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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 (Article in Progress)
Abstract
Accurate position estimation is critical for the effectiveness of automatic 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, allowing 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). 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,” Int. Conf. ICT Converg., pp. 349–354, 2024, doi: 10.1109/ICTC62082.2024.10827762.
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- 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 Trans. Ind. Electron., 2019.
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- C. Jia;, J. Ma;, and W. M. Kouw, “Multiple Variational Kalman-GRU for Ship Trajectory Prediction With Uncertainty,” vol. 61, no. 2, 2025, doi: 10.1109/TAES.2024.3491053.
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- Tao Zou; Weixiang Zeng; Wenlin Yang; Muk Chen Ong; Yunting Wang; Weilun Situ, “An Adaptive Robust Cubature Kalman Filter Based on Sage-Husa Estimator for Improving Ship Heave Measurement Accuracy,” 2023.
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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,” Int. Conf. ICT Converg., pp. 349–354, 2024, doi: 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 Appl., vol. 14, no. 11, pp. 1424–1437, 2020, doi: 10.1049/iet-cta.2019.1121.
A. P. A. Mohinder S Grewal, Kalman Filtering -- theory and practicing using matlab -- edisi 3. 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, doi: 10.3390/aerospace10121023.
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, doi: 10.1109/TIM.2022.3197775.
B. Cole and G. Schamberg, “Unscented Kalman filter for long-distance vessel tracking in geodetic coordinates,” Appl. Ocean Res., vol. 124, 2022, doi: 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 Intell. Veh. Symp., 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 Trans. Ind. Electron., 2019.
S. Zollo and B. Ristic, “On polar and versus Cartesian coordinates for target tracking,” ISSPA 1999 - Proc. 5th Int. Symp. Signal Process. Its Appl., vol. 2, no. February 1999, pp. 499–502, 1999, doi: 10.1109/ISSPA.1999.815719.
A. Budiyono, “Principles of GNSS, Inertial, and Multi-sensor Integrated Navigation Systems,” Ind. Robot An Int. J., vol. 39, no. 3, 2013, doi: 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., vol. 15, no. 1, 2023, doi: 10.3390/rs15010180.
A. Li and Z. Qiang, “Multi-sensor data fusion method based on adaptive Kalman filtering,” 2024, pp. 306–311. doi: 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, doi: 10.1109/LSENS.2021.3058119.
A. Y. N. and S. T. Pieter Abbeel, Adam Coates, Michael Montemerlo, “Discriminative Training of Kalman Filters,” J. Neurochem., vol. 52, no. 5, pp. 1401–1406, 2005, doi: 10.1111/j.1471-4159.1989.tb09186.x.
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. doi: 10.1109/OCEANS51537.2024.10682352.
L. Tian;, W. Xue;, and L. Cheng, “Hand Position Tracking based on Optimized Consistent Extended Kalman Filter,” 2022. doi: 10.1109/CCDC55256.2022.10033812.
H. S. Darling, “Do you have a standard way of interpreting the standard deviation? A narrative review,” Cancer Res. Stat. Treat., vol. 5, no. 4, pp. 728–733, 2022, doi: 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 Conf., vol. 232, pp. 1–4, 2018, doi: 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, doi: 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, doi: 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. doi: 10.1109/ICCMS.2010.444.
D. J. McLaughlin, “Gap free CONUS surveillance using dense networks of short range radars,” 2010. doi: 10.1109/ARRAY.2010.5613393.
Tao Zou; Weixiang Zeng; Wenlin Yang; Muk Chen Ong; Yunting Wang; Weilun Situ, “An Adaptive Robust Cubature Kalman Filter Based on Sage-Husa Estimator for Improving Ship Heave Measurement Accuracy,” 2023.
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, doi: 10.3390/s19061371.
G. Yu;, C. Li;, and B. Lu, “Processing 3D Flight Trajectory Data with Adaptive Kalman Filtering,” 2024. doi: 10.1109/ICCASIT62299.2024.10828030.