基于紧耦合TDOA/AOA的鲁棒SOP定位算法

doi:10.7527/S1000-6893.2025.32402

Abstract

Abstract:

The navigation and positioning system based on Signals of Opportunity （SOP） serves as an effective supplement when the Global Navigation Satellite System （GNSS） is unavailable. Accurate SOP positioning is foundational to achieving high-precision navigation. To this end， this paper proposes a robust positioning algorithm， namely the Two-step Geometrical Bias-restrain （Ts-GBr）， which integrates Time Difference of Arrival （TDOA） and Angle of Arrival （AOA）. Firstly， we linearly express the nonlinear positioning equations using a tightly coupled method， thereby mitigating the influence of initial values on convergence in traditional nonlinear positioning algorithms. Secondly， given that the Two-step Weighted Least Squares （Ts-WLS） method approximates weighting matrices and overlooks higher-order error terms， resulting in poor robustness and threshold effects， this paper proposes a more robust algorithm Ts-GBr. Ts-GBr constructs weighting matrices by evaluating positioning errors from heterogeneous measurements， enhancing the accuracy of initial positioning. To further eliminate the impact of measurement errors on positioning performance， Ts-GBr recalculates the covariance matrix of positioning errors in the second step and establishes constraints for the tightly coupled positioning model. Theoretical analysis proves that Ts-GBr is an unbiased estimation algorithm and can approximate the Cramer Rao Bound （CRB）. Finally， the performance of Ts-GBr is tested under conditions involving different measurement errors， receiver motion trajectories， and far/near-range SOP scenarios. Results indicate that Ts-GBr exhibits greater robustness. Based on the Root Mean Square Error （RMSE） analysis of the average results for far/near-range SOP positioning errors， Ts-GBr improves positioning performance by approximately 50% compared to Ts-WLS.

Key words: SOP, TDOA/AOA, bias-suppression, CRB, tightly coupled

CLC Number:

V249.32⁺4

Hao XU, Rongling LANG, Ya FAN. Robust SOP positioning algorithm based on tightly coupled TDOA/AOA[J]. Acta Aeronautica et Astronautica Sinica, 2026, 47(5): 332402.

Figures/Tables 11

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Fig.7

Table 1

Location performance of Ts-WLS and Ts-GBr for near-range SOP

$σ θ$ /（°）	Ts-WLS算法RMSE，Ts-GBr算法RMSE/m
$σ θ$ /（°）	$σ τ 2 = 100$	$σ τ 2 = 300$	$σ τ 2 = 500$	$σ τ 2 = 700$	$σ τ 2 = 900$
0.5	3.1，1.4	5.4，1.5	7.8，1.5	10.5，1.5	12.5，1.6
2.5	9.4，7.3	11.3，7.6	15.9，7.6	17.5，7.7	21.6，8.2
4.5	15.7，13.3	18.4，13.5	22.1，14.1	24.3，14.6	27.6，14.8
6.5	25.4，19.1	28.2，19.7	31.3，21.5	34.5，21.8	38.1，22.3
8.5	36.2，27.7	39.7，28.4	41.6，28.8	45.2，28.6	48.3，29.3

Table 1

Table 2

Location performance of Ts-WLS and Ts-GBr for far-rang SOP

$σ θ$ /（°）	Ts-WLS算法RMSE，Ts-GBr算法RMSE/m
$σ θ$ /（°）	$σ τ 2 = 100$	$σ τ 2 = 300$	$σ τ 2 = 500$	$σ τ 2 = 700$	$σ τ 2 = 900$
0.5	88.3，52.5	94.1，61.2	117.6，79.1	136.8，99.3	151.3，110.5
2.5	165.9，58.4	172.7，73.4	186.1，88.7	199.4，107.8	207.8，121.6
4.5	284.3，69.5	287.5，88.2	292.3，102.7	308.9，112.2	319.2，148.4
6.5	495.2，87.7	510.4，106.2	536.9，114.3	550.7，127.1	558.7，145.6
8.5	615.4，123.7	627.6，138.5	643.5，203.2	676.1，218.9	702.4，245.3

