Journal of Radio Electronics. eISSN 1684-1719. 2026. ¹3
Full text in Russian (pdf)
DOI: https://doi.org/10.30898/1684-1719.2026.3.4
Mathematical Model of a Small-Scale UAV
A.A. Mironov, M.E. Solovyev
Yaroslavl State Technical University (YaSTU),
150023, Russia, Yaroslavl, Moskovskii prospekt, 88
The paper was received December 23, 2025.
Abstract. The article presents the first comprehensive mathematical model of a small unmanned aerial vehicle (UAV) developed in domestic practice, designed as an architecturally complete cyber-physical agent for digital simulation of attack scenarios against civilian infrastructure and their countermeasures using man-portable air-defense systems (MANPADS). The model integrates four key aspects of UAV behavior: flight dynamics (physical layer), sensor perception and tactical situation assessment (cognitive layer), offensive payload, and multispectral observability – including radar, acoustic, visual, radiofrequency, and infrared signatures. For the first time within a unified framework, the model incorporates: 1) the ability to define arbitrary observer and radar positions with dynamic line-of-sight (LoS) visibility calculations; 2) AI-based implementations of sensor perception and tactical evaluation functions; 3) a closed-loop digital twin architecture with feedback from air-defense systems and dynamic environment model updates. The scientific novelty of this work lies in the proposed original cyber-physical agent architecture, formalized as a state evolution operator that accounts for UAV interactions with complex urban environments–including buildings, electromagnetic interference, GNSS spoofing tactics, and multispectral detection channels. The model enables the generation of realistic UAV trajectories and signature profiles and is intended for use in airspace digital twins, simulation-based training systems, and quantitative assessment tools for air-defense effectiveness.
Key words: UAV, cyber-physical system, digital twin, air defense, multispectral signature, mathematical modeling, tactical behavior, radar cross-section, acoustic signature, modeling architecture.
Corresponding author: Mironov Artem Anatolievich, am-a79@mail.ru
References
1. Voronov E.M., Repkin A.L., Khromov F.M., Timofeev D.A. & Geraskin A.U. Mathematical model for simulation-based implementation of the air defence system of a surface ship formation. Herald of the Bauman Moscow State Technical University. “Instrument Engineering” Series, 2022, no. 1 https://doi.org/10.18698/0236-3933-2022-1-62-84
2. Geneva Convention relative to the Protection of Civilian Persons in Time of War, entered into force 21 October 1950. Ratified by the Presidium of the Supreme Soviet of the USSR on 17 April 1954, entered into force after ratification on 10 November 1954 https://docs.cntd.ru/document/1901071
3. Makarenko S.I., Starostin A.V. Country’s air defence against strikes by UAVs and cruise missiles: new threats, problematic issues, technical and economic analysis of architecture options. Systems of Control, Communication and Security, 2024, no. 2. https://doi.org/10.24412/2410-9916-2024-2-086-148
4. Makarenko S.I., Timoshenko A.V., Vasilchenko A.S. Analysis of means and methods for countering UAVs. Part 1. UAV as an object of detection and engagement. Systems of Control, Communication and Security, 2020, no. 1. https://doi.org/10.24411/2410-9916-2020-10105
5. Tikshaev V.N., Barvinenko V.V. The problem of countering UAVs and possible solutions. Military Thought, 2021, no. 1.
6. Chen Z., Yang J., Ma B., Shi K., Yu K. & Yuan W. Research on open-source simulation platforms for multi-copter UAV swarms. Robotics, 2023, vol. 12, no. 2, p. 53. https://doi.org/10.3390/robotics12020053
7. Aker C. & Kalkan S. Using deep networks for drone detection. In: 2017 14th IEEE International Conference on Advanced Video and Signal Based Surveillance (AVSS). IEEE, 2017. https://doi.org/10.1109/AVSS.2017.8078539
