{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T18:28:12Z","timestamp":1785954492304,"version":"3.56.0"},"reference-count":34,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,7]],"date-time":"2023-08-07T00:00:00Z","timestamp":1691366400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"China Postdoctoral Science Special Foundation","award":["2021TQ0102"],"award-info":[{"award-number":["2021TQ0102"]}]},{"name":"China Postdoctoral Science Special Foundation","award":["62203158"],"award-info":[{"award-number":["62203158"]}]},{"name":"China Postdoctoral Science Special Foundation","award":["kq2202175"],"award-info":[{"award-number":["kq2202175"]}]},{"name":"China Postdoctoral Science Special Foundation","award":["2023JJ40182"],"award-info":[{"award-number":["2023JJ40182"]}]},{"name":"National Natural Science Foundation of China","award":["2021TQ0102"],"award-info":[{"award-number":["2021TQ0102"]}]},{"name":"National Natural Science Foundation of China","award":["62203158"],"award-info":[{"award-number":["62203158"]}]},{"name":"National Natural Science Foundation of China","award":["kq2202175"],"award-info":[{"award-number":["kq2202175"]}]},{"name":"National Natural Science Foundation of China","award":["2023JJ40182"],"award-info":[{"award-number":["2023JJ40182"]}]},{"name":"Changsha Natural Science Foundation","award":["2021TQ0102"],"award-info":[{"award-number":["2021TQ0102"]}]},{"name":"Changsha Natural Science Foundation","award":["62203158"],"award-info":[{"award-number":["62203158"]}]},{"name":"Changsha Natural Science Foundation","award":["kq2202175"],"award-info":[{"award-number":["kq2202175"]}]},{"name":"Changsha Natural Science Foundation","award":["2023JJ40182"],"award-info":[{"award-number":["2023JJ40182"]}]},{"name":"National Natural Science Foundation of Hunan Province","award":["2021TQ0102"],"award-info":[{"award-number":["2021TQ0102"]}]},{"name":"National Natural Science Foundation of Hunan Province","award":["62203158"],"award-info":[{"award-number":["62203158"]}]},{"name":"National Natural Science Foundation of Hunan Province","award":["kq2202175"],"award-info":[{"award-number":["kq2202175"]}]},{"name":"National Natural Science Foundation of Hunan Province","award":["2023JJ40182"],"award-info":[{"award-number":["2023JJ40182"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This article primarily investigates nonlinear disturbance observer-based bearing-only formation tracking control for unmanned aerial vehicle (UAV) systems that encounter uncertainties and disturbances. The employed distributed control strategy relies solely on the relative bearing information of neighboring UAVs. To tackle the challenges posed by unknown disturbances and system uncertainties, a novel nonlinear disturbance observer is proposed to effectively mitigate their impact. Moreover, the issue of unknown controller orientation arising from controller singularities is addressed by introducing a Butterworth low-pass filter. This filter ensures a consistent controller gain and enhances disturbance suppression, ultimately transforming the controller gain function into a constant value of 1. Subsequently, a bearing-only formation tracking controller is developed using the backstepping control approach. The stability of the closed-loop control systems is rigorously proven using Lyapunov theory. Finally, numerical simulations are conducted to validate the effectiveness of the proposed scheme in achieving formation control objectives.<\/jats:p>","DOI":"10.3390\/axioms12080768","type":"journal-article","created":{"date-parts":[[2023,8,8]],"date-time":"2023-08-08T12:45:53Z","timestamp":1691498753000},"page":"768","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Nonlinear Disturbance Observer-Based Bearing-Only Unmanned Aerial Vehicle Formation Control"],"prefix":"10.3390","volume":"12","author":[{"given":"Can","family":"Ding","sequence":"first","affiliation":[{"name":"College of Electrical and Information Engineering, Hunan University, Changsha 211105, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jing","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Electrical and Information Engineering, Hunan University, Changsha 211105, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5350-0795","authenticated-orcid":false,"given":"Zhe","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Electrical and Information Engineering, Hunan University, Changsha 211105, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"106332","DOI":"10.1016\/j.ast.2020.106332","article-title":"A new consensus theory-based method for formation control and obstacle avoidance of UAVs","volume":"107","author":"Wu","year":"2020","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3671","DOI":"10.1016\/j.jfranklin.2022.02.034","article-title":"Event-triggered-based adaptive dynamic programming for distributed formation control of mul-ti-UAV","volume":"359","author":"Dou","year":"2022","journal-title":"J. Frankl. Inst."