SSUKF-FA-RBF: A Kalman-Enhanced High-Precision Positioning Framework for BeiDou Navigation Using Firefly-Optimized Neural Estimation
Abstract
This study addresses the high-precision positioning requirements of the BeiDou Navigation System (BDS) by focusing on the commonly adopted BDS/Inertial Navigation System integrated navigation mode. A novel Spherical Simplex Unscented Kalman Filter (SSUKF) algorithm is proposed, featuring an improved sigma-point sampling strategy that enhances filtering accuracy while reducing computational overhead. In parallel, the Time Difference of Arrival (TDOA) method is combined with the Firefly Algorithm (FA) to optimize a Radial Basis Function (RBF) neural network, further enhancing positioning precision. Evaluation is conducted using an Ultra-Wideband TDOA dataset. Results show that the SSUKF algorithm significantly reduces positioning error. Specifically, the root means square error (RMSE) achieved by SSUKF is 0.1614 m-a reduction of 62.2% compared to the Extended Kalman Filter and 52.1% compared to the Unscented Kalman Filter. When integrated with the FA-optimized RBF neural network, the hybrid SSUKF-FA-RBF model achieves an RMSE of 0.127 m under high-noise conditions, demonstrating strong robustness and accuracy. In addition to its accuracy, the SSUKF algorithm offers improved computational efficiency, making it suitable for real-time, high-precision applications. Error analysis confirms the robustness and stability of the SSUKF-FA-RBF model across various environments. Under zero standard deviation noise, the model achieves 96.4% accuracy, 95.6% precision, and a 96.1% recall ratesubstantially outperforming comparative models. This study contributes an enhanced Kalman filtering method and an optimized positioning framework, advancing both accuracy and computational efficiency for the BDS. The proposed approach offers effective technical support for a wide range of high-precision positioning applications.References
Gao W, Zhou W, Tang C, et al. High-precision services of BeiDou navigation satellite system (BDS): Current state, achievements, and future directions. Satellite Navigation, 2024, 5(1): 20.
Xiao Y, Wang Z, Chao N, et al. Gravity field recovery of inter-satellite links between Beidou navigation satellite system (BDS) and LEO based on geodesy and time reference in space (GETRIS). Advances in Space Research, 2024, 73(12): 5889-5909.
Zhang S, Tu R, Gao Z, et al. LEO-enhanced GNSS/INS tightly coupled integration based on factor graph optimization in the urban environment. Remote Sensing, 2024, 16(10): 1782.
Zhang L, Lou Y, Song W, et al. Performance enhancement of PPP/SINS tightly coupled navigation based on improved robust maximum correntropy kalman filtering. Advances in Space Research, 2024, 74(5): 2078-2091.
Pang S, Zhang B, Lu J, et al. Application of IMU/GPS integrated navigation system based on adaptive unscented kalman filter algorithm in 3D positioning of forest rescue personnel. Sensors, 2024, 24(18): 5873.
Chen W, Wang T, Yao Z, et al. Analysis of the gain factors of 5G-assisted BDS RTK positioning in urban environments. Satellite Navigation, 2024, 5(1): 28.
Li F, Tu R, Geng F, et al. Combined BDS/5G seamless positioning scheme based on joint switching strategy of satellite elevation angle and CNR in complex urban environments. Measurement Science and Technology, 2024, 35(12): 126307.
Liaquat S, Faizan M, Chattha J N, et al. A framework for preventing unauthorized drone intrusions through radar detection and GPS spoofing. Ain Shams Engineering Journal, 2024, 15(5): 102707.
Chen W, Jing Y, Zhao S, et al. A distributed collaborative navigation strategy based on adaptive extended kalman filter integrated positioning and model predictive control for global navigation satellite system/inertial navigation system dual-robot. Remote Sensing, 2025, 17(4): 721.
Park G. Optimal vehicle position estimation using adaptive unscented Kalman filter based on sensor fusion. Mechatronics, 2024, 99(1): 103144.
Yin Y, Zhang J, Guo M, et al. Sensor fusion of GNSS and IMU data for robust localization via smoothed error state Kalman filter. Sensors, 2023, 23(7): 3676.
Wu Q, Li C, Shen T, et al. Improved adaptive iterated extended kalman filter for GNSS/INS/UWB-integrated fixed-point positioning. CMES-Computer Modeling in Engineering & Sciences, 2023, 134(3): 1.
Yuan Y, Li F, Chen J, et al. An improved Kalman filter algorithm for tightly GNSS/INS integrated navigation system. Math. Biosci. Eng, 2024, 21(1): 963-983.
