分布，均值为0，方差为$\boldsymbol{R}_{i}$。
根据卡尔曼滤波基本方程，可以得到预测结果如下：
$\left\{\begin{array}{l}\hat{\boldsymbol{X}}_{i}^{-}=\boldsymbol{\varPhi}_{i / i-1} \hat{\boldsymbol{X}}_{i-1} \\\boldsymbol{P}_{i}^{-}=\boldsymbol{\varPhi}_{i / i-1} \boldsymbol{P}_{i-1} \boldsymbol{\varPhi}_{i / i-1}^{\mathrm{T}}+\boldsymbol{\varGamma}_{i / i-1} \boldsymbol{Q}_{i-1} \boldsymbol{\varGamma}_{i / i-1}^{\mathrm{T}}\end{array}\right.$
式中，$\hat{\boldsymbol{X}_{i}}^{-}$为当前时刻的预测值，$\hat{\boldsymbol{X}}_{i-1}$为上一时刻的最优估计值；$\boldsymbol{P}_{i}^{-}$为当前时刻的预测值协方差，$\boldsymbol{P}_{i-1}$为上一时刻的最优估计值协方差。
同样有卡尔曼滤波更新方程如下：
$\left\{\begin{array}{l}\boldsymbol{K}_{i}=\boldsymbol{P}_{i}^{-} \boldsymbol{H}_{i}^{\mathrm{T}}\left(\boldsymbol{H}_{i} \boldsymbol{P}_{i}^{-} \boldsymbol{H}_{i}^{\mathrm{T}}+\boldsymbol{R}_{i}\right)^{-1} \\\hat{\boldsymbol{X}}_{i}=\hat{\boldsymbol{X}}_{i}^{-}+\boldsymbol{K}_{i}\left(\boldsymbol{Y}_{i}-\boldsymbol{H}_{i} \hat{\boldsymbol{X}}_{i}^{-}\right) \\\boldsymbol{P}_{i}=\left(\boldsymbol{I}-\boldsymbol{K}_{i} \boldsymbol{H}_{i}\right) \boldsymbol{P}_{i}^{-}\end{array}\right.$
式中，$\boldsymbol{K}_{i}$为卡尔曼滤波增益系数；$\hat{\boldsymbol{X}}_{i}$为当前时刻最优估计值；$\boldsymbol{P}_{i}$为当前时刻最优估计值的协方差。
由于卫星状态最优估计值协方差矩阵$\boldsymbol{P}_{i}$一定是正定的，利用Cholesky可将其分解成2个三角矩阵的乘积，同时这2个三角矩阵互为转置。也就是说，对于$\boldsymbol{P}_{i}=\boldsymbol{\varDelta}_{i} \boldsymbol{\varDelta}_{i}^{\mathrm{T}}, \boldsymbol{\varDelta}_{i}$一定可以写成下三角矩阵的形式，代入式(21)可得：
$\begin{gather*}\boldsymbol{\varDelta}_{i} \boldsymbol{\varDelta}_{i}^{\mathrm{T}}=\boldsymbol{P}_{i}= \\\boldsymbol{\varDelta}_{i}^{-}\left(\boldsymbol{I}-\boldsymbol{\rho}_{i}^{-2} \boldsymbol{\varDelta}_{i}^{-\mathrm{T}} \boldsymbol{H}_{i}^{\mathrm{T}} \boldsymbol{H}_{i} \boldsymbol{\varDelta}_{i}^{-}\right) \boldsymbol{\varDelta}_{i}^{-\mathrm{T}} \end{gather*}$
