{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,19]],"date-time":"2026-06-19T16:31:45Z","timestamp":1781886705101,"version":"3.54.5"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,9,3]],"date-time":"2025-09-03T00:00:00Z","timestamp":1756857600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,9,3]],"date-time":"2025-09-03T00:00:00Z","timestamp":1756857600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"MOST \u2013 Sustainable Mobility National Research Center","award":["PIANO NAZIONALE DI RIPRESA E RESILIENZA (PNRR) \u2013 MISSIONE 4 COMPONENTE 2, INVESTIMENTO 1.4 \u2013 D.D. 1033 17\/06\/2022, CN00000023"],"award-info":[{"award-number":["PIANO NAZIONALE DI RIPRESA E RESILIENZA (PNRR) \u2013 MISSIONE 4 COMPONENTE 2, INVESTIMENTO 1.4 \u2013 D.D. 1033 17\/06\/2022, CN00000023"]}]},{"name":"FAIR - Future Artificial Intelligence Research","award":["PIANO NAZIONALE DI RIPRESA E RESILIENZA (PNRR) \u2013 MISSIONE 4 COMPONENTE 2, INVESTIMENTO 1.3 \u2013 D.D. 1555 11\/10\/2022, PE00000013"],"award-info":[{"award-number":["PIANO NAZIONALE DI RIPRESA E RESILIENZA (PNRR) \u2013 MISSIONE 4 COMPONENTE 2, INVESTIMENTO 1.3 \u2013 D.D. 1555 11\/10\/2022, PE00000013"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Intell Robot Syst"],"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>Traditional adaptive and robust control methods sometimes struggle to address the complex and dynamic uncertainties inherent in quadrotor flight, frequently resulting in overly conservative or unstable behavior due to reliance on presumed uncertainty bounds. To overcome these limitations, we propose a unified online-learning-based control framework that integrates data-driven uncertainty estimation with a nominal flight controller. Furthermore, we systematically evaluate four representative learning methods, namely Extreme Learning Machines (ELMs), Radial Basis Function Neural Networks (RBFNNs), Echo State Networks (ESNs), and Gaussian Processes (GPs), spanning both parametric and non-parametric techniques. Through extensive simulation and hardware experiments, we benchmark these methods in terms of (i) uncertainty estimation accuracy, (ii) trajectory tracking performance, and (iii) real-time computational efficiency. Our results reveal key trade-offs: GPs yield highest accuracy but require approximations for real-time use. ELMs and RBFNNs enable fast inference but suffer initialization sensitivity. ESNs provide the best balance, offering both stability and real-time performance. This work underscores the critical role of model selection in learning-based flight control, providing insights for deploying these algorithms in resource-constrained aerial systems operating under uncertainty.<\/jats:p>","DOI":"10.1007\/s10846-025-02305-5","type":"journal-article","created":{"date-parts":[[2025,9,3]],"date-time":"2025-09-03T10:47:22Z","timestamp":1756896442000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Learning uncertainties online for quadrotor flight control: A comparative study"],"prefix":"10.1007","volume":"111","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5083-9408","authenticated-orcid":false,"given":"Weibin","family":"Gu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiance","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2386-3146","authenticated-orcid":false,"given":"Alessandro","family":"Rizzo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,9,3]]},"reference":[{"key":"2305_CR1","doi-asserted-by":"publisher","first-page":"2520","DOI":"10.1109\/ICRA.2011.5980409","volume":"2011","author":"D Mellinger","year":"2011","unstructured":"Mellinger, D., Kumar, V.R.: Minimum snap trajectory generation and control for quadrotors. IEEE International Conference on Robotics and Automation. 