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Firstly, the dynamic models of UAVs and USVs are introduced, and a unified heterogeneous multi-agent system model with actuator faults is established. Then, a distributed fault-tolerant formation controller is proposed for the unified model of UAVs and USVs in the XY plane by using adaptive updating laws and radial basis function neural network. After that, a decentralized formation-tracking controller is designed for the altitude control system of UAVs. Based on the Lyapunov stability theory, it can be proved that the formation errors and tracking errors are uniformly ultimately bounded which means that the expected time-varying formation is achieved. Finally, a simulation study is given to demonstrate the effectiveness of the proposed scheme.<\/jats:p>","DOI":"10.3390\/s22166212","type":"journal-article","created":{"date-parts":[[2022,8,18]],"date-time":"2022-08-18T23:28:41Z","timestamp":1660865321000},"page":"6212","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Adaptive Fault-Tolerant Formation Control of Heterogeneous Multi-Agent Systems under Directed Communication Topology"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7671-0605","authenticated-orcid":false,"given":"Shangkun","family":"Liu","sequence":"first","affiliation":[{"name":"College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China"},{"name":"Jiangsu Key Laboratory of Internet of Things and Control Technologies, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9153-4360","authenticated-orcid":false,"given":"Bin","family":"Jiang","sequence":"additional","affiliation":[{"name":"College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China"},{"name":"Jiangsu Key Laboratory of Internet of Things and Control Technologies, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3189-5359","authenticated-orcid":false,"given":"Zehui","family":"Mao","sequence":"additional","affiliation":[{"name":"College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China"},{"name":"Jiangsu Key Laboratory of Internet of Things and Control Technologies, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8040-4085","authenticated-orcid":false,"given":"Yajie","family":"Ma","sequence":"additional","affiliation":[{"name":"College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China"},{"name":"Jiangsu Key Laboratory of Internet of Things and Control Technologies, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/j.isatra.2020.07.005","article-title":"Distributed adaptive fault-tolerant close formation flight control of multiple trailing fixed-wing UAVs","volume":"106","author":"Yu","year":"2020","journal-title":"ISA Trans."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1646","DOI":"10.1109\/TCSI.2021.3049347","article-title":"Distributed Fault-Tolerant Consensus Tracking Control of Multi-Agent Systems under Fixed and Switching Topologies","volume":"68","author":"Liu","year":"2021","journal-title":"IEEE Trans. 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