Table 2

Table 3

Table 4

References 42

[1]	JADE MORTON Y T， VAN DIGGELEN F， SPILKER J J Jr， et al. Position， navigation， and timing technologies in the 21st century： Integrated satellite navigation， sensor systems， and civil applications［M］. New York： John Wiley & Sons， 2020： 1115-1120.
[2]	DU Y S， QIN H L， ZHAO C. LEO satellites/INS integrated positioning framework considering orbit errors based on FKF［J］. IEEE Transactions on Instrumentation and Measurement， 2024， 73： 5501714.
[3]	ZHANG X D， WANG F Z， LI H B. An efficient method for cooperative multi-target localization in automotive radar［J］. IEEE Signal Processing Letters， 2021， 29： 16-20.
[4]	PSIAKI M L， SLOSMAN B D. Tracking digital FM OFDM signals for the determination of navigation observables［J］. NAVIGATION： Journal of the Institute of Navigation， 2022， 69（2）： navi.521.
[5]	YANG C， SOLOVIEV A. Mobile positioning with signals of opportunity in urban and urban canyon environments［C］∥2020 IEEE/ION Position， Location and Navigation Symposium （PLANS）. Piscataway： IEEE Press， 2020： 1043-1059.
[6]	NEINAVAIE M， KHALIFE J， KASSAS Z M. Cognitive opportunistic navigation in private networks with 5G signals and beyond［J］. IEEE Journal of Selected Topics in Signal Processing， 2022， 16（1）： 129-143.
[7]	ABDALLAH A A， KASSAS Z M. UAV navigation with 5G carrier phase measurements［J］. Proceedings of the 34th International Technical Meeting of the Satellite Division of the Institute of Navigation， 2021： 3294-3306.
[8]	NGUYEN A A， SHADRAM Z， KASSAS Z M. A lower bound for the error covariance of radio SLAM with terrestrial signals of opportunity［J］. Proceedings of the 34th International Technical Meeting of the Satellite Division of the Institute of Navigation， 2021： 2294-2306.
[9]	SHEN J Y， MOLISCH A F， SALMI J. Accurate passive location estimation using TOA measurements［J］. IEEE Transactions on Wireless Communications， 2012， 11（6）： 2182-2192.
[10]	CAO S， CHEN X， ZHANG X， et al. Combined weighted method for TDOA-based localization［J］. IEEE Transactions on Instrumentation and Measurement， 2020， 69（5）： 1962-1971.
[11]	YAMASAKI R， OGINO A， TAMAKI T， et al. TDOA location system for IEEE 802.11b WLAN［C］∥IEEE Wireless Communications and Networking Conference， 2005. Piscataway： IEEE Press， 2005： 2338-2343.
[12]	LANG R L， XU H， GAO F， et al. Improving DOA estimation of GNSS interference through sparse non-uniform array reconfiguration［J］. Chinese Journal of Aeronautics， 2025， 38（8）： 103384.
[13]	ZHENG Y， SHENG M， LIU J Y， et al. Exploiting AoA estimation accuracy for indoor localization： A weighted AoA-based approach［J］. IEEE Wireless Communications Letters， 2019， 8（1）： 65-68.
[14]	LANG R L， XU H， GAO F. Robust direction estimation of terrestrial signal via sparse non-uniform array reconfiguration under perturbations［J］. Remote Sensing， 2024， 16（18）： 3482.
[15]	TANG Z W， LANG R L， XU H. Sparse DOA estimation method for unknown terrestrial signals of opportunity［J］. Journal of Shanghai Jiaotong University （Science）， 2024： 1-11.