2011, 2520\u20132525 (2011)","journal-title":"IEEE International Conference on Robotics and Automation."},{"key":"2305_CR2","doi-asserted-by":"crossref","unstructured":"Conyers, S.A., Rutherford, M.J., Valavanis, K.P.: An empirical evaluation of ground effect for small-scale rotorcraft. In: 2018 IEEE international conference on robotics and automation (ICRA). pp. 1244\u20131250. IEEE (2018)","DOI":"10.1109\/ICRA.2018.8461035"},{"key":"2305_CR3","doi-asserted-by":"publisher","unstructured":"David Du Mutel\u00a0de Pierrepont\u00a0Franzetti, I., Parin, R., Capello, E., Rutherford, M., Valavanis, K.: Ground, ceiling and wall effect evaluation of small quadcopters in pressure-controlled environments. J. Intell. Robotic Syst. 08, 110. https:\/\/doi.org\/10.1007\/s10846-024-02155-7","DOI":"10.1007\/s10846-024-02155-7"},{"key":"2305_CR4","doi-asserted-by":"crossref","unstructured":"Bauersfeld, L., Kaufmann, E., Foehn, P., Sun, S., Scaramuzza, D.: NeuroBEM: Hybrid Aerodynamic Quadrotor Model. (2021). ArXiv:abs\/2106.08015","DOI":"10.15607\/RSS.2021.XVII.042"},{"issue":"3","key":"2305_CR5","doi-asserted-by":"publisher","first-page":"7209","DOI":"10.1109\/LRA.2022.3181755","volume":"7","author":"R Penicka","year":"2022","unstructured":"Penicka, R., Song, Y., Kaufmann, E., Scaramuzza, D.: Learning Minimum-Time Flight in Cluttered Environments. IEEE Robotics and Automation Letters. 7(3), 7209\u20137216 (2022). https:\/\/doi.org\/10.1109\/LRA.2022.3181755","journal-title":"IEEE Robotics and Automation Letters."},{"key":"2305_CR6","doi-asserted-by":"crossref","unstructured":"Gu, W., Hu, D., Cheng, L., Cao, Y., Rizzo, A., Valavanis, K.P.: Autonomous wind turbine inspection using a quadrotor. In: 2020 international conference on unmanned aircraft systems (ICUAS). pp. 709\u2013715 (2020)","DOI":"10.1109\/ICUAS48674.2020.9214066"},{"key":"2305_CR7","doi-asserted-by":"publisher","first-page":"1924","DOI":"10.1109\/TCST.2012.2209887","volume":"21","author":"T Lee","year":"2013","unstructured":"Lee, T.: Robust Adaptive Attitude Tracking on SO(3) With an Application to a Quadrotor UAV. IEEE Trans. Control Syst. Technol. 21, 1924\u20131930 (2013)","journal-title":"IEEE Trans. Control Syst. Technol."},{"key":"2305_CR8","first-page":"1","volume":"2019","author":"TW Ou","year":"2019","unstructured":"Ou, T.W., Liu, Y.: Adaptive backstepping tracking control for quadrotor aerial robots subject to uncertain dynamics. Am. Control Conf. (ACC) 2019, 1\u20136 (2019)","journal-title":"Am. Control Conf. (ACC)"},{"key":"2305_CR9","doi-asserted-by":"crossref","unstructured":"Zhu, B., Huo, W.: Adaptive backstepping control for a miniature autonomous helicopter. In: 2011 50th IEEE conference on decision and control and european control conference. pp. 5413\u20135418 (2011)","DOI":"10.1109\/CDC.2011.6160275"},{"key":"2305_CR10","doi-asserted-by":"publisher","DOI":"10.1016\/j.robot.2025.104942","volume":"188","author":"W Gu","year":"2025","unstructured":"Gu, W., Primatesta, S., Rizzo, A.: Robust adaptive control for aggressive quadrotor maneuvers via SO(3) and backstepping techniques. Robot. Auton. Syst. 188, 104942 (2025)","journal-title":"Robot. Auton. Syst."},{"key":"2305_CR11","doi-asserted-by":"publisher","DOI":"10.1115\/1.4030419","volume":"137","author":"FA Goodarzi","year":"2014","unstructured":"Goodarzi, F.A., Lee, D., Lee, T.: Geometric adaptive tracking control of a quadrotor unmanned aerial vehicle on SE(3) for agile maneuvers. J. Dyn. Syst. Meas. Control-Trans. Asme. 137, 091007 (2014)","journal-title":"J. Dyn. Syst. Meas. Control-Trans. Asme."