式中，$\boldsymbol{\rho}_{i}^{2}=\boldsymbol{H}_{i} \varDelta_{i}^{-} \varDelta_{i}^{-\mathrm{T}} \boldsymbol{H}_{i}^{\mathrm{T}}+\boldsymbol{R}_{i}$，求$\boldsymbol{P}_{i}$的根等价于求$\boldsymbol{I}-\boldsymbol{\rho}_{i}^{-2} \boldsymbol{\varDelta}_{i}^{-\mathrm{T}} \boldsymbol{H}_{i}^{\mathrm{T}} \boldsymbol{H}_{i} \boldsymbol{\varDelta}_{i}^{-}$的根。利用待定系数法可求得当前时刻最优估计值的协方差平方根：
$\begin{equation*}\boldsymbol{\varDelta}_{i}=\boldsymbol{\varDelta}_{i}^{-}\left(\boldsymbol{I}-\boldsymbol{r}_{i}^{-1} \boldsymbol{\varDelta}_{i}^{-\mathrm{T}} \boldsymbol{H}_{i}^{\mathrm{T}} \boldsymbol{H}_{i} \boldsymbol{\varDelta}_{i}^{-}\right)\end{equation*}$
其中,
$\begin{equation*}\boldsymbol{r}_{i}^{-1}=\frac{2 \boldsymbol{\rho}_{i}^{2} \pm \sqrt{4 \boldsymbol{\rho}_{i}^{4}-4 \boldsymbol{\rho}_{i}^{2}\left(\boldsymbol{\rho}_{i}^{2}-\boldsymbol{R}_{i}\right)}}{2}\end{equation*}$
至此完成了EKF的量测更新。由于当前时刻预测值的协方差平方根$\boldsymbol{\varDelta}_{i}^{-}$未知，引入时间更新，式(20)中第2个公式可转化为：
$\begin{gather*}\boldsymbol{P}_{i}^{-}=\boldsymbol{\varDelta}_{i}^{-} \boldsymbol{\varDelta}_{i}^{-\mathrm{T}}=\boldsymbol{\varPhi}_{i / i-1} \boldsymbol{\varDelta}_{i-1} \boldsymbol{\varDelta}_{i-1}^{\mathrm{T}} \boldsymbol{\varPhi}_{i / i-1}^{\mathrm{T}}+ \\\boldsymbol{\varGamma}_{i / i-1} \boldsymbol{Q}_{i-1}^{1 / 2}\left(\boldsymbol{Q}_{i-1}^{1 / 2}\right)^{\mathrm{T}} \boldsymbol{\varGamma}_{i / i-1}^{\mathrm{T}}= \\{\left[\begin{array}{ll}\boldsymbol{\varPhi}_{i / i-1} \boldsymbol{\varDelta}_{i-1} & \boldsymbol{\varGamma}_{i / i-1} \boldsymbol{Q}_{i-1}^{1 / 2}\end{array}\right]\left[\begin{array}{c}\boldsymbol{\varDelta}_{i-1}^{\mathrm{T}} \boldsymbol{\varPhi}_{i / i-1}^{\mathrm{T}} \\\left(\boldsymbol{Q}_{i-1}^{1 / 2}\right)^{\mathrm{T}} \boldsymbol{\varGamma}_{i / i-1}^{\mathrm{T}}\end{array}\right]} \end{gather*}$
令$\boldsymbol{A}=\left[\begin{array}{c}\boldsymbol{\varDelta}_{i-1}^{\mathrm{T}} \boldsymbol{\varPhi}_{i / i-1}^{\mathrm{T}} \\ \left(\boldsymbol{Q}_{i-1}^{1 / 2}\right)^{\mathrm{T}} \boldsymbol{\varGamma}_{i / i-1}^{\mathrm{T}}\end{array}\right]$，根据Householder变换，矩阵$\boldsymbol{A}$一定可以进行QR正交分解，即$\boldsymbol{A}= \boldsymbol{Q}\left[\begin{array}{c}\boldsymbol{D} \\ 0\end{array}\right]$，其中$\boldsymbol{Q}$为正交矩阵，$\boldsymbol{D}$为上三角矩阵，因此$\boldsymbol{\varDelta}_{i}^{-\mathrm{T}}=\boldsymbol{D}$。通过$\boldsymbol{\varDelta}_{i-1}$计算出$\boldsymbol{\varDelta}_{i}^{-}$，即完成了EKF的时间更新。