[16]	LI J Z， GUO F C， JIANG W L. A linear-correction least-squares approach for geolocation using FDOA measurements only［J］. Chinese Journal of Aeronautics， 2012， 25（5）： 709-714.
[17]	XU H， CHANG Q， LANG R L. A two-step algorithm for locating the emitter by FDOA［C］∥2023 IEEE International Conference on Consumer Electronics （ICCE）. Piscataway： IEEE Press， 2023.
[18]	TORRIERI D J. Statistical theory of passive location systems［J］. IEEE Transactions on Aerospace and Electronic Systems， 1984， AES-20（2）： 183-198.
[19]	CHAN Y T， HO K C. A simple and efficient estimator for hyperbolic location［J］. IEEE Transactions on Signal Processing， 1994， 42（8）： 1905-1915.
[20]	张月霞，段世茹. 智能反射面辅助NOMA系统AOA定位算法［J］. 半导体光电， 2024， 45（5）： 823-829.
	ZHANG Y X， DUAN S R. Reconfigurable intelligence surface assisted NOMA system for an angles of arrival localization algorithm［J］. Semiconductor Optoelectronics， 2024， 45（5）： 823-829 （in Chinese）.
[21]	李敬花，李亚鹏，宋得宁，等. 船舶舱室环境双级多元感知机蓝牙AoA定位方法［J］. 制造业自动化， 2024， 46（6）： 113-119.
	LI J H， LI Y P， SONG D N， et al. A bluetooth AoA positioning method based on dual-level MLP for location in ship onboard environment［J］. Manufacturing Automation， 2024， 46（6）： 113-119 （in Chinese）.
[22]	SAFAVI S， KHAN U A， KAR S， et al. Distributed localization： A linear theory［J］. Proceedings of the IEEE， 2018， 106（7）： 1204-1223.
[23]	GHOLAMI M R， GEZICI S， STROM E G. A concave-convex procedure for TDOA based positioning［J］. IEEE Communications Letters， 2013， 17（4）： 765-768.
[24]	BAYAT S， AMIRI R. Advances in UAV-assisted localization： Joint source and UAV parameter estimation［J］. IEEE Transactions on Vehicular Technology， 2023， 72（11）： 14268-14278.
[25]	XIONG W X， MOHANTY S， SCHINDELHAUER C， et al. Convex relaxation approaches to robust RSS-TOA based source localization in NLOS environments［J］. IEEE Transactions on Vehicular Technology， 2023， 72（8）： 11068-11073.
[26]	TOMIC S， BEKO M， TUBA M. A linear estimator for network localization using integrated RSS and AOA measurements［J］. IEEE Signal Processing Letters， 2019， 26（3）： 405-409.
[27]	ZHENG Q F， LUO L G， SONG H， et al. A RSSI-AOA-based UHF partial discharge localization method using MUSIC algorithm［J］. IEEE Transactions on Instrumentation and Measurement， 2021， 70： 9002309.
[28]	吴龙文，王宝莹，魏俊杰，等. 基于AOA的双机无源定位模型及其解算方法［J］. 系统工程与电子技术， 2020， 42（5）： 978-986.
	WU L W， WANG B Y， WEI J J， et al. AOA based dual-aircraft passive positioning model and its location methods［J］. Systems Engineering and Electronics， 2020， 42（5）： 978-986 （in Chinese）.
[29]	李志刚，吕杰，董小飞，等. TDOA、AOA算法在车辆出入管理中的应用研究［J］. 重庆交通大学学报（自然科学版）， 2019， 38（7）： 14-19.
	LI Z G， LV J， DONG X F， et al. Application of TDOA and AOA algorithm in vehicle access management［J］. Journal of Chongqing Jiaotong University （Natural Science）， 2019， 38（7）： 14-19 （in Chinese）.
[30]	WU H， CHEN S X， ZHANG Y H， et al. Robust structured total least squares algorithm for passive location［J］. Journal of Systems Engineering and Electronics， 2015， 26（5）： 946-953.
