Interacting Multiple Model Adaptive Robust Kalman Filter for Position Estimation for Swarm Drones under Hybrid Noise Conditions

Abstract

This study evaluates the Interacting Multiple Model Adaptive Robust Kalman Filter (IMM-ARKF) for accurate position estimation in a leader-follower swarm of nine drones, consisting of one leader and eight followers following distinct trajectories. The evaluation is conducted under hybrid noise conditions combining Gaussian and Student’s t-distributions at 10%, 30%, and 50% ratios. The IMM-ARKF, which relies solely on its adaptive robust filtering mechanism, is compared with standard Interacting Multiple Model Kalman Filter (IMM-KF) and Extended Kalman Filter (IMM-EKF) methods. Simulations show that IMM-ARKF provides better accuracy, reducing root mean square error (RMSE) by up to 43.9% compared to IMM-EKF and 34.9% compared to IMM-KF across different noise conditions, due to its ability to adapt to hybrid noise. However, this improved performance comes with a computational cost, increasing processing time by up to 148% compared to IMM-EKF and 92.1% compared to IMM-KF, reflecting the complexity of its adaptive approach. These results demonstrate the effectiveness of IMM-ARKF in enhancing navigation accuracy and robustness for multi-drone systems in challenging environments.

Keywords

Interacting multiple model / Adaptive robust kalman filter / Swarm drones / Position estimation / Trajectory tracking / Hybrid noise modelling / Webots / Student’s t-distribution

Statement of the Use of Generative Al and Al-Assisted Technologies in the Writing Process

During the preparation of this manuscript, author U.A. used Gemini (Google) in order to improve the grammar, clarity, and readability of the text. After using this tool/service, the authors reviewed and edited the content as needed and takes full responsibility for the content of the published article.

Author Contributions

Conceptualization, F.C. and U.A.; Methodology, F.C. and U.A.; Software, U.A.; Validation, F.C., U.A. and S.P.; Formal Analysis, U.A.; Investigation, U.A.; Visualization, U.A.; Writing—Original Draft Preparation, U.A.; Writing—Review & Editing, F.C., U.A. and S.P.; Supervision, F.C.

Ethics Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

No datasets were used or analyzed in this study.

Funding

This research received no external funding.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References

[1] Chen M, Xiong Z, Xiong J, Wang R. A hybrid cooperative navigation method for UAV swarm based on factor graph and Kalman filter. Int. J. Distrib. Sens. Netw. 2022, 18, 155014772110647. doi:10.1177/15501477211064758.

[2] Zhao L, Chen B, Hu F. Research on swarm control based on complementary collaboration of unmanned aerial vehicle swarms under complex conditions. Drones 2025, 9, 119. doi:10.3390/drones9020119.

[3] Yang B, Wang H, Shi Z. Interacting multiple model adaptive robust Kalman filter for process and measurement modeling errors simultaneously. Signal Process. 2024, 227, 109743. doi:10.1016/j.sigpro.2024.109743.

[4] Gadsden SA, Wilkerson S, Alshabi M. Fault Detection of a Mechatronic System Using Interacting Operating Modes. In Proceedings of the Canadian Society for Mechanical Engineering International Congress, Charlottetown, PE, Canada, 27-30 June 2021; Volume 3. doi:10.32393/csme.2021.202.

[5] Radhika MN, Mallick M, Tian X. IMM Filtering Algorithms for a highly maneuvering fighter aircraft: An overview. Algorithms 2024, 17, 399. doi:10.3390/a17090399.

[6] Candan F, Beke A, Shen C, Mihaylova L. An Interacting Multiple Model Correntropy Kalman Filter Approach for Unmanned Aerial Vehicle Localisation. In Proceedings of the 2021 International Conference on INnovations in Intelligent SysTems and Applications (INISTA), Kocaeli, Turkey, 25-27 August 2021; pp. 1-6. doi:10.1109/inista55318.2022.9894214.

[7] Rodríguez AJ, Sanjurjo E, Pastorino R, Naya MÁ. Multibody-Based input and state observers using adaptive extended Kalman filter. Sensors 2021, 21, 5241. doi:10.3390/s21155241.

[8] Al-Absi MA, Fu R, Kim K-H, Lee YS, Al-Absi AA, Lee H-J. Tracking unmanned aerial vehicles based on the Kalman filter considering uncertainty and error aware. Electronics 2021, 10, 3067. doi:10.3390/electronics10243067.

[9] Huai L, Li B, Yun P, Song C, Wang J. Weighted Maximum Correntropy Criterion-Based Interacting Multiple-Model filter for maneuvering target tracking. Remote Sens. 2023, 15, 4513. doi:10.3390/rs15184513.

[10] Zhang J, Zhou W, Wang X. UAV swarm navigation using dynamic adaptive Kalman filter and network navigation. Sensors 2021, 21, 5374. doi:10.3390/s21165374.

[11] Zhang L, Sidoti D, Bienkowski A, Pattipati KR, Bar-Shalom Y, Kleinman DL. On the Identification of Noise Covariances and Adaptive Kalman Filtering: A New Look at a 50 Year-Old Problem. IEEE Access 2020, 8, 59362-59388. doi:10.1109/access.2020.2982407.

[12] Hematulin W, Kamsing P, Torteeka P, Somjit T, Phisannupawong T, Jarawan T. Trajectory planning for multiple UAVs and hierarchical collision avoidance based on nonlinear Kalman filters. Drones 2023, 7, 142. doi:10.3390/drones7020142.

[13] Liu X, Liu X, Zhang W, Yang Y. Interacting multiple model UAV navigation algorithm based on a robust Cubature Kalman filter. IEEE Access 2020, 8, 81034-81044. doi:10.1109/access.2020.2991032.

[14] Lee AS, Wu Y, Gadsden SA, AlShabi M. Interacting multiple model estimators for fault detection in a magnetorheological damper. Sensors 2023, 24, 251. doi:10.3390/s24010251.

[15] Zhao F, Cai R. Adaptive particle filter for state estimation with application to non-linear system. IET Signal Process. 2022, 16, 1023-1033. doi:10.1049/sil2.12147.

[16] Cosme LB, D’Angelo MFSV, Caminhas WM, Camargos MO, Palhares RM. An adaptive approach for estimation of transition probability matrix in the interacting multiple model filter. J. Intell. Fuzzy Syst. 2021, 41, 155-166. doi:10.3233/jifs-201129.

[17] Cyberbotics. Webots Main Window. Available online: https://cyberbotics.com/#cyberbotics (accessed on 23 September 2025).

[18] Bitcraze. pid_controller.py. GitHub Repository. Available online: https://github.com/bitcraze/crazyflie-simulation/blob/main/controllers_shared/python_based/pid_controller.py (accessed on 23 September 2025).

[19] Lizzio FF, Bugaj M, Rostáš J, Primatesta S. Comparison of multiple models in decentralized target estimation by a UAV swarm. Drones 2023, 8, 5. doi:10.3390/drones8010005.

[20] Huang H, Zhou H, Zheng M, Xu C, Zhang X, Xiong W. Cooperative Collision Avoidance Method for Multi-UAV Based on Kalman Filter and Model Predictive Control. In Proceedings of the 2019 IEEE International Conference on Unmanned Systems and Artificial Intelligence (ICUSAI), Xi’an, China, 22-24 November 2019; pp. 1-7. doi:10.1109/icusai47366.2019.9124863.