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Article(id=1322940888955106184, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322940829001724376, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2024.06.278, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1718121600000, receivedDateStr=2024-06-12, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793343286979, onlineDateStr=2026-10-30, pubDate=1747238400000, pubDateStr=2025-05-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793343286979, onlineIssueDateStr=2026-10-30, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793343286979, creator=13701087609, updateTime=1793343286979, updator=13701087609, issue=Issue{id=1322940829001724376, tenantId=1146029695717560320, journalId=1281212831689347082, year='2025', volume='45', issue='5', pageStart='441', pageEnd='550', issueExtLink='null', onlineDate='null', pubDate='1747238400000', pubDateStr='2025-05-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793343272686, creator='13701087609', updateTime=1793343272686, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=482, endPage=487, ext={EN=ArticleExt(id=1322940890205008778, articleId=1322940888955106184, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=Research on Clock Correction Positioning Method Based on Low Earth Orbit Single Satellite, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

Aiming at the problems of poor positioning accuracy, slow convergence speed, and large influence of clock drift in the case of single-satellite positioning methods, a joint pseudo-ranging/Doppler positioning model with clock-difference correction is proposed, which combines the mathematical model of clock-difference with the pseudo-ranging observation equations and Doppler observation equations, and eliminates the influence of the change of the receiver's clock-difference on the positioning accuracy through the simultaneous estimation of clock drift by the mathematical model of clock-difference and the Doppler observation equations. The proposed method is validated in a simulation scenario, and the results show that the optimal positioning accuracy of the pseudorange/Doppler joint positioning with clock difference correction can reach 219 m under different working conditions, which is 59.4% higher than that of the Doppler localization method; when the pseudorange measurement error is 1 m, the positioning accuracy can be up to 169 m, and the convergence time is around 300 s; when the frequency measurement error is 0.5 Hz, the positioning accuracy can be around 140 m.

, authors=Jia SU1, Xiaohui JIN1, Qingwu YI2, 3, *, Hongchi SUN3, authorsList=Jia SU, Xiaohui JIN, Qingwu YI, Hongchi SUN, authorCompany=null, correspAuthors=Qingwu YI, authorNote=

SU Jia, PhD, professor, majorsin wireless communication technology andtarget detection andtracking, E-mail:

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针对传统定位方法在单星情况下定位精度差、收敛速度慢、受钟漂影响较大的问题, 提出钟差修正的伪距/多普勒联合定位模型, 将钟差数学模型与伪距观测方程、多普勒观测方程结合, 通过钟差数学模型与多普勒观测方程同时估测钟漂, 消除接收机钟漂变化对定位精度的影响。在仿真场景下, 对所提方法进行验证, 结果表明, 钟差修正的伪距/多普勒联合定位在不同工况下的最优定位精度可达219 m, 相较于多普勒定位方法提升59.4%;伪距测量误差为1 m时, 定位精度达169 m, 收敛时间在300 s左右; 频率测量误差为0.5 Hz时, 定位精度在140 m左右。

, authors=苏佳1, 靳晓会1, 易卿武2, 3, *, 孙洪驰3, authorsList=苏佳, 靳晓会, 易卿武, 孙洪驰, authorCompany=null, correspAuthors=易卿武, authorNote=

苏佳, 博士, 教授, 主要研究方向为无线通信技术、目标检测与跟踪, E-mail:

, correspAuthorsNote=

易卿武, 研究员, 主要从事卫星导航系统装备及完好性技术研究, E-mail:

YI Qingwu, researcher, majors in satellite navigation system equipment and integrity technology, E-mail:

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SU Jia, PhD, professor, majorsin wireless communication technology andtarget detection andtracking, E-mail:

