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Adaptive EKF-Based Ship Trajectory Estimation with Earth Curvature Modeling and Dynamic Noise Tuning
Corresponding Author(s) : Eddy Bambang Soewono
Kinetik: Game Technology, Information System, Computer Network, Computing, Electronics, and Control,
Vol. 11, No. 1, February 2026
Abstract
Accurate position estimation is critical for the effectiveness of automated weapon and navigation systems. Standard Extended Kalman Filter (EKF) models typically adopt flat-Earth assumptions and static noise covariances, which limit their accuracy in operational environments. This study proposes an optimized EKF framework that integrates two complementary approaches. First, ship trajectories are represented in Earth-Centered Earth-Fixed (ECEF) coordinates with a WGS-84 reference to account for Earth’s curvature. Second, process (Q) and measurement (R) covariances are adaptively determined using Joint Likelihood Maximization (JLM) with logarithmic scale exploration, enabling the filter to automatically identify the most accurate configuration. Each Q/R setting is evaluated within the EKF framework using root mean square error (RMSE) derived from radar data logs. The method was tested under short-history scenarios (5 and 10 data points) within an operational range of ±15 km, reflecting conditions commonly encountered in Combat Management Systems (CMS). The results show that while coordinate transformation alone provides only marginal improvements at short ranges, the combination of curvature modelling and adaptive Q/R tuning significantly reduces RMSE, achieving average errors approaching zero with high repeatability as measured by standard deviation. This research demonstrates a novel integration of geometric and statistical optimization in EKF design and highlights its applicability to ship trajectory estimation and defence systems.
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- Q. A. M. Thi, C. Lee, T. M. Tao, and C. H. Youn, “Tracking Vessel Activities with AIS Data using an Adaptive Extended Kalman Filter,” International Conference on ICT Convergence, pp. 349–354, 2024. https://doi.org/10.1109/ICTC62082.2024.10827762
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- T. Zou;, W. Zeng;, W. Yang;, M. C. Ong;, and Y. W. W. Situ, “An Adaptive Robust Cubature Kalman Filter Based on Sage-Husa Estimator for Improving Ship Heave Measurement Accuracy,” IEEE Sens J, vol. 23, no. 9, 2023. https://doi.org/10.1109/JSEN.2023.3260300
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References
Q. A. M. Thi, C. Lee, T. M. Tao, and C. H. Youn, “Tracking Vessel Activities with AIS Data using an Adaptive Extended Kalman Filter,” International Conference on ICT Convergence, pp. 349–354, 2024. https://doi.org/10.1109/ICTC62082.2024.10827762
C. Yang, Z. Gao, X. Huang, and T. Kan, “Hybrid extended-cubature Kalman filters for non-linear continuous-time fractional-order systems involving uncorrelated and correlated noises using fractional-order average derivative,” IET Control Theory and Applications, vol. 14, no. 11, pp. 1424–1437, 2020. https://doi.org/10.1049/iet-cta.2019.1121
M. S. Grewal and A. P. Andrews, Kalman Filtering -- theory and practicing using matlab -- edisi 3, 3rd ed. 2008.
A. Wondosen, Y. Debele, S. K. Kim, H. Y. Shi, B. Endale, and B. S. Kang, “Bayesian Optimization for Fine-Tuning EKF Parameters in UAV Attitude and Heading Reference System Estimation,” Aerospace, vol. 10, no. 12, 2023. https://doi.org/10.3390/aerospace10121023
S. Fossen and T. I. Fossen, “Extended kalman filter design and motion prediction of ships using live automatic identification system (AIS) Data,” Proceedings - 2018 2nd European Conference on Electrical Engineering and Computer Science, EECS 2018, no. December, pp. 464–470, 2018. https://doi.org/10.1109/EECS.2018.00092
A. Werries and J. M. Dolan, “Adaptive Kalman Filtering Methods for Low-Cost GPS / INS Localization for Autonomous Vehicles,” pp. 1–9, 2016.
D. F. Crouse, “Simulating aerial targets in 3D accounting for the earth’s curvature,” Journal of Advances in Information Fusion, vol. 10, no. 1, pp. 31–57, 2015.
B. Or and I. Klein, “A Hybrid Model and Learning-Based Adaptive Navigation Filter,” IEEE Trans Instrum Meas, vol. 71, no. Dvl, pp. 1–11, 2022. https://doi.org/10.1109/TIM.2022.3197775
B. Cole and G. Schamberg, “Unscented Kalman filter for long-distance vessel tracking in geodetic coordinates,” Applied Ocean Research, vol. 124, 2022. https://doi.org/10.1016/j.apor.2022.103205
B. Boulkroune, K. Geebelen, J. Wan, and E. van Nunen, “Auto-tuning extended Kalman filters to improve state estimation,” IEEE Intelligent Vehicles Symposium (IV), 2023.