[31]	闫雷兵，陆音，张业荣. 基于改进最小二乘算法的TDOA/AOA定位方法［J］. 电波科学学报， 2016， 31（2）： 394-400.
	YAN L B， LU Y， ZHANG Y R. Improved least-squares algorithm for TDOA/AOA-based localization［J］. Chinese Journal of Radio Science， 2016， 31（2）： 394-400 （in Chinese）.
[32]	张怡，席彦彪，李刚伟，等. 基于卡尔曼滤波的TDOA/AOA混合定位算法［J］. 计算机工程与应用， 2015， 51（20）： 62-66.
	ZHANG Y， XI Y B， LI G W， et al. TDOA/AOA hybrid positioning algorithm based on Kalman filter in NLOS environment［J］. Computer Engineering and Applications， 2015， 51（20）： 62-66 （in Chinese）.
[33]	JIANG H Y， ZHANG K， SHEN C， et al. Hybrid location algorithm of TDOA/AOA based on extended Kalman filter［C］∥2021 IEEE 21st International Conference on Communication Technology （ICCT）. Piscataway： IEEE Press， 2021： 413-417.
[34]	胡爱华. 基于灾后复杂环境的无线定位关键技术研究［D］. 北京：北京邮电大学， 2022.
	HU A H. Research on key technology of wireless location based on post-disaster complex environment［D］. Beijing： Beijing University of Posts and Telecommunications， 2022 （in Chinese）.
[35]	ZHANG X H， TAN SIHONG S， HONG T YEW. An integrated closed-form fusion algorithm for TDOA/AOA［C］∥2016 19th International Conference on Information Fusion （FUSION）. Piscataway： IEEE Press， 2016： 1622-1629.
[36]	NUR-A-ALAM M， HAQUE M M. A least square approach for TDOA/AOA wireless location in WCDMA system［C］∥2008 11th International Conference on Computer and Information Technology. Piscataway： IEEE Press， 2008： 686-690.
[37]	MA C L， KLUKAS R， LACHAPELLE G. An enhanced two-step least squared approach for TDOA/AOA wireless location［C］∥IEEE International Conference on Communications， 2003. Piscataway： IEEE Press， 2003： 987-991.
[38]	CONG L， ZHUANG W H. Hybrid TDOA/AOA mobile user location for wideband CDMA cellular systems［J］. IEEE Transactions on Wireless Communications， 2002， 1（3）： 439-447.
[39]	YI P， HOU F X， ZHAO J S. Distributed cooperative localization based on a mix of single-site AOA and multi-site TDOA with heterogeneous sensors［C］∥2024 IEEE International Conference on Unmanned Systems （ICUS）. Piscataway： IEEE Press， 2024： 1191-1196.
[40]	LIU P H， WU Z， YAO Z， et al. Robust single-point localization technique using downlink TDOA-AOA fusion［C］∥2024 14th International Conference on Indoor Positioning and Indoor Navigation （IPIN）. Piscataway： IEEE Press， 2024.
[41]	MAHBAS A， COSMAS J， MUEEN S， et al. LOS analysis for localization in high frequency systems［J］. IEEE Transactions on Wireless Communications， 2024， 23（6）： 6425-6437.
[42]	陈永光，李昌锦，李修和. 三站时差定位的精度分析与推算模型［J］. 电子学报， 2004， 32（9）： 1452-1455.
	CHEN Y G， LI C J， LI X H. A precision analyzing & reckoning model in tri-station TDOA location［J］. Acta Electronica Sinica， 2004， 32（9）： 1452-1455 （in Chinese）.

参数	直线	圆形	半圆形	抛物线
Ts-GBr的定位误差/m	2.9	1.3	2.2	2.3
Ts-WLS的定位误差/m	6.5	3.2	3.7	4.5

参数	直线	圆形	半圆形	抛物线
Ts-GBr的定位误差/m	66.6	31.6	47.9	43.8
Ts-WLS的定位误差/m	110.8	44.2	60.8	57.5

Robust SOP positioning algorithm based on tightly coupled TDOA/AOA

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