},{"key":"2305_CR12","doi-asserted-by":"publisher","first-page":"11043","DOI":"10.3182\/20140824-6-ZA-1003.01860","volume":"47","author":"T Chingozha","year":"2014","unstructured":"Chingozha, T., Nyandoro, O.T.: Adaptive sliding backstepping control of quadrotor UAV attitude. IFAC Proc 47, 11043\u201311048 (2014)","journal-title":"IFAC Proc"},{"issue":"5","key":"2305_CR13","doi-asserted-by":"publisher","first-page":"2891","DOI":"10.1109\/TIE.2014.2364982","volume":"62","author":"B Zhao","year":"2015","unstructured":"Zhao, B., Xian, B., Zhang, Y., Zhang, X.: Nonlinear robust adaptive tracking control of a quadrotor UAV via immersion and invariance methodology. IEEE Trans. Industr. Electron. 62(5), 2891\u20132902 (2015). https:\/\/doi.org\/10.1109\/TIE.2014.2364982","journal-title":"IEEE Trans. Industr. Electron."},{"key":"2305_CR14","unstructured":"Ioannou, P.A., Sun, J.: Robust Adaptive Control. (2012)"},{"key":"2305_CR15","doi-asserted-by":"publisher","first-page":"1469","DOI":"10.1007\/s10846-020-01227-8","volume":"100","author":"W Gu","year":"2020","unstructured":"Gu, W., Valavanis, K.P., Rutherford, M.J., Rizzo, A.: UAV model-based flight control with artificial neural networks: A survey. J. Intell. Robotic Syst. 100, 1469\u20131491 (2020)","journal-title":"J. Intell. Robotic Syst."},{"key":"2305_CR16","doi-asserted-by":"crossref","unstructured":"Mohajerin, N., Mozifian, M., Waslander, S.: Deep learning a quadrotor dynamic model for multi-step prediction. In: 2018 IEEE international conference on robotics and automation (ICRA), p.p 2454\u20132459 (2018)","DOI":"10.1109\/ICRA.2018.8460840"},{"issue":"4","key":"2305_CR17","doi-asserted-by":"publisher","first-page":"10256","DOI":"10.1109\/LRA.2022.3192609","volume":"7","author":"A Saviolo","year":"2022","unstructured":"Saviolo, A., Li, G., Loianno, G.: Physics-inspired temporal learning of quadrotor dynamics for accurate model predictive trajectory tracking. IEEE Robotics Autom. Lett. 7(4), 10256\u201310263 (2022)","journal-title":"IEEE Robotics Autom. Lett."},{"key":"2305_CR18","doi-asserted-by":"publisher","DOI":"10.1016\/j.robot.2023.104569","volume":"171","author":"W Gu","year":"2024","unstructured":"Gu, W., Primatesta, S., Rizzo, A.: Physics-informed neural network for quadrotor dynamical modeling. Robot. Auton. Syst. 171, 104569 (2024). https:\/\/doi.org\/10.1016\/j.robot.2023.104569","journal-title":"Robot. Auton. Syst."},{"key":"2305_CR19","first-page":"9784","volume":"2018","author":"G Shi","year":"2019","unstructured":"Shi, G., Shi, X., O\u2019Connell, M., Yu, R., Azizzadenesheli, K., Anandkumar, A., et al.: Neural Lander: Stable Drone Landing Control Using Learned Dynamics. Int. Conf. Robotics Autom (ICRA) 2018, 9784\u20139790 (2019)","journal-title":"Int. Conf. Robotics Autom (ICRA)"},{"key":"2305_CR20","doi-asserted-by":"publisher","first-page":"1063","DOI":"10.1109\/TRO.2021.3098436","volume":"38","author":"G Shi","year":"2020","unstructured":"Shi, G., Honig, W., Shi, X., Yue, Y., Chung, S.J.: Neural-swarm2: Planning and control of heterogeneous multirotor swarms using learned interactions. IEEE Trans. Rob. 38, 1063\u20131079 (2020)","journal-title":"IEEE Trans. Rob."},{"key":"2305_CR21","doi-asserted-by":"crossref","unstructured":"O\u2019Connell, M., Shi, G., Shi, X., Azizzadenesheli, K., Anandkumar, A., Yue, Y., et\u00a0al.: Neural-Fly enables rapid learning for agile flight in strong winds. Sci. Robotics. 7 (2022)","DOI":"10.1126\/scirobotics.abm6597"},{"key":"2305_CR22","doi-asserted-by":"publisher","unstructured":"Deshpande, A.M,. Minai, A.A., Kumar, M.: Robust deep reinforcement learning for quadcopter control. IFAC-PapersOnLine. 