{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T19:52:52Z","timestamp":1765828372432,"version":"3.37.3"},"reference-count":24,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2024,3,18]],"date-time":"2024-03-18T00:00:00Z","timestamp":1710720000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,3,18]],"date-time":"2024-03-18T00:00:00Z","timestamp":1710720000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Intell Robot Syst"],"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Based on hierarchical inner-outer loop strategy, the tracking control for the helicopter system could be designed individually for the position loop and for the attitude loop, thus simplifying the underactuated control problem. However, due to the nonlinear coupling between the position dynamics and rotation dynamics, the performance of the position control is affected by attitude errors, especially when the attitude control can not tracks the reference attitude instantaneously. This work provides a hierarchical trajectory tracking control design for the helicopter with model uncertainties, ensuring the stability of the overall system considering the perturbation caused by attitude tracking errors and the nonlinear coupling. The attitude of the helicopter is descried by unit-quaternion, for which anti-unwinding control design is presented. Besides, the criteria for avoidance of singularity in generation of the reference attitude is derived. Simulation results demonstrate the effectiveness of the design.<\/jats:p>","DOI":"10.1007\/s10846-024-02072-9","type":"journal-article","created":{"date-parts":[[2024,3,18]],"date-time":"2024-03-18T07:01:57Z","timestamp":1710745317000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Nonsingular Hierarchical Approach for Trajectory Tracking Control of Miniature Helicopter with Model Uncertainties"],"prefix":"10.1007","volume":"110","author":[{"given":"Ce","family":"Liu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,3,18]]},"reference":[{"issue":"2","key":"2072_CR1","doi-asserted-by":"publisher","first-page":"210","DOI":"10.1049\/iet-cta.2015.0565","volume":"10","author":"Y Zou","year":"2016","unstructured":"Zou, Y., Huo, W.: Singularity-free nonlinear controller for a model-scaled autonomous helicopter[J]. IET Control Theory Appl 10(2), 210\u2013219 (2016)","journal-title":"IET Control Theory Appl"},{"key":"2072_CR2","doi-asserted-by":"publisher","first-page":"1363","DOI":"10.1007\/s11071-021-06332-3","volume":"104","author":"K Eliker","year":"2021","unstructured":"Eliker, K., Grouni, S., et al.: Quadcopter nonsingular finite-time adaptive robust saturated command-filtered control system under the presence of uncertainties and input saturation. Non Dyn 104, 1363\u20131387 (2021)","journal-title":"Non Dyn"},{"key":"2072_CR3","doi-asserted-by":"publisher","first-page":"2171","DOI":"10.1007\/s11071-015-2473-7","volume":"83","author":"Y Zou","year":"2016","unstructured":"Zou, Y.: Adaptive trajectory tracking control approach for a model-scaled helicopter. Non Dyn 83, 2171\u20132181 (2016)","journal-title":"Non Dyn"},{"key":"2072_CR4","first-page":"3788","volume-title":"Non-singular trajectory tracking control of a pitch-constrained quad-rotorcraft using integral barrier Lyapunov function, American Control Conference","author":"R Dasgupta","year":"2020","unstructured":"Dasgupta, R., Roy, S.B., Bhasim, S.: Non-singular trajectory tracking control of a pitch-constrained quad-rotorcraft using integral barrier Lyapunov function, American Control Conference, pp. 3788\u20133796. Denver, CO, USA (2020)"},{"issue":"2","key":"2072_CR5","doi-asserted-by":"publisher","first-page":"466","DOI":"10.1109\/TCST.2010.2042450","volume":"19","author":"IA Raptis","year":"2011","unstructured":"Raptis, I.A., Valavanic, K.P.: A novel nonlinear backstepping controller design for helicopters using the rotation matrix. IEEE Trans Control Syst Technol 19(2), 466\u2013473 (2011)","journal-title":"IEEE Trans Control Syst Technol"},{"issue":"6","key":"2072_CR6","doi-asserted-by":"publisher","first-page":"1053","DOI":"10.1016\/j.automatica.2010.03.010","volume":"46","author":"A Abdessameud","year":"2010","unstructured":"Abdessameud, A., Tayebi, A.: Global trajectory tracking control of VTOL-UAVs without linear velocity measurements. Automatica 46(6), 1053\u20131059 (2010)","journal-title":"Automatica"},{"key":"2072_CR7","doi-asserted-by":"crossref","unstructured":"Mayhew CG, Sanfelice RG, Teel AR (2011) On quaternion-based attitude control and the unwinding phenomenon. In Proceedings of the 2011 american control conference, 299-304","DOI":"10.1109\/ACC.2011.5991127"},{"issue":"11","key":"2072_CR8","doi-asserted-by":"publisher","first-page":"2555","DOI":"10.1109\/TAC.2011.2108490","volume":"56","author":"CG Mayhew","year":"2011","unstructured":"Mayhew, C.G., Sanfelice, R.G., Teel, A.R.: Quaterion-based hybrid control for robust global attitude tracking. IEEE Trans Aut Con 56(11), 2555\u20132566 (2011)","journal-title":"IEEE Trans Aut Con"},{"issue":"9","key":"2072_CR9","first-page":"1004","volume":"42","author":"H Cui","year":"2012","unstructured":"Cui, H., Cheng, X.: Anti-unwinding attitude maneuver control of spacecraft considering bounded disturbance and input saturation. Sci China: Technol Sci 42(9), 1004\u20131015 (2012)","journal-title":"Sci China: Technol Sci"},{"key":"2072_CR10","doi-asserted-by":"crossref","unstructured":"Zhao S., Dong W., Farrell