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with measurement errors at different frequencies for single satellite, figureFileSmall=tYLrsPMaMpXpwWLIN5kZ5A==, figureFileBig=GXHH03HAt3tWORs4OT5UBw==, tableContent=null), ArticleFig(id=1322940905287726027, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322940888955106184, language=CN, label=图5, caption=单星不同频率测量误差的定位精度, figureFileSmall=tYLrsPMaMpXpwWLIN5kZ5A==, figureFileBig=GXHH03HAt3tWORs4OT5UBw==, tableContent=null), ArticleFig(id=1322940905359029196, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322940888955106184, language=EN, label=Tab. 1, caption=

Positioning accuracy under clock drift variations

, figureFileSmall=null, figureFileBig=null, tableContent=
钟漂变化情况 RMSE/m
钟漂变化时 65
钟漂不变时 5.9×10-9
), ArticleFig(id=1322940905602298829, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322940888955106184, language=CN, label=表1, caption=

钟差变化下的定位精度

, figureFileSmall=null, figureFileBig=null, tableContent=
钟漂变化情况 RMSE/m
钟漂变化时 65
钟漂不变时 5.9×10-9
), ArticleFig(id=1322940905690379214, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322940888955106184, language=EN, label=Tab. 2, caption=

Positioning accuracy under different relative transit trajectories of satellites for single satellite

, figureFileSmall=null, figureFileBig=null, tableContent=
卫星过境相对轨迹 RMSE/m
钟差修正的伪距/多普勒联合定位 未钟差修正的伪距/多普勒联合定位 多普勒定位 伪距定位
工况1 885 3 450 1 826 3 817
工况2 489 1 830 1 470 2 098
工况3 219 2 155 540 2 425
), ArticleFig(id=1322940905753293775, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322940888955106184, language=CN, label=表2, caption=

单星不同卫星过境相对轨迹的定位精度

, figureFileSmall=null, figureFileBig=null, tableContent=
卫星过境相对轨迹 RMSE/m
钟差修正的伪距/多普勒联合定位 未钟差修正的伪距/多普勒联合定位 多普勒定位 伪距定位
工况1 885 3 450 1 826 3 817
工况2 489 1 830 1 470 2 098
工况3 219 2 155 540 2 425
), ArticleFig(id=1322940905812014032, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322940888955106184, language=EN, label=Tab. 3, caption=

Experimental results of positioning accuracy with different pseudorange measurement accuracy

, figureFileSmall=null, figureFileBig=null, tableContent=
伪距测量误差/m RMSE/m
钟差修正的伪距/多普勒联合定位 未钟差修正的伪距/多普勒联合定位 多普勒定位 伪距定位
1 169 3 488 736 3 516
2 213 3 504 739 3 532
3 298 3 525 754 3 554
4 391 3 546 633 3 578
), ArticleFig(id=1322940905870734289, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322940888955106184, language=CN, label=表3, caption=

不同伪距测量精度定位精度实验结果

, figureFileSmall=null, figureFileBig=null, tableContent=
伪距测量误差/m RMSE/m
钟差修正的伪距/多普勒联合定位 未钟差修正的伪距/多普勒联合定位 多普勒定位 伪距定位
1 169 3 488 736 3 516
2 213 3 504 739 3 532
3 298 3 525 754 3 554
4 391 3 546 633 3 578
), ArticleFig(id=1322940905933648850, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322940888955106184, language=EN, label=Tab. 4, caption=

Experimental results of positioning accuracy with measurement errors at different frequencies

, figureFileSmall=null, figureFileBig=null, tableContent=
多普勒测量误差/Hz RMSE/m
钟差修正的伪距/多普勒联合定位 未钟差修正的伪距/多普勒联合定位 多普勒定位 伪距定位
0.5 140 3 495 335 3 521
1 156 3 509 656 3 536
2 185 3 512 1 064 3 538
3 191 3 515 2 719 3 530
), ArticleFig(id=1322940906088838099, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322940888955106184, language=CN, label=表4, caption=