Neusypin K, Kupriyanov A, Maslennikov A, et al. Investigation into the nonlinear Kalman filter to correct the INS/GNSS integrated navigation system. GPS Solutions, 2023, 27(2): 91.
Zhang T, Guo S, Fan L, et al. Improving BeiDou global navigation satellite system (BDS-3)-derived station coordinates using calibrated satellite antennas and station inter-system translation parameters. Remote Sensing, 2025, 17(3): 1.
Fu X, Zhao K, Sun Y, et al. A novel cascading partial ambiguity resolution method of BDS triple-frequency with inertial aiding for kinematic-to-kinematic relative positioning. Measurement Science and Technology, 2024, 36(1): 016316.
Cao Y, Lian W, Yang J, et al. Design of high-precision displacement safety monitoring and three-dimensional spatial alarm system for Beidou based on intelligent algorithms. Measurement: Sensors, 2024, 33(1): 101099.
Khodarahmi M, Maihami V. A review on Kalman filter models. Archives of Computational Methods in Engineering, 2023, 30(1): 727-747.
Bai Y, Yan B, Zhou C, et al. State of art on state estimation: Kalman filter driven by machine learning. Annual Reviews in Control, 2023, 56(1): 100909.
Feng S, Li X, Zhang S, et al. A review: State estimation based on hybrid models of Kalman filter and neural network. Systems Science & Control Engineering, 2023, 11(1): 2173682.
Rosafalco L, Conti P, Manzoni A, et al. EKF–SINDy: Empowering the extended Kalman filter with sparse identification of nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering, 2024, 431(1): 117264.
Takyi-Aninakwa P, Wang S, Liu G, et al. Enhanced extended-input LSTM with an adaptive singular value decomposition UKF for LIB SOC estimation using full-cycle current rate and temperature data. Applied Energy, 2024, 363(1): 123056.
Moradi-Sarvestani S, Jooshaki M, Fotuhi-Firuzabad M, et al. Incorporating direct load control demand response into active distribution system planning. Applied Energy, 2023, 339(2): 120897.
Peng S, Zuo J, Xu W, et al. Fractional moments based adaptive scaled unscented transformation for probabilistic power flow of AC-DC hybrid grids. IEEE Transactions on Power Systems, 2024, 39(5): 6249-6262.
Wang L, Chen T, Zou C. The TSCR method for precision estimation of ill-posed mixed additive and multiplicative random error model. Communications in Statistics-Simulation and Computation, 2024, 53(9): 4581-4595.
Zhang X, Kang J, Yu H. Intelligent Navigation system based on big data traffic system. scalable computing: Practice and experience, 2024, 25(2): 1.
Okada T, Dong S, Kuzuno R, et al. State observer of multibody systems formulated using differential algebraic equations. Multibody System Dynamics, 2024, 1(1): 1-31.
Kirmaz A, Şahin T, Michalopoulos D S, et al. ToA and TDoA estimation using artificial neural networks for high-accuracy ranging. IEEE Journal on Selected Areas in Communications, 2023, 41(12): 3816-3830.
Agbasi J C, Egbueri J C. Prediction of potentially toxic elements in water resources using MLP-NN, RBF-NN, and ANFIS: a comprehensive review. Environmental Science and Pollution Research, 2024, 31(21): 30370-30398.
Huo D, Chen J, Zhang H, et al. Intelligent prediction for digging load of hydraulic excavators based on RBF neural network. Measurement, 2023, 206(1): 112210.
DOI:
https://doi.org/10.31449/inf.v49i31.8787Downloads
Published
How to Cite
Issue
Section
License
I assign to Informatica, An International Journal of Computing and Informatics ("Journal") the copyright in the manuscript identified above and any additional material (figures, tables, illustrations, software or other information intended for publication) submitted as part of or as a supplement to the manuscript ("Paper") in all forms and media throughout the world, in all languages, for the full term of copyright, effective when and if the article is accepted for publication. This transfer includes the right to reproduce and/or to distribute the Paper to other journals or digital libraries in electronic and online forms and systems.
I understand that I retain the rights to use the pre-prints, off-prints, accepted manuscript and published journal Paper for personal use, scholarly purposes and internal institutional use.
In certain cases, I can ask for retaining the publishing rights of the Paper. The Journal can permit or deny the request for publishing rights, to which I fully agree.
I declare that the submitted Paper is original, has been written by the stated authors and has not been published elsewhere nor is currently being considered for publication by any other journal and will not be submitted for such review while under review by this Journal. The Paper contains no material that violates proprietary rights of any other person or entity. I have obtained written permission from copyright owners for any excerpts from copyrighted works that are included and have credited the sources in my article. I have informed the co-author(s) of the terms of this publishing agreement.
Copyright © Slovenian Society Informatika