J.A. (2013) Quaternion-based trajectory tracking control of VTOL-UAVs using command filtered backstepping. Proceedings of the 2013 American Control Conference, Washington, DC, 1018-1023","DOI":"10.1109\/ACC.2013.6579970"},{"issue":"5","key":"2072_CR11","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 quadrotor UAV. IEEE Trans Control Syst Technol 21(5), 1924\u20131930 (2013)","journal-title":"IEEE Trans Control Syst Technol"},{"key":"2072_CR12","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1016\/j.sysconle.2011.10.017","volume":"61","author":"T Lee","year":"2012","unstructured":"Lee, T.: Exponential stability of an attitude tracking system on SO(3) for large-angle rotational maneuvers. Syst Control Lett 61, 231\u2013237 (2012)","journal-title":"Syst Control Lett"},{"key":"2072_CR13","doi-asserted-by":"crossref","unstructured":"Krishna, A.B., Sen, A., Kothari, M.: Robust geometric control of a helicopter using sliding mode control. International Conference on Unmanned Aircraft Systems (ICUAS), 737\u2013743 (2020)","DOI":"10.1109\/ICUAS48674.2020.9213949"},{"issue":"5","key":"2072_CR14","doi-asserted-by":"publisher","first-page":"902","DOI":"10.2514\/1.G006310","volume":"45","author":"AB Krishna","year":"2022","unstructured":"Krishna, A.B., Sen, A., Kothari, M.: Robust geometric trajectory tracking control of a variable-pitch quadrotor. J Guid Control Dyn 45(5), 902\u2013920 (2022)","journal-title":"J Guid Control Dyn"},{"issue":"5","key":"2072_CR15","doi-asserted-by":"publisher","first-page":"1927","DOI":"10.1109\/TCST.2013.2296519","volume":"22","author":"B Zhu","year":"2014","unstructured":"Zhu, B.: 3-D path-following control for a model-scaled autonomous helicopter. IEEE Trans Control Syst Technol 22(5), 1927\u20131934 (2014)","journal-title":"IEEE Trans Control Syst Technol"},{"key":"2072_CR16","unstructured":"Zou Y (2016) Trajectory tracking control developments for miniature unmanned helicopters, PhD thesis, BUAA, China"},{"issue":"3","key":"2072_CR17","doi-asserted-by":"publisher","first-page":"566","DOI":"10.1109\/TCST.2011.2121907","volume":"20","author":"W Dong","year":"2012","unstructured":"Dong, W., et al.: Command filtered adaptive backstepping. IEEE Trans Control Syst Technol 20(3), 566\u2013580 (2012)","journal-title":"IEEE Trans Control Syst Technol"},{"issue":"6","key":"2072_CR18","doi-asserted-by":"publisher","first-page":"1391","DOI":"10.1109\/TAC.2009.2015562","volume":"54","author":"J Farrell","year":"2009","unstructured":"Farrell, J., Polycarpou, M., Dong, W.: Command filtered backstepping. IEEE Trans Autom Control 54(6), 1391\u20131395 (2009)","journal-title":"IEEE Trans Autom Control"},{"issue":"5","key":"2072_CR19","doi-asserted-by":"publisher","first-page":"803","DOI":"10.1016\/j.mechatronics.2011.02.002","volume":"21","author":"GW Cai","year":"2011","unstructured":"Cai, G.W., et al.: Design and implementation of a robust and nonlinear flight control system for an unmanned helicopter. Mechatronics 21(5), 803\u2013820 (2011)","journal-title":"Mechatronics"},{"key":"2072_CR20","doi-asserted-by":"crossref","unstructured":"Ren, B., Ge, S., et al.: Adaptive neural control for output feedback nonlinear systems using a barrier Lyapunov function. IEEE Trans Neural Netw 21(8), 1339\u20131345 (2010)","DOI":"10.1109\/TNN.2010.2047115"},{"issue":"4","key":"2072_CR21","first-page":"376","volume":"26","author":"F Zhang","year":"2002","unstructured":"Zhang, F., et al.: A new large-scale transformation algorithm of quaternion to Euler angles. J Nanjing Univ Nat Sci Technol 26(4), 376\u2013380 (2002)","journal-title":"J Nanjing Univ Nat Sci Technol"},{"key":"2072_CR22","unstructured":"Liu C (2022) Research on Under-actuated control for small-scale unmanned helicopters, PhD thesis, BUAA, China"},{"key":"2072_CR23","unstructured":"Kahan, W.: IEEE Standard 754 for binary floating-point arithmetic. Lecture Notes on the Status of IEEE 754, (1997)"},{"key":"2072_CR24","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3785-1","volume-title":"Identification Modeling and Characteristics of Miniature Rotorcraft","author":"B Mettler","year":"2003","unstructured":"Mettler, B.: Identification Modeling and Characteristics of Miniature Rotorcraft. Kluwer, Norwell, MA (2003)"}],"container-title":["Journal of Intelligent &amp; Robotic Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10846-024-02072-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10846-024-02072-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10846-024-02072-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,18]],"date-time":"2024-07-18T10:06:45Z","timestamp":1721297205000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10846-024-02072-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,18]]},"references-count":24,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["2072"],"URL":"https:\/\/doi.org\/10.1007\/s10846-024-02072-9","relation":{},"ISSN":["1573-0409"],"issn-type":[{"type":"electronic","value":"1573-0409"}],"subject":[],"published":{"date-parts":[[2024,3,18]]},"assertion":[{"value":"19 January 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 February 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 March 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no relevant financial or non-financial interests to disclose.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of interest\/Competing interests"}}],"article-number":"48"}}