不同频率测量误差的定位精度实验结果

, figureFileSmall=null, figureFileBig=null, tableContent=
多普勒测量误差/Hz RMSE/m
钟差修正的伪距/多普勒联合定位 未钟差修正的伪距/多普勒联合定位 多普勒定位 伪距定位
0.5 140 3 495 335 3 521
1 156 3 509 656 3 536
2 185 3 512 1 064 3 538
3 191 3 515 2 719 3 530
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基于低轨单星的钟差修正定位方法
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苏佳 1 , 靳晓会 1 , 易卿武 2, 3, * , 孙洪驰 3
大地测量与地球动力学 | 现代大地测量 2025,45(5): 482-487
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大地测量与地球动力学 |现代大地测量 2025 , 45 (5) : 482 -487
基于低轨单星的钟差修正定位方法
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苏佳1 , 靳晓会1, 易卿武2, 3, * , 孙洪驰3
作者信息
  • 1 河北科技大学信息科学与工程学院, 石家庄市裕翔街26号, 050018
  • 2 西安电子科技大学雷达信号处理全国重点实验室, 西安市太白南路2号, 710071
  • 3 卫星导航系统与装备技术国家重点实验室, 石家庄市中山西路589号, 050081
通讯作者:

易卿武, 研究员, 主要从事卫星导航系统装备及完好性技术研究, E-mail:

YI Qingwu, researcher, majors in satellite navigation system equipment and integrity technology, E-mail:

作者简介:

苏佳, 博士, 教授, 主要研究方向为无线通信技术、目标检测与跟踪, E-mail:

Research on Clock Correction Positioning Method Based on Low Earth Orbit Single Satellite
Jia SU1 , Xiaohui JIN1, Qingwu YI2, 3, * , Hongchi SUN3
Affiliations
  • 1 School of Information Science and Engineering, Hebei University of Science and Technology, 26 Yuxiang Street, Shijiazhuang 050018, China
  • 2 National Key Laboratory of Radar Single Processing, Xidian University, 2 South-Taibai Road, Xi'an 710071, China
  • 3 State Key Laboratory of Satellite Navigation System and Equipment Technology, 589 West-Zhongshan Road, Shijiazhuang 050081, China

    • SU Jia, PhD, professor, majorsin wireless communication technology andtarget detection andtracking, E-mail:

出版时间: 2025-05-15 doi: 10.14075/j.jgg.2024.06.278
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摘要
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针对传统定位方法在单星情况下定位精度差、收敛速度慢、受钟漂影响较大的问题, 提出钟差修正的伪距/多普勒联合定位模型, 将钟差数学模型与伪距观测方程、多普勒观测方程结合, 通过钟差数学模型与多普勒观测方程同时估测钟漂, 消除接收机钟漂变化对定位精度的影响。在仿真场景下, 对所提方法进行验证, 结果表明, 钟差修正的伪距/多普勒联合定位在不同工况下的最优定位精度可达219 m, 相较于多普勒定位方法提升59.4%;伪距测量误差为1 m时, 定位精度达169 m, 收敛时间在300 s左右; 频率测量误差为0.5 Hz时, 定位精度在140 m左右。

低轨卫星  /  钟差修正  /  伪距  /  多普勒

Aiming at the problems of poor positioning accuracy, slow convergence speed, and large influence of clock drift in the case of single-satellite positioning methods, a joint pseudo-ranging/Doppler positioning model with clock-difference correction is proposed, which combines the mathematical model of clock-difference with the pseudo-ranging observation equations and Doppler observation equations, and eliminates the influence of the change of the receiver's clock-difference on the positioning accuracy through the simultaneous estimation of clock drift by the mathematical model of clock-difference and the Doppler observation equations. The proposed method is validated in a simulation scenario, and the results show that the optimal positioning accuracy of the pseudorange/Doppler joint positioning with clock difference correction can reach 219 m under different working conditions, which is 59.4% higher than that of the Doppler localization method; when the pseudorange measurement error is 1 m, the positioning accuracy can be up to 169 m, and the convergence time is around 300 s; when the frequency measurement error is 0.5 Hz, the positioning accuracy can be around 140 m.