M. Sato and M. Toda, “Adaptive Algorithms of Tuning and Switching Kalman and H∞ Filters and Their Application to Estimation of Ship Oscillation with Time-Varying Frequencies,” IEEE Transactions on Industrial Electronics, 2019.
S. Zollo and B. Ristic, “On polar and versus Cartesian coordinates for target tracking,” ISSPA 1999 - Proceedings of the 5th International Symposium on Signal Processing and Its Applications, vol. 2, no. February 1999, pp. 499–502, 1999. https://doi.org/10.1109/ISSPA.1999.815719
A. Budiyono, “Principles of GNSS, Inertial, and Multi-sensor Integrated Navigation Systems,” Industrial Robot: An International Journal, vol. 39, no. 3, 2013. https://doi.org/10.1108/ir.2012.04939caa.011
K. N. Baasch, L. Icking, F. Ruwisch, and S. Schön, “Coordinate Frames and Transformations in GNSS Ray-Tracing for Autonomous Driving in Urban Areas,” Remote Sens (Basel), vol. 15, no. 1, 2023. https://doi.org/10.3390/rs15010180
P. Abbeel, A. Coates, M. Montemerlo, A. Y. Ng, and S. Thrun, “Discriminative Training of Kalman Filters,” J Neurochem, vol. 52, no. 5, pp. 1401–1406, 2005.
A. Li and Z. Qiang, “Multi-sensor data fusion method based on adaptive Kalman filtering,” 2024, pp. 306–311. https://doi.org/10.1145/3638782.3638829
Y. Chen, W. Li;, and Y. Wang, “Online Adaptive Kalman Filter for Target Tracking With Unknown Noise Statistics,” vol. 5, no. 3, 2021. https://doi.org/10.1109/LSENS.2021.3058119
Q. Dong;, N. Wang;, C. Zou;, and L. H. K. Qu, “An Adaptive Order Variation Mathematical Modeling of Ship Maneuvering Motion Under Environmental Changes,” 2024. https://doi.org/10.1109/OCEANS51537.2024.10682352
L. Tian;, W. Xue;, and L. Cheng, “Hand Position Tracking based on Optimized Consistent Extended Kalman Filter,” 2022. https://doi.org/10.1109/CCDC55256.2022.10033812
H. S. Darling, “Do you have a standard way of interpreting the standard deviation? A narrative review,” Cancer Research, Statistics, and Treatment, vol. 5, no. 4, pp. 728–733, 2022. https://doi.org/10.4103/crst.crst_284_22
S. Hu and B. Yan, “Ship Tracking with Static Electric Field Based on Adaptive Progressive Update Extended Kalman Filter,” MATEC Web of Conferences, vol. 232, pp. 1–4, 2018. https://doi.org/10.1051/matecconf/201823204063
F. Deng, H.-L. Yang, and L.-J. Wang, “Adaptive Unscented Kalman Filter Based Estimation and Filtering for Dynamic Positioning with Model Uncertainties,” Int J Control Autom Syst, vol. 117, pp. 667–687, 2019. https://doi.org/10.1007/s12555-018-9503-4
C. Jia;, J. Ma;, and W. M. Kouw, “Multiple Variational Kalman-GRU for Ship Trajectory Prediction with Uncertainty,” vol. 61, no. 2, 2025. https://doi.org/10.1109/TAES.2024.3491053
W. Lv;, L. Wang;, and S. Jiang, “A Trajectory Simulation Model of the Short-Range Anti-ship Missile Based on Considering Curvature of the Earth,” 2010. https://doi.org/10.1109/ICCMS.2010.444
D. J. McLaughlin, “Gap free CONUS surveillance using dense networks of short range radars,” 2010. https://doi.org/10.1109/ARRAY.2010.5613393
T. Zou;, W. Zeng;, W. Yang;, M. C. Ong;, and Y. W. W. Situ, “An Adaptive Robust Cubature Kalman Filter Based on Sage-Husa Estimator for Improving Ship Heave Measurement Accuracy,” IEEE Sens J, vol. 23, no. 9, 2023. https://doi.org/10.1109/JSEN.2023.3260300
B. Ge, H. Zhang, L. Jiang, Z. Li, and M. M. Butt, “Adaptive unscented kalman filter for target tracking with unknown time-varying noise covariance,” Sensors (Switzerland), vol. 19, no. 6, 2019. https://doi.org/10.3390/s19061371
G. Yu;, C. Li;, and B. Lu, “Processing 3D Flight Trajectory Data with Adaptive Kalman Filtering,” 2024. https://doi.org/10.1109/ICCASIT62299.2024.10828030