8. Al-Sa’d, M.F. et al. RF-based drone detection and identification using deep learning approaches: An initiative towards a large open source drone database. Future Generation Computer Systems, 2019, vol. 100, pp. 86–97. https://doi.org/10.1016/j.future.2019.05.007
9. Bernardini A., Mangiatordi F., Pallotti E., Capodiferro L. Drone detection by acoustic signature identification. Electronic Imaging, 2017, vol. 2017, no. 10, pp. 60–64. https://doi.org/10.2352/ISSN.2470-1173.2017.10.IMAWM-168
10. Boeing G. Modeling and Analyzing Urban Networks and Amenities with OSMnx. Geographical Analysis, 2025
11. Bradley M.S., Szwejkowski C., Szczesniak M. Optical Measurement of the Reflectance Behavior of Z93, the Thermal Coating on the International Space Station. Spectroscopy, 2020, vol. 35, no. S3, pp. 6–11
12. Catanzaro B., Lorenz J. & Dombrowski M. Compact CMOS multispectral/polarimetric camera. Proceedings of SPIE, 2006, vol. 6233, art. 62330O. https://doi.org/10.1117/12.666494
13. Cavanaugh D.B. et al. VNIR hypersensor camera system. Proceedings of SPIE, 2009, vol. 7457, art. 74570B. https://doi.org/10.1117/12.833539
14. Dombrowski M. et al. Object discrimination and optical performance of a real-time 2–5 μm hyperspectral imager. Proceedings of SPIE, 2006, vol. 6233, art. 62330N. https://doi.org/10.1117/12.665906
15. Dombrowski M. et al. Performance and application of a very high-speed 2–12 μm ultra spectral FTIR imager. Proceedings of SPIE, 2006, vol. 6233, art. 62330P. https://doi.org/10.1117/12.666063
16. Dombrowski M. & Catanzaro B. Design of dual-band SWIR/MWIR and MWIR/LWIR imagers. Proceedings of SPIE, 2004, vol. 5563, art. 55630B. https://doi.org/10.1117/12.543875
17. Dombrowski M., Catanzaro B. & Lorenz J. Hyperspectral sensor test bed for real-time algorithm evaluation. Proceedings of SPIE, 2002, vol. 4725, art. 47250C. https://doi.org/10.1117/12.453358
18. Dombrowski M., Catanzaro B. & Lorenz J. Manufacturing and performance evaluation of a refractive real-time MWIR hyperspectral imager. Proceedings of SPIE, 2003, vol. 5078, art. 50780B. https://doi.org/10.1117/12.498142
19. Dombrowski M., Catanzaro B. & Lorenz J. Progress towards a refractive real-time MWIR hyperspectral imager. Proceedings of SPIE, 2004, vol. 5563, art. 55630C. https://doi.org/10.1117/12.543871
20. Farlik J., Kratky M., Časar J. & Starý V. Multispectral Detection of Commercial Unmanned Aerial Vehicles. Sensors, 2019, vol. 19, no. 7, p. 1517. https://doi.org/10.3390/s19071517
21. GreyB. Thermal Imaging for Low Emissivity UAV Detection. 2024. https://xray.greyb.com/drones/thermal-imaging-low-emissivity
22. Güvenç İ. et al. Detection localization and tracking of unauthorized UAS and jammers. In: 2017 IEEE/AIAA 36th Digital Avionics Systems Conference (DASC). IEEE, 2017, pp. 1–10. https://doi.org/10.1109/DASC.2017.8102043
23. Ibrahem Mouhamad & Weijia Zhang. A Deep Neural Network Based UAV assisted Wireless Network. International Journal of Open Information Technologies, 2025, no. 1. https://cyberleninka.ru/article/n/a-deep-neural-network-based-uav-assisted-wireless-network
24. Jafolla J.C. The Prediction and Measurement of the Optical Properties of Complex Surfaces. Paper presented at ITBMS – International IR Target and Background Modeling & Simulation Workshop, Banyuls-sur-mer, France, 2018
25. Jafolla J. & Vaitekunas D.A. Handheld directional reflectometer: an angular imaging device to measure BRDF and HDR in real time. Proceedings of SPIE, 1998, vol. 3438, art. 34380D. https://doi.org/10.1117/12.328461
26. Jafolla J., Vaitekunas D.A. Theory and measurement of bidirectional reflectance for signature analysis. Proceedings of SPIE, 1999, vol. 3699, art. 36990B. https://doi.org/10.1117/12.352935
27. Ki M., Cha J., Lyu H. Detect and avoid system based on multi sensor fusion for UAV. In: 2018 International Conference on ICT Convergence (ICTC), 2018, pp. 1107–1109. https://doi.org/10.1109/ICTC.2018.8539587
28. Kim B.K., Kang H., Park S. Drone classification using convolutional neural networks with merged doppler images. IEEE Geoscience and Remote Sensing Letters, 2017, vol. 14, no. 1, pp. 38–42 https://doi.org/10.1109/LGRS.2016.2624820