low earth orbit satellite  /  clock bias correction  /  pseudorange  /  Doppler
苏佳, 靳晓会, 易卿武, 孙洪驰. 基于低轨单星的钟差修正定位方法. 大地测量与地球动力学, 2025 , 45 (5) : 482 -487 . DOI: 10.14075/j.jgg.2024.06.278
Jia SU, Xiaohui JIN, Qingwu YI, Hongchi SUN. Research on Clock Correction Positioning Method Based on Low Earth Orbit Single Satellite[J]. Journal of Geodesy and Geodynamics, 2025 , 45 (5) : 482 -487 . DOI: 10.14075/j.jgg.2024.06.278
正文
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目前，利用低轨星座实现全球互联网通信、导航等的构想受到越来越多的关注，Iridium[1]、SpaceX[2]、Centi-Space、GW等大型低轨星座项目被相继提出[3]。低轨卫星不仅可以增强卫星导航信号，作为全球导航卫星系统(global navigation satellite system, GNSS)的有效提升与补充，还可以通过通信系统和导航系统的融合，播发独立测距信号，形成备份的定位导航能力。低轨卫星定位技术的研究为解决GNSS不可用情况下的导航定位问题提供了有效途径。在低轨定位方面，国内外学者进行了研究。文献[4]提出三星编队卫星TDOA定位方法，验证编队构型对信号源定位误差分布有直接影响。实验结果表明，单历元信号源定位的定位误差在510 m量级，多历元信号源定位误差在110 m量级。文献[5]提出一种获取低轨卫星多普勒测量值的接收机结构，在25颗低轨卫星时可以实现11 m的定位精度。文献[6]验证无人机利用2颗Orbcomm卫星的多普勒测量值数据辅助惯性导航系统，实现定位误差8.9 m。文献[7]设计一种轻量级接收设备，接收6颗星链卫星的信号，定位误差达28.9 m。文献[8]采用单程双向测距方法消除时钟误差，通过随机切换卫星的方法进行测量，提升单星定位精度。文献[9]提出利用多次频率测量联合估计辐射源位置的方法，实现低轨双星无源定位。文献[10]对铱星STL突发信号体制进行研究，并采用伪距和多普勒测量值联合定位方法进行位置解算，实现了优于100 m的定位精度。但用户终端大部分时间段可见星为2~4颗，并未考虑只有单星的情况。文献[11]通过对观测量和定位精度的性能分析验证低轨通导融合信号的安全定位技术体制，在单星情况下，钟漂变化时的定位精度为200~300 m，钟漂不变时的定位精度为50~100 m。
综上所述，目前对低轨卫星的研究主要集中在多星定位，文献[4-7, 9-10]均是在多星情况下进行实验，单星定位的研究还不够全面，文献[11]仅讨论了钟漂变化对定位精度的影响，并未提出解决办法。本文针对目前低轨单星定位存在的受钟漂变化影响较大、定位精度差的问题，提出钟差修正的伪距/多普勒联合定位方法。利用从低轨卫星信号中提取的多普勒测量值和伪距，同时利用钟差数学模型约束钟漂变化进行联合定位，以降低在单星情况下钟漂变化带来的影响。
1 定位算法原理
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1.1 多普勒定位算法
接收机第i时刻的多普勒观测方程可表示为：
$f_i(\boldsymbol{u})=\frac{f^s}{c} \boldsymbol{v}_i \cdot \frac{\boldsymbol{x}_r-\boldsymbol{x}_i^s}{\left|\boldsymbol{x}_r-\boldsymbol{x}_i^s\right|}+\delta f_{r_i}+\varepsilon_{f_i}$
式中，u为接收机位置坐标和接收机钟漂4个未知量矢量；fs为卫星发射信号频率；c为光速；vi为卫星运动速度矢量；xr=[x, y, z]T为接收机位置矢量；$\boldsymbol{x}_i^s=\left[x_i^s, y_i^s, z_i^s\right]^{\mathrm{T}}$为卫星位置矢量；δfri为第i时刻接收机的钟漂；εfi为第i时刻多普勒频率测量误差。
1.2 钟差修正的伪距/多普勒联合定位原理
钟差数学模型为：
