Algorithm for estimating ballistic object trajectories using an electro-optical tracking system
DOI:
https://doi.org/10.24425/mms.2025.154667Abstract
This paper presents an estimation algorithm designed for tracking aerial ballistic objects using measurements from an electro-optical tracking system. Building upon our previous research, which focused on estimating the trajectory and flight parameters of an unguided short-range ballistic missile with motion constrained to two dimensions, this study introduces a more advanced and practical solution. The new approach uses a flight dynamics model formulated in a three-dimensional coordinate system. Unlike the previously developed algorithm, the one described in this paper accurately determines the object’s location within a geographically oriented horizontal reference frame. It also eliminates the need for prior knowledge of the shooting direction, which would be challenging to establish in practice, and more realistically models the influence of wind on the object’s motion in three dimensions. The paper includes the mathematical model of the tracking system, the extended Kalman filter used for estimating the ballistic object’s position and other flight parameters as well as simulation results for the proposed system.References
[1] Smagowski, P. (2020). Estymacja parametrów lotu niekierowanego pocisku rakietowego z wykorzystaniem optoelektronicznego systemu trajektograficznego. [Doctoral dissertation, Wojskowa Akademia Techniczna]. https://bip.wat.edu.pl/bip/dokumenty/postepowania-awansowe/psmagowski/rozprawa_doktorska_piotr_smagowski.pdf
[2] Kaniewski, P., Smagowski, P., & Konatowski, S. (2019). Ballistic Target Tracking with Use of Cinetheodolites. International Journal of Aerospace Engineering, 2019, 1-13. https://doi.org/10.1155/2019/3240898
[3] Kim, J. (2023b). Computationally efficient Ground-to-Air missile seeker based on camera images. IEEE Access, 11, 104839-104845. https://doi.org/10.1109/access.2023.3318745
[4] Moon, K., Kwon, H., Ryoo, C., & Sim, H. (2018). Trajectory estimation for a ballistic missile in ballistic phase using IR images. In Proceedings of the 9th International Conference on Mechanical and Aerospace Engineering (ICMAE) (pp. 173-177). https://doi.org/10.1109/icmae.2018.8467635
[5] Słowak, P., & Kaniewski, P. (2023). Homography augmented particle filter SLAM. Metrology and Measurement Systems. https://doi.org/10.24425/mms.2023.146420
[6] Wang, X., & Li, C. (2021). Cooperative Tracking of Hypersonic Target with Bearing-Only Measurements. Proceedings of the 40th Chinese Control Conference (CCC) (pp. 5212-5216). https://doi.org/10.23919/ccc52363.2021.9549405
[7] Paś, J., Rosiński, A., & Białek, K. (2021). A reliability-operational analysis of a track-side CCTV cabinet taking into account interference. Bulletin of the Polish Academy of Sciences. Technical Sciences, 136747. https://doi.org/10.24425/bpasts.2021.136747
[8] Costa, P., & Moore, W. (2002). Extended Kalman-Bucy filters for radar tracking and identification. Proceedings of the IEEE National Radar Conference, 127-131. https://doi.org/10.1109/nrc.1991.114744
[9] Farina, A., Del Gaudio, M., D’Elia, U., Immediata, S., Ortenzi, L., Timmoneri, L., & Toma. (2004). Detection and tracking of ballistic target. Proceedings of the IEEE National Radar Conference (pp. 450-456). https://doi.org/10.1109/nrc.2004.1316467
[10] Aditya, P., Apriliani, E., Arif, D. K., & Baihaqi, K. (2018). Estimation of three-dimensional radar tracking using modified extended Kalman filter. Journal of Physics Conference Series, 974, 012071. https://doi.org/10.1088/1742-6596/974/1/012071
[11] Abreu, J. a. P., Neto, J. V. F., & Oliveira, R. C. L. (2011). Ballistic Rockets tracking: Kalman versus 𝛼𝛽𝛾 filters. Proceedings of the 13th International Conference on Modelling and Simulation (UKSim) (pp. 313-318). https://doi.org/10.1109/uksim.2011.66
[12] Brown, R. G., & Hwang, P. Y. C. (1992). Introduction to random signals and applied Kalman filtering. In John Wiley and Sons eBooks. http://ci.nii.ac.jp/ncid/BA17030473
[13] Blackman, S. S., & Populi, R. (1999). Design and Analysis of Modern Tracking Systems. http://ci.nii.ac.jp/ncid/BA43213153
[14] Hull, D. G. (2007). Fundamentals of Airplane Flight Mechanics. In Springer eBooks. https://doi.org/10.1007/978-3-540-46573-7
[15] Siouris, G. M. (2004). Missile guidance and control systems. Applied Mechanics Reviews, 57(6), B32. https://doi.org/10.1115/1.1849174
[16] Sahbon, N., & Welcer, M. (2024). Comparison of two aerodynamic models for projectile trajectory simulation. Aerospace, 11(3), 189. https://doi.org/10.3390/aerospace11030189
[17] Świderski, W., Kaczmarzyk, J., & Szklarski, A. (2014). Wybrane zagadnienia projektowania kierowanego imitatora celu powietrznego ICP 12S6. Problemy Techniki Uzbrojenia. http://yadda.icm.edu.pl/baztech/element/bwmeta1.element.baztech-187b3033-ae97-4c81-8996-4851d352bcc1
[18] National Aeronautics and Space Administration. (1976). U.S. Standard Atmosphere, 1976 (NASA-TM-X-74335). NASA Technical Reports Server. https://ntrs.nasa.gov/api/citations/19770009539/downloads/19770009539.pdf
[19] Kailath, T., Sayed, A. H., & Hassibi, B. (2000). Linear Estimation. Prentice Hall.
[20] Zhang, X., Lei, H., Li, J., & Zhang, D. (2014). Ballistic missile trajectory prediction and the solution algorithms for impact point prediction. In Proceedings of the 2014 IEEE Chinese Guidance, Navigation and Control Conference (pp. 879-883). https://doi.org/10.1109/cgncc.2014.7007325
[21] Lewicka, O. (2023). Method for accuracy assessment of topo-bathymetric surface models based on geospatial data recorded by UAV and USV vehicles. Metrology and Measurement Systems, 461-480. https://doi.org/10.24425/mms.2023.146421