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2514","DOI":"10.1049\/iet-cta.2020.0342","article-title":"Dynamic event-triggered cooperative formation control for UAVs subject to time-varying disturbances","volume":"14","author":"Wei","year":"2020","journal-title":"IET Control Theory Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"111495","DOI":"10.1016\/j.oceaneng.2022.111495","article-title":"A novel path following approach for autonomous ships based on fast marching method and deep reinforcement learning","volume":"257","author":"Wang","year":"2022","journal-title":"Ocean Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2583","DOI":"10.1007\/s11071-020-06050-2","article-title":"High-order sliding mode observer-based trajectory tracking control for a quadrotor UAV with uncertain dynamics","volume":"102","author":"Zhao","year":"2020","journal-title":"Nonlinear Dyn."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.fss.2022.03.005","article-title":"Fuzzy-based dynamic event triggering formation control for nonstrict-feedback nonlinear MASs","volume":"452","author":"Cao","year":"2023","journal-title":"Fuzzy Sets Syst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"820","DOI":"10.1007\/s11424-021-0150-0","article-title":"Distributed event-triggered formation control of USVs with prescribed performance","volume":"35","author":"Chen","year":"2022","journal-title":"J. Syst. Sci. Complex."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"111067","DOI":"10.1016\/j.oceaneng.2022.111067","article-title":"A fixed-time distributed formation control of marine surface vehicles with actuator input saturation and time-varying ocean currents","volume":"251","author":"Fu","year":"2022","journal-title":"Ocean Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"107600","DOI":"10.1016\/j.ast.2022.107600","article-title":"Distributed robust adaptive formation control of fixed-wing UAVs with unknown uncertainties and disturbances","volume":"126","author":"Zhi","year":"2022","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cja.2021.04.022","article-title":"A review on fault-tolerant cooperative control of multiple unmanned aerial vehicles","volume":"35","author":"Ziquan","year":"2022","journal-title":"Chin. J. Aeronaut."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Xu, Z., Yan, T., Yang, S.X., and Gadsden, S.A. (IEEE Trans. Ind. Inform., 2023). Distributed Leader Follower Formation Control of Mobile Robots based on Bioinspired Neural Dynamics and Adaptive Sliding Innovation Filter, IEEE Trans. Ind. Inform., in press.","DOI":"10.1109\/TII.2023.3272666"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"5198","DOI":"10.1109\/TSMC.2019.2946127","article-title":"Distributed Lyapunov-based model predictive formation tracking control for autonomous underwater vehicles subject to disturbances","volume":"51","author":"Wei","year":"2019","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1016\/j.neucom.2021.12.056","article-title":"Vision-based neural formation tracking control of multiple autonomous vehicles with visibility and performance constraints","volume":"492","author":"He","year":"2022","journal-title":"Neurocomputing"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2401","DOI":"10.1109\/LCSYS.2022.3159128","article-title":"Bearing-constrained formation tracking control of nonholonomic agents without inter-agent communication","volume":"6","author":"Kim","year":"2022","journal-title":"IEEE Control Syst. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"111672","DOI":"10.1016\/j.oceaneng.2022.111672","article-title":"Bearing-based formation tracking control of AUVs with optimal gains tuning","volume":"258","author":"Su","year":"2022","journal-title":"Ocean Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"110310","DOI":"10.1016\/j.automatica.2022.110310","article-title":"Gradient-based bearing-only formation control: An elevation angle approach","volume":"141","author":"Chen","year":"2022","journal-title":"Automatica"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1261","DOI":"10.1080\/00207179.2019.1641626","article-title":"Bearing-only formation control of multi-agent systems in local reference frames","volume":"94","author":"Luo","year":"2021","journal-title":"Int. J. Control"},{"key":"ref_18","first-page":"3316","article-title":"Finite-time bearing-only formation tracking of heterogeneous mobile robots with collision avoidance","volume":"68","author":"Wu","year":"2021","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/LCSYS.2020.2999972","article-title":"Bearing-only formation tracking control of multi-agent systems with local reference frames and con-stant-velocity leaders","volume":"5","author":"Zhao","year":"2020","journal-title":"IEEE Control Syst. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"6556","DOI":"10.1002\/rnc.6146","article-title":"Distributed bearing-based formation control and network localization with exogenous disturbances","volume":"32","author":"Bae","year":"2022","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1255","DOI":"10.1109\/TAC.2015.2459191","article-title":"Bearing rigidity and almost global bearing-only formation stabilization","volume":"61","author":"Zhao","year":"2015","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"4541","DOI":"10.1109\/TAC.2019.2903290","article-title":"Bearing-only formation tracking control of multiagent systems","volume":"64","author":"Zhao","year":"2019","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"7552","DOI":"10.1109\/TCYB.2020.3042491","article-title":"Adaptive bearing-only formation tracking control for nonholonomic multiagent systems","volume":"52","author":"Li","year":"2021","journal-title":"IEEE Trans. Cybern."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1231","DOI":"10.1109\/LCSYS.2022.3231413","article-title":"Localization and tracking control of autonomous vehicles in time-varying bearing formation","volume":"7","author":"Tang","year":"2022","journal-title":"IEEE Control Syst. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Su, H., Chen, C., Yang, Z., Zhu, S., and Guan, X. (IEEE Control Syst. Lett., 2022). Bearing-Based Formation Tracking Control with Time-Varying Velocity Estimation, IEEE Control Syst. Lett., in press.","DOI":"10.1109\/TCYB.2022.3169891"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3897","DOI":"10.1007\/s10462-021-10097-x","article-title":"Consensus in multi-agent systems: A review","volume":"55","author":"Amirkhani","year":"2022","journal-title":"Artif. Intell. Rev."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1080\/21642583.2022.2074169","article-title":"Recent advances on cooperative control of heterogeneous multi-agent systems subject to constraints: A survey","volume":"10","author":"Bao","year":"2022","journal-title":"Syst. Sci. Control Eng."},{"key":"ref_28","first-page":"4620","article-title":"Bearing-only formation control with prespecified convergence time","volume":"52","author":"Li","year":"2020","journal-title":"IEEE Trans. Cybern."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"13363","DOI":"10.1109\/TCYB.2021.3124827","article-title":"Finite-time cooperative control for bearing-defined leader-following formation of multiple dou-ble-integrators","volume":"52","author":"Zhao","year":"2021","journal-title":"IEEE Trans. Cybern."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1016\/j.automatica.2016.03.010","article-title":"Localizability and distributed protocols for bearing-based network localization in arbitrary dimensions","volume":"69","author":"Zhao","year":"2016","journal-title":"Automatica"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1109\/TCST.2014.2314460","article-title":"Time-varying formation control for unmanned aerial vehicles: Theories and applications","volume":"23","author":"Dong","year":"2014","journal-title":"IEEE Trans. Control Syst. Technol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1739","DOI":"10.1016\/j.jfranklin.2015.01.014","article-title":"Tracking differentiator design for the robust backstepping control of a flexible air-breathing hy-personic vehicle","volume":"352","author":"Bu","year":"2015","journal-title":"J. Frankl. Inst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1109\/TCYB.2020.2978981","article-title":"Adaptive formation control of networked robotic systems with bearing-only measurements","volume":"51","author":"Li","year":"2020","journal-title":"IEEE Trans. Cybern."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2449","DOI":"10.1109\/JSYST.2017.2778063","article-title":"Trajectory tracking by multiple agents in for-mation with collision avoidance and connectivity assurance","volume":"12","author":"Mondal","year":"2017","journal-title":"IEEE Syst. J."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/8\/768\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:27:35Z","timestamp":1760128055000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/8\/768"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,7]]},"references-count":34,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,8]]}},"alternative-id":["axioms12080768"],"URL":"https:\/\/doi.org\/10.3390\/axioms12080768","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,7]]}}}