$\Delta t_r=\delta t_r+\left(t-t_0\right) \delta \dot{t}_r$
式中，Δtr为接收机钟差；δtr为接收机钟差初值；$\delta \dot{t}_r$为接收机钟漂；t为接收机收到当前信号的时间；t0为接收机时间初值。
多历元定位时，伪距定位受接收机钟漂的影响，钟漂不再是一个固定参数，利用传统的伪距单点定位无法求解。所以，将钟差模型与伪距/多普勒定位联合，用户定位的待估参数变为[xr, δtr, $\delta \dot{t}_r$]，通过多普勒观测方程与伪距观测方程同时约束钟漂与钟差，求解用户位置。钟差修正的伪距/多普勒联合定位算法数学模型如下：
$\left\{ {\begin{array}{*{20}{l}}{{\rho _1} = \sqrt {{{\left( {x_1^s - x} \right)}^2} + {{\left( {y_1^s - y} \right)}^2} + {{\left( {z_1^s - z} \right)}^2}} + }\\\begin{array}{l}\delta {t_{{r_1}}} + \left( {{t_1} - t} \right)\delta {{\dot t}_{{r_1}}}\quad \\\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{}&{}&{}&{}&{}\end{array}}&{}&{}&{}&{}&{}&{}\end{array} \vdots \end{array}\\{{\rho _n} = \sqrt {{{\left( {x_n^s - x} \right)}^2} + {{\left( {y_n^s - y} \right)}^2} + {{\left( {z_n^s - z} \right)}^2}} + }\\{\delta {t_{{r_n}}} + \left( {{t_n} - t} \right)\delta {{\dot t}_{{r_n}}}}\\{{f_1} = \frac{{{f^s}}}{c}\boldsymbol{v}_1^s \cdot \frac{{{\boldsymbol{x}_r} - \boldsymbol{x}_1^s}}{{\left\| {{\boldsymbol{x}_r} - \boldsymbol{x}_1^s} \right\|}} + \delta {{\dot t}_{{r_1}}}}\\{\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{}&{}\end{array}}&{}&{}\end{array}}&{}&{}\end{array}}&{}\end{array} \vdots }\\{{f_n} = \frac{{{f^s}}}{c}\boldsymbol{v}_n^s \cdot \frac{{{\boldsymbol{x}_r} - \boldsymbol{x}_n^s}}{{\left\| {{\boldsymbol{x}_r} - \boldsymbol{x}_n^s} \right\|}} + \delta {{\dot t}_{{r_n}}}}\end{array}} \right.$
式中，前n行方程为伪距定位的观测方程，ρ1~ρn为伪距测量值；后n行方程为多普勒定位观测方程，f1~fn为多普勒频率测量值。
令初始解为u0=[$x_0, y_0, z_0, \delta \dot{t}_{r_0}, \delta t_{r_0}$]T, 将式(3)按照一阶泰勒方程展开，得到如下方程：
$\left\{\begin{array}{l}\rho_i\left(x, y, z, \delta \dot{t}_r, \delta t_r\right)=\rho_i\left(x_0, y_0, z_0, \delta \dot{t}_{r_0}, \delta t_{r_0}\right)+ \\\quad \frac{\partial \rho_i}{\partial x}\left(x-x_0\right)+\frac{\partial \rho_i}{\partial y}\left(y-y_0\right)+\frac{\partial \rho_i}{\partial z}\left(z-z_0\right)+ \\\quad\left(t-t_0\right)\left(\delta \dot{t}_{r_i}-\delta \dot{t}_{r_0}\right)+\left(\delta