29. Li Y. Modeling Open Experiment Teaching of UAV Based on Robot Operating System.
30. Martins B., Holland A., Silkoset A. Countering the Drone Threat: Implications of C-UAS Technology for Norway in an EU and NATO Context. 2020
31. McCormick K., Nascimento J.M., Hendricks L. Advanced Imaging System with Multiple Optical Sensing Modes. Proceedings of SPIE, 2018, vol. 10644, art. 106441S. https://doi.org/10.1117/12.2303843
32. Mehrubeoglu M., Van Sickle A., McLauchlan L. Borrowing least squares analysis from spectral unmixing to classify plastics in SWIR hyperspectral images. Proceedings of SPIE, 2020, vol. 11576, art. 115760B. https://doi.org/10.1117/12.2584007
33. Mehrubeoglu M., Van Sickle A., McLauchlan L. Detection and Identification of Plastics using SWIR Hyperspectral Imaging. Proceedings of SPIE, 2020, vol. 11504, art. 115040G. https://doi.org/10.1117/12.2570040
34. Nguyen P. et al. Cost-effective and passive RF-based drone presence detection and characterization. GetMobile: Mobile Computing and Communications Review, 2018, vol. 21, no. 4, pp. 30–34 https://doi.org/10.1145/3191789.3191800
35. Pinel-Lamotte A. et al. QuietDrone2020: Acoustic Monitoring of Small UAVs. MicroDB Project Report. France: MicroDB, 2024. 48 p. https://microdb.fr/wp-content/uploads/sites/3/2024/04/4-Monitoring_A108_QuietDrone2020_Pinel-Lamotte-final.pdf
36. Podder P., Zawodniok M. & Madria S. Deep Learning for UAV Detection and Classification via Radio Frequency Signal Analysis. In: 2024 25th IEEE International Conference on Mobile Data Management (MDM). IEEE, 2024, pp. 235–240. https://doi.org/10.1109/MDM61037.2024.00040
37. Sawyer C.W. et al. A John White Alexander painting: A comparison of imaging technologies for resolving a painting under another painting. Journal of the American Institute for Conservation, 2019, vol. 58, no. 3, pp. 303–324. https://doi.org/10.1080/01971360.2018.1556542
38. Shi X. et al. Anti-drone system with multiple surveillance technologies: Architecture, implementation, and challenges. IEEE Communications Magazine, 2018, vol. 56, no. 4, pp. 68–74. https://doi.org/10.1109/MCOM.2018.1700430
39. Solomitckii D. et al. Technologies for efficient amateur drone detection in 5G millimeter-wave cellular infrastructure. IEEE Communications Magazine, 2018, vol. 56, no. 1, pp. 43–50 https://doi.org/10.1109/MCOM.2017.1700450
40. Soria E., Schiano F. & Floreano D. SwarmLab: a Matlab Drone Swarm Simulator. In: 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2020, pp. 8005–8011. https://doi.org/10.1109/IROS45743.2020.9340854
41. Twesme J. & Corzine A. Naval Air Systems Command (NAVAIR) Unmanned Aerial Vehicle (UAV) Unmanned Combat Aerial Vehicle (UCAV) Distributed Simulation Infrastructure. In: AIAA Modeling and Simulation Technologies Conference, 2003. AIAA 2003-6612. https://doi.org/10.2514/6.2003-6612
42. Utebayeva D.Z. Research of effective UAV detection using acoustic data recognition: PhD thesis. Almaty: Nazarbayev University, 2024. 142 p
43. Uzoagba C. project proposal topic: intelligent real-time uav autonomy for compromised uav/uas system data due to cyber-attacks cu led. 2023
44. Vaitekunas D.A. et al. Measurement and analysis of optical surface properties for input to ShipIR. Proceedings of SPIE, 2009, vol. 7300, art. 73000M. https://doi.org/10.1117/12.820055
45. Vaitekunas D.A. & Jafolla J. Bidirectional reflectance measurements for high-resolution signature modeling. Proceedings of SPIE, 2004, vol. 5431, art. 54310M. https://doi.org/10.1117/12.548085
46. Wu M. et al. Real-time drone detection using deep learning approach. In: Machine Learning and Intelligent Communications. Springer, 2018, pp. 22–32 https://doi.org/10.1007/978-3-030-00557-3_3
47. Yang J., Huang C. & Wang J. The study and development of UAV digital twin system. Journal of Physics: Conference Series, 2022, vol. 2366, art. 012038. https://doi.org/10.1088/1742-6596/2366/1/012038
48. Yuan Z. et al. Experimental Analysis and Modeling of Monostatic UAV RCS for ISAC Channels. IEEE Antennas and Wireless Propagation Letters, 2025, vol. 24, no. 1, pp. 222–226. https://doi.org/10.1109/LAWP.2024.3492502
49. Zeng Y., Morris J. & Dombrowski M. Validation of a new method for measuring and continuously monitoring the efficiency of industrial flares. Journal of the Air & Waste Management Association, 2016, vol. 66, no. 1, pp. 9–18. https://doi.org/10.1080/10962247.2015.1114045
50. Zhang Y. et al. A Unified RCS Modeling of Typical Targets for 3GPP ISAC Channel Standardization and Experimental Analysis. arXiv preprint arXiv:2505.20673, 2025. https://doi.org/10.1109/JSAC.2025.3608732
Mironov A.A., Solovyev M.E. Mathematical model of a small-scale UAV // Journal of Radio Electronics. – 2026. – ¹. 3. https://doi.org/10.30898/1684-1719.2026.3.4 (In Russian)