综合量测更新和时间更新，可将上一时刻最优估计值作为当前时刻输入量计算出当前时刻最优估计值，同时利用矩阵的平方根减小过程中的计算量。由于卫星状态向量维数庞大，利用序贯滤波处理，最终快速实现卫星状态向量的逐次滤波。
2 实时精密定轨实验
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为验证本文实时精密定轨算法精度，本文使用DSP作为算法运行核心处理器，外挂接口管理FPGA、FLASH、CAN芯片，联合星务计算机、GNSS接收机和地面数据处理系统搭建实时精密定轨硬件平台，具体参数如表 1所示。
外挂FPGA芯片主要用于精细控制时钟频率、实现数据交互以及完成CAN总线通信功能，DSP与FPGA之间通过EMIF总线进行高效的信息交互；FLASH芯片用于固化启动代码、应用程序以及配置文件，FPGA芯片通过引导操作将这些数据加载至DSP芯片，以启动定轨计算任务。为了确保数据的可靠性和正确性，FPGA芯片对FLASH中的数据实施三模冗余校验机制；DDR3RAM作为代码运行区，存储主控代码和有效数据，通过软件编程的方式实现三模冗余校验；星务计算机模拟软件通过CAN总线发送各项总线指令，用于控制定轨的启停以及钟差信息的传输；GNSS接收机则通过CAN总线接收目标卫星的广播星历和观测数据；DSP综合处理钟差、广播星历和观测数据，经过质量控制、轨道积分以及参数估计等步骤，实现对在轨卫星的实时精密定轨以及轨道预报功能；地面数据处理系统模拟数据传输系统，通过LVDS接收定轨结果，并对定轨精度进行分析评估。
表 2给出逐次解算某低轨卫星精密轨道的相关内容和处理策略。采样时间为7 d，采样间隔为30 s，观测量为北斗双频伪距和载波相位；通过使用地面站计算的钟差改正数，得到采样时间内的实际钟差，再交由星务计算机传输；考虑轨道高度较低，将EIGEN_GL04C重力模型截断到120阶×120阶以满足描述地球的非球面重力势的需求；大气阻力对在轨卫星的影响可以不考虑；对于太阳辐射压，采用Box-wing模型描述卫星结构特征，并在滤波器中引入太阳光压系数，以弥补太阳光压模型的不足；为了补偿建模误差或未建模的动力学误差对定轨精度的影响，引入沿切向和法向的经验力模型；另外，为最大程度降低模糊度对轨道精度的影响，采用常数模型估计的策略。
3 结果分析
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3.1 实时轨道精度分析
为验证本文算法计算的轨道精度，本文利用某同步轨道卫星作为在轨卫星，选取2018-02-01起后续7 d的观测量计算精密轨道，根据《北斗卫星导航系统RNSS公开服务性能评估方法》评估实时精密轨道计算精度; 将对应时间的事后精密产品作为参考轨道，两者对比进行精度分析。本文定轨算法每2 h定轨一次，单次计算时间小于35 s，内存占用小于4 MB，满足星载资源约束。
图 1为02-01—02-07一周内在国际地球参考框架(ITRF)下切向(A)、法向(C)、径向(R)和三维(3D)定轨误差以及相应的均方根值。
由图 1可知，02-01定轨初始阶段无原始数据积累，算法收敛较慢，定轨精度较差，其定轨三维误差RMS远高于后续定轨。随着定轨时间增加，原始数据积累，定轨误差在任意方向均迅速收敛，最终三维RMS收敛至13 cm左右，充分满足卫星在轨任务轨道精度需求；24 h原始数据积累即可使误差下降至20 cm以下，可满足紧急任务轨道精度需求。图 2为7 d内EKF算法卫星定轨残差和均值。
EKF定轨残差为实际观测量和基于上一时刻最优估计的预测值之间的差异，通常可用于评估EKF滤波器性能和状态估计的准确性。图 2中A、C、R三个方向残差均值量级均在10-4cm，基本围绕零均值波动，3D残差也仅为0.95 cm。若剔除图像中的异常值，均值可进一步缩小，说明本文EKF滤波器估计较为准确。图 3为定轨7 d内每天的定轨误差RMS统计，表 3为具体数据统计结果。