t_{r_i}-\delta t_{r_0}\right)+\text { h. o. } \mathrm{t} \\f_i\left(x, y, z, \delta \dot{t}_r, \delta t_r\right)=f_i\left(x_0, y_0, z_0, \delta \dot{t}_{r_0}, \delta t_{r_0}\right)+ \\\quad \frac{\partial f_i}{\partial x}\left(x-x_0\right)+\frac{\partial f_i}{\partial y}\left(y-y_0\right)+\frac{\partial f_i}{\partial z}\left(z-z_0\right)+ \\\quad\left(\delta \dot{t}_{r_i}-\delta \dot{t}_{r_0}\right)+\text { h. o. } \mathrm{t}\end{array}\right.$
忽略高阶项(h.o.t)，将多个观测方程联立并利用矩阵形式表示：
$\Delta \boldsymbol{y}=\boldsymbol{H} \Delta \boldsymbol{u}+\boldsymbol{\varepsilon}$
式中，Δy =[Δρ, Δf]T为伪距和频率测量值与估计值之差；Δu=[$\Delta x, \Delta y, \Delta z, \Delta \delta \dot{t}_r, \Delta \delta t_r$]T=[x－x0, y－y0, z－z0, $\dot{\delta t_r}-\dot{\delta t_{r_0}}$, δtr－δtr0]T为接收机位置分量、接收机钟差与接收机钟漂的状态更新量；ε为误差向量；状态转移矩阵H=[L, G]T，其中：
$\boldsymbol{L}=\left[\begin{array}{ccccc}l_{x_1} & l_{y_1} & l_{z_1} & t-t_0 & 1 \\l_{x_2} & l_{y_2} & l_{z_2} & t-t_0 & 1 \\\vdots & \vdots & \vdots & \vdots & \vdots \\l_{x_n} & l_{y_n} & l_{z_n} & t-t_0 & 1\end{array}\right]$
$\boldsymbol{G}=\left[\begin{array}{ccccc}g_{x_1} & g_{y_1} & g_{z_1} & 1 & 0 \\g_{x_2} & g_{y_2} & g_{z_2} & 1 & 0 \\\vdots & \vdots & \vdots & \vdots & \vdots \\g_{x_n} & g_{y_n} & g_{z_n} & 1 & 0\end{array}\right]$
式中，$l_{x_i}=\frac{x_i^s-x_0}{\left|\boldsymbol{x}_r-\boldsymbol{x}_i^s\right|}, l_{y_i}=\frac{y_i^s-y_0}{\left|\boldsymbol{x}_r-\boldsymbol{x}_i^s\right|},$$\begin{aligned}& l_{z_i}=\frac{z_i^s-z_0}{\left|\boldsymbol{x}_r-\boldsymbol{x}_i^s\right|} ; g_{x_i}=\frac{v_{i_x}}{\left|\boldsymbol{x}_r-\boldsymbol{x}_i^s\right|}- \\& \frac{\left(x_0-x_i^s\right)\left(\boldsymbol{x}_0-\boldsymbol{x}_i^s\right)^{\mathrm{T}} \boldsymbol{v}_i}{\left|\boldsymbol{x}_r-\boldsymbol{x}_i^s\right|^3}, g_{y_i}=\frac{v_{i_y}}{\left|\boldsymbol{x}_r-\boldsymbol{x}_i^s\right|}- \\& \frac{\left(y_0-y_i^s\right)\left(\boldsymbol{x}_0-\boldsymbol{x}_i^s\right)^{\mathrm{T}} \boldsymbol{v}_i}{\left|\boldsymbol{x}_r-\boldsymbol{x}_i^s\right|^3}, g_{z_i}=\frac{v_{i_z}}{\left|\boldsymbol{x}_r-\boldsymbol{x}_i^s\right|}- \\& \frac{\left(z_0-z_i^s\right)\left(\boldsymbol{x}_0-\boldsymbol{x}_i^s\right)^{\mathrm{T}} \boldsymbol{v}_i}{\left|\boldsymbol{x}_r-\boldsymbol{x}_i^s\right|^3} 。\end{aligned}$