54(20):90\u201395. Modeling, Estimation and Control Conference MECC 2021. (2021). https:\/\/doi.org\/10.1016\/j.ifacol.2021.11.158","DOI":"10.1016\/j.ifacol.2021.11.158"},{"key":"2305_CR23","doi-asserted-by":"crossref","unstructured":"Song, Y., Steinweg, M., Kaufmann, E., Scaramuzza, D. :Autonomous drone racing with deep reinforcement learning. In: 2021 IEEE\/RSJ international conference on intelligent robots and systems (IROS), pp. 1205\u20131212 (2021)","DOI":"10.1109\/IROS51168.2021.9636053"},{"key":"2305_CR24","doi-asserted-by":"crossref","unstructured":"Gu, W., Rizzo, A.: Online residual learning using interpretable reservoir computing for quadrotor control. In: 2024 international conference on unmanned aircraft systems (ICUAS), pp. 23\u201330 (2024)","DOI":"10.1109\/ICUAS60882.2024.10557060"},{"key":"2305_CR25","doi-asserted-by":"crossref","unstructured":"Shi, L., Mucchiani, C., Karydis, K.: Online modeling and control of soft multi-fingered grippers via koopman operator theory. In: 2022 IEEE 18th international conference on automation science and engineering (CASE), pp. 1946\u20131952 (2022)","DOI":"10.1109\/CASE49997.2022.9926464"},{"key":"2305_CR26","unstructured":"Jiahao, T.Z., Chee, K.Y., Hsieh, M.A.: Online dynamics learning for predictive control with an application to aerial robots. In: Liu, K., Kulic, D., Ichnowski, J., (eds.) Proceedings of the 6th conference on robot learning of proceedings of machine learning research, vol. 205, pp. 2251\u20132261. PMLR (2023). Available from: https:\/\/proceedings.mlr.press\/v205\/jiahao23a.html"},{"key":"2305_CR27","doi-asserted-by":"crossref","unstructured":"Zheng, L., Yang, R., Pan, J., Cheng, H.: Safe learning-based tracking control for quadrotors under wind disturbances. In: American control conference (ACC). 2021, 3638\u20133643. IEEE (2021)","DOI":"10.23919\/ACC50511.2021.9482929"},{"key":"2305_CR28","doi-asserted-by":"crossref","unstructured":"Munoz, E., Kalaria, D., Lin, Q., Dolan, J.M.: Online adaptive compensation for model uncertainty using extreme learning machine-based control barrier functions. In: 2022 IEEE\/RSJ international conference on intelligent robots and systems (IROS). pp. 10959\u201310966 IEEE (2022)","DOI":"10.1109\/IROS47612.2022.9981680"},{"key":"2305_CR29","doi-asserted-by":"publisher","unstructured":"S\u00f6nmez, S., Rutherford, M.J., Valavanis, K.P.: A survey of offline- and online-learning-based algorithms for multirotor UAVs. Drones. 8(4) (2024). https:\/\/doi.org\/10.3390\/drones8040116","DOI":"10.3390\/drones8040116"},{"issue":"1\u20133","key":"2305_CR30","doi-asserted-by":"publisher","first-page":"489","DOI":"10.1016\/j.neucom.2005.12.126","volume":"70","author":"GB Huang","year":"2006","unstructured":"Huang, G.B., Zhu, Q.Y., Siew, C.K.: Extreme Learning Machine: Theory and Applications. Neurocomputing 70(1\u20133), 489\u2013501 (2006)","journal-title":"Neurocomputing"},{"issue":"2","key":"2305_CR31","doi-asserted-by":"publisher","first-page":"246","DOI":"10.1162\/neco.1991.3.2.246","volume":"3","author":"J Park","year":"1991","unstructured":"Park, J., Sandberg, I.W.: Universal Approximation using Radial-Basis-Function Networks. Neural Comput. 3(2), 246\u2013257 (1991)","journal-title":"Neural Comput."},{"key":"2305_CR32","unstructured":"Jaeger, H.: The echo state approach to analysing and training recurrent neural networks; (2001). Available from: https:\/\/api.semanticscholar.org\/CorpusID:15467150"},{"key":"2305_CR33","doi-asserted-by":"crossref","unstructured":"Rasmussen, C.E.: Gaussian processes in machine learning. In: Summer school on machine learning. pp. 63\u201371. Springer (2003)","DOI":"10.1007\/978-3-540-28650-9_4"},{"key":"2305_CR34","doi-asserted-by":"crossref","unstructured":"Martini, S., Stefanovic, M., Rizzo, A., Rutherford, M.J., Livreri, P., Valavanis, K.P.: A benchmark framework for testing, evaluation, and comparison of quadrotor linear and nonlinear controllers. In: 2023 international conference on unmanned aircraft systems (ICUAS). pp. 471\u2013478 (2023)","DOI":"10.1109\/ICUAS57906.2023.10156106"},{"key":"2305_CR35","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1016\/j.arcontrol.2023.03.009","volume":"55","author":"A Saviolo","year":"2023","unstructured":"Saviolo, A., Loianno, G.: Learning quadrotor dynamics for precise, safe, and agile flight control. Annu. Rev. Control. 