根据图 3和表 3的分析结果，定轨误差在第1天后迅速收敛，并在第4天基本达到收敛极限。这表明，仅积累3 d的原始数据即可使本文提出的算法精度达到最高水平。在实际的在轨应用中，若做好单粒子防护措施，该算法可作为常加电单机在轨运行的依据。具体操作为：在保留3 d原始数据的基础上，每2 h丢弃最初2 h的数据，然后循环运行。这种运行方式不仅可以持续保持算法的最高精度，还能有效节省缓存空间，进而有助于缩小硬件的体积。
3.2 轨道预报精度分析
卫星并不需要一直进行轨道预报，在轨运行时应通过任务触发进行突发计算，可极大节省硬件算力。本文选取同一颗在轨卫星，起始时间相同，利用预报起始时间前最多3 d原始数据作为初值进行2 h轨道预报，并将相应时间的事后精密产品作为参考轨道，两者对比进行精度分析。本文轨道预报算法单次计算时间小于35 s，内存占用小于4 MB，满足星载资源约束和在轨任务的时间要求。
图 4、图 5和图 6分别给出了02-01、02-02和02-03 06:00—08:00轨道预报与事后精密产品在ITRF下A、C、R、3D方向的误差以及相应的RMS。
由图 4~6可知，随着原始数据积累，轨道预报误差同样呈现缩小趋势，三维轨道预报误差在02-01为100.79 cm，02-02缩小至29.94 cm，02-03收敛于18.75 cm，相比于利用实时观测量计算的实时轨道，预报轨道精度略有下降，但仍收敛于20 cm以下。轨道预报精度与滤波器训练状态有关，原始轨道数据积累时间越长，滤波器训练精度越高，预报得到的轨道精度越高。同时发现，滤波器在更短的时间内效果更佳，轨道预报在前1 h误差较小，后1 h逐渐增大，最后时刻有明显发散趋势。
图 7为每日的轨道预报误差RMS统计，表 4根据不同预报时间统计了轨道预报误差的RMS值。
综合图 4~7和表 4分析，以72 h定轨积分器为基础预报2 h，误差可收敛在18 cm左右，满足基本任务精度要求。
72 h预报轨道配合§3.1的72 h实时定轨，在轨卫星只需满足轨道数据存储空间和计算算力需求，滚动执行即可满足卫星在轨参数自主解算及成像任务需求。
4 结语
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本文基于EKF算法实现了卫星实时轨道解算与短时轨道预报，并对其精度开展误差分析。实验结果表明，在完成初始数据积累后，实时轨道解算精度呈现快速收敛特性，72 h定轨三维位置精度达15.18 cm；当轨道确定滤波器完成充分训练后，系统可支持2 h有效预报窗口，其预报轨道三维精度达18.75 cm，满足卫星在轨参数自主解算及成像任务需求。本算法基于DSP硬件平台成功实现，验证了星载嵌入式系统在实时轨道处理方面的工程可行性。该研究成果为实现在轨卫星参数自主解算-成像规划-数据处理全链路闭环提供了技术支撑，可有效缓解星地传输资源受限问题，对提升卫星自主运行能力具有一定工程应用价值。
基金
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  • 中科卫星(E2H3160103)
文献
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2026年第46卷第1期
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doi: 10.14075/j.jgg.2024.12.577
  • 接收时间：2024-12-23
  • 首发时间：2026-10-29
  • 出版时间：2026-01-15
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  • 收稿日期：2024-12-23
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    1 中国科学院空天信息创新研究院, 北京, 100094

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费玉龙, 副研究员, 主要从事合成孔径雷达成像算法和控制研究, E-mail:
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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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