利用最小二乘法求解，可得：
$\Delta \boldsymbol{u}=\left(\boldsymbol{H}^{\mathrm{T}} \boldsymbol{H}\right)^{-1} \boldsymbol{H}^{\mathrm{T}} \Delta \boldsymbol{y}$
$\|\Delta \boldsymbol{u}\|<\sigma$
式中，σ表示设定的阈值，根据经验，σ设置为10-6 m。在第1次求解完成之后，可以用求得的结果作为下一次求解的初始值进行迭代运算，直至2次迭代后的解算结果之差小于预设的迭代门限值，则求解过程收敛，求得的坐标即为用户终端的最终估计值。
综上所述，钟差修正的伪距/多普勒联合定位算法流程如图1所示。
2 实验结果分析
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通过仿真实验验证钟差修正后的定位性能。设置的仿真低轨星座参考铱星星座设计，卫星位于近地极地轨道的6个平面上，每个轨道面上均匀分布着10颗卫星。轨道高度为780 km，轨道倾角为30°。60颗卫星部分运动弧段分布如图2所示。仿真过程如下：首先采用轨道根数拟合低轨广播星历，获得精密星历；然后在观测量仿真时加入观测误差；最后将仿真数据进行钟差修正的伪距/多普勒联合定位(简称钟差修正的联合定位)，评估定位性能。根据目前实际测量器件的实际测量精度，在仿真实验中，接收机钟差初值设置为10~8 s，钟漂初值设置为10-9 s/s，钟漂变化率设置为10-13 s/(s/s)。
为了验证接收机钟漂变化对定位精度的影响，进行仿真实验。在仿真过程中，其他误差因素设置为0。定位精度用三维均方根误差(root mean square error, RMSE)评价：
$w=\sqrt{\frac{1}{N} \sum\limits_{i=1}^N\left[\left(x-x_r\right)^2+\left(y-y_r\right)^2+\left(z-z_r\right)^2\right]}$
式中，w表示三维均方根误差；N表示仿真次数；(xr, yr, zr)表示接收机实际位置。
由表1可知，在其他误差为0时，接收机钟漂变化时的定位精度为65 m，接收机钟漂不变时的定位精度为5.9×10-9 m。可见接收机钟漂变化对定位精度影响较大。
2.1 卫星过境相对轨迹对定位精度的影响
通过实验验证不同工况下定位精度。图3(a)、3(b)、3(c)为3种工况的卫星过境轨迹，最大仰角分别为15°、35°、46°，表示3种情况下卫星与接收机相对位置；图3(d)、3(e)、3(f)为3种工况下对应的定位精度。由表2可知，在工况3时，即卫星仰角在46°情况下，钟差修正的联合定位的定位精度最高，在219 m左右；在工况1时，钟差修正的联合定位的定位精度最差，为885 m。但在3种工况下，钟差修正的联合定位效果优于未经钟差修正的伪距/多普勒联合定位(简称未钟差修正的联合定位)、多普勒定位以及伪距定位，验证了所提方法的有效性，同时证明卫星与接收机的相对位置会对定位精度产生影响。
2.2 观测量精度对定位精度的影响
2.2.1 伪距测量精度对定位精度的影响
在伪距观测值的仿真中，分别引入不同的伪距测量误差，得到伪距测量精度不同的伪距观测值进行定位。引入的伪距测量随机误差服从正态分布，均值为0，标准差分别为1 m、2 m、3 m、4 m，多普勒测距误差设置为1 Hz。定位结果采用RMSE进行评价，实验结果如图4所示。对于不同的标准差随机生成20组伪距测量误差，取20次定位结果的平均值作为最终定位结果，数据见表3。
由图4可见，随着伪距测量误差的增大，钟差修正的伪距/多普勒联合定位的定位精度逐渐变差，但定位精度仍优于其他3种方法。由表3可知，在伪距测量误差为1 m时，定位精度为169 m左右(收敛时间在300 s左右)，提升效果最为显著，定位精度较未钟差修正的联合定位、多普勒定位以及伪距定位分别提升95.15%、77.04%、95.22%。伪距测量误差增大导致伪距定位精度变差，在钟差修正的联合定位中，伪距定位误差的增大影响整体定位结果，导致定位精度变差。在伪距测量误差为4 m时，定位精度较未钟差修正的联合定位、多普勒定位以及伪距定位分别提升89%、38.25%、89.08%。