55, 45\u201360 (2023). https:\/\/doi.org\/10.1016\/j.arcontrol.2023.03.009","journal-title":"Annu. Rev. Control."},{"key":"2305_CR36","doi-asserted-by":"crossref","unstructured":"Folkestad, C., Pastor, D., Burdick, J.W.: Episodic Koopman learning of nonlinear robot dynamics with application to fast multirotor landing. In: 2020 IEEE international conference on robotics and automation (ICRA). pp. 9216\u20139222. IEEE (2020)","DOI":"10.1109\/ICRA40945.2020.9197510"},{"issue":"620","key":"2305_CR37","doi-asserted-by":"publisher","first-page":"982","DOI":"10.1038\/s41586-023-06419-4","volume":"08","author":"E Kaufmann","year":"2023","unstructured":"Kaufmann, E., Bauersfeld, L., Loquercio, A., Mueller, M., Koltun, V., Scaramuzza, D.: Champion-level drone racing using deep reinforcement learning. Nature 08(620), 982\u2013987 (2023). https:\/\/doi.org\/10.1038\/s41586-023-06419-4","journal-title":"Nature"},{"issue":"3","key":"2305_CR38","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1016\/j.cosrev.2009.03.005","volume":"3","author":"M Luko\u0161evi\u010dius","year":"2009","unstructured":"Luko\u0161evi\u010dius, M., Jaeger, H.: Reservoir computing approaches to recurrent neural network training. Comput. Sci. Rev. 3(3), 127\u2013149 (2009). https:\/\/doi.org\/10.1016\/j.cosrev.2009.03.005","journal-title":"Comput. Sci. Rev."},{"key":"2305_CR39","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1007\/s12559-017-9461-9","volume":"9","author":"C Gallicchio","year":"2017","unstructured":"Gallicchio, C., Micheli, A.: Echo state property of deep reservoir computing networks. Cogn. Comput. 9, 337\u2013350 (2017)","journal-title":"Cogn. Comput."},{"key":"2305_CR40","unstructured":"Bull, A.D.: Convergence rates of efficient global optimization algorithms. J. Mach. Learn. Res. 2011;12(10)"},{"key":"2305_CR41","doi-asserted-by":"publisher","unstructured":"Panerati, J., Zheng, H., Zhou, S., Xu, J., Prorok, A., Schoellig, A.P.: Learning to fly\u2014a gym environment with pybullet physics for reinforcement learning of multi-agent quadcopter control. pp. 7512\u20137519. (2021). https:\/\/doi.org\/10.1109\/IROS51168.2021.9635857","DOI":"10.1109\/IROS51168.2021.9635857"},{"key":"2305_CR42","unstructured":"Hensman, J., Fusi, N., Lawrence, N.D.: Gaussian processes for big data. (2013) arXiv preprint arXiv:1309.6835"}],"container-title":["Journal of Intelligent &amp; Robotic Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10846-025-02305-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10846-025-02305-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10846-025-02305-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,6]],"date-time":"2025-10-06T04:14:03Z","timestamp":1759724043000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10846-025-02305-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,3]]},"references-count":42,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["2305"],"URL":"https:\/\/doi.org\/10.1007\/s10846-025-02305-5","relation":{},"ISSN":["1573-0409"],"issn-type":[{"value":"1573-0409","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,3]]},"assertion":[{"value":"25 October 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 August 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 September 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval and consent to participate"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}},{"value":"Alessandro Rizzo is a Senior Editor-at-Large of the Journal of Intelligent and Robotic Systems. The authors have no other conflicts of interest that are relevant to the contnt of this article.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of Interest"}}],"article-number":"98"}}