2.2.2 多普勒测量精度对定位精度的影响
在多普勒观测值的仿真中，分别引入不同的频率测量误差，得到频率测量精度不同的多普勒观测值进行定位。引入的多普勒测量随机误差服从正态分布，均值为0，标准差分别为0.5 Hz、1 Hz、2 Hz、3 Hz，伪距测距误差设置为1 m。实验结果如图5所示，最终定位结果取20次随机生成频率测量误差定位结果的平均值，定位结果采用RMSE进行评价。
从图5可以看出，钟差修正的联合定位相较于其他3种方法收敛速度更快、定位精度更高。由表4可知，在频率测量误差为0.5 Hz时，钟差修正的联合定位的定位精度在140 m左右，定位精度较未钟差修正的联合定位、多普勒定位以及伪距定位分别提升96.0%、58.32%以及96.03%。频率测量误差增大导致多普勒定位精度变差，在钟差修正的联合定位中，多普勒定位误差的增大影响整体定位结果，导致定位精度变差。
3 结语
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本文针对可见卫星不足时定位精度受接收机钟漂影响的问题，提出钟差修正模型，结合伪距/多普勒联合定位进行仿真实验，验证了模型的有效性。主要结论如下：
1) 卫星过境相对轨迹影响卫星空间几何构型，在工况3时，定位精度最好，并且钟差修正的联合定位的定位精度优于未钟差修正的联合定位、多普勒定位以及伪距定位，验证了本文所提方法的有效性。
2) 通过多组伪距测量误差对比验证，伪距测量误差越小，钟差修正的联合定位精度越好，在伪距测量误差为1 m时，定位精度可达169 m左右。
3) 通过多组频率测量误差对比实验可知，频率测量误差越小，钟差修正的联合定位精度越好，在频率测量误差为0.5 Hz时，定位精度在140 m左右。
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  • 国家重点研发计划(2021YFF0603903)
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2025年第45卷第5期
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doi: 10.14075/j.jgg.2024.06.278
  • 接收时间：2024-06-12
  • 首发时间：2026-10-30
  • 出版时间：2025-05-15
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  • 收稿日期：2024-06-12
基金
National Key Research and Development Program of China(2021YFF0603903)
国家重点研发计划(2021YFF0603903)
作者信息
    1 河北科技大学信息科学与工程学院, 石家庄市裕翔街26号, 050018
    2 西安电子科技大学雷达信号处理全国重点实验室, 西安市太白南路2号, 710071
    3 卫星导航系统与装备技术国家重点实验室, 石家庄市中山西路589号, 050081

通讯作者:

易卿武, 研究员, 主要从事卫星导航系统装备及完好性技术研究, E-mail:

YI Qingwu, researcher, majors in satellite navigation system equipment and integrity technology, E-mail:

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https://castjournals.cast.org.cn/joweb/ddcl/CN/10.14075/j.jgg.2024.06.278
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2种不同金属材料的力学参数
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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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