{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T16:49:50Z","timestamp":1773161390660,"version":"3.50.1"},"reference-count":42,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,4,19]],"date-time":"2024-04-19T00:00:00Z","timestamp":1713484800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia (FCT)","doi-asserted-by":"publisher","award":["2020.06736.BD"],"award-info":[{"award-number":["2020.06736.BD"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia (FCT)","doi-asserted-by":"publisher","award":["MYRG 2022-00205-FST"],"award-info":[{"award-number":["MYRG 2022-00205-FST"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia (FCT)","doi-asserted-by":"publisher","award":["MYRG-GRG2023-00107-FST-UMDF"],"award-info":[{"award-number":["MYRG-GRG2023-00107-FST-UMDF"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Institute for Mechanical Engineering (IDMEC)","award":["2020.06736.BD"],"award-info":[{"award-number":["2020.06736.BD"]}]},{"name":"Institute for Mechanical Engineering (IDMEC)","award":["MYRG 2022-00205-FST"],"award-info":[{"award-number":["MYRG 2022-00205-FST"]}]},{"name":"Institute for Mechanical Engineering (IDMEC)","award":["MYRG-GRG2023-00107-FST-UMDF"],"award-info":[{"award-number":["MYRG-GRG2023-00107-FST-UMDF"]}]},{"name":"LAETA Programatic Funding","award":["2020.06736.BD"],"award-info":[{"award-number":["2020.06736.BD"]}]},{"name":"LAETA Programatic Funding","award":["MYRG 2022-00205-FST"],"award-info":[{"award-number":["MYRG 2022-00205-FST"]}]},{"name":"LAETA Programatic Funding","award":["MYRG-GRG2023-00107-FST-UMDF"],"award-info":[{"award-number":["MYRG-GRG2023-00107-FST-UMDF"]}]},{"name":"Institute for Systems and Robotics (ISR)","award":["2020.06736.BD"],"award-info":[{"award-number":["2020.06736.BD"]}]},{"name":"Institute for Systems and Robotics (ISR)","award":["MYRG 2022-00205-FST"],"award-info":[{"award-number":["MYRG 2022-00205-FST"]}]},{"name":"Institute for Systems and Robotics (ISR)","award":["MYRG-GRG2023-00107-FST-UMDF"],"award-info":[{"award-number":["MYRG-GRG2023-00107-FST-UMDF"]}]},{"name":"University of Macau, Macau, China","award":["2020.06736.BD"],"award-info":[{"award-number":["2020.06736.BD"]}]},{"name":"University of Macau, Macau, China","award":["MYRG 2022-00205-FST"],"award-info":[{"award-number":["MYRG 2022-00205-FST"]}]},{"name":"University of Macau, Macau, China","award":["MYRG-GRG2023-00107-FST-UMDF"],"award-info":[{"award-number":["MYRG-GRG2023-00107-FST-UMDF"]}]},{"name":"European Union under the Next Generation EU","award":["2020.06736.BD"],"award-info":[{"award-number":["2020.06736.BD"]}]},{"name":"European Union under the Next Generation EU","award":["MYRG 2022-00205-FST"],"award-info":[{"award-number":["MYRG 2022-00205-FST"]}]},{"name":"European Union under the Next Generation EU","award":["MYRG-GRG2023-00107-FST-UMDF"],"award-info":[{"award-number":["MYRG-GRG2023-00107-FST-UMDF"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Drones"],"abstract":"<jats:p>This paper introduces a quadrotor trajectory tracking controller comprising a steady-state optimal position controller with a normed input saturation and modular integrative action coupled with a backstepping attitude controller. First, the translational and rotational dynamical models are designed in the body-fixed frame to avoid external rotations and are partitioned into an underactuated position system and a quaternion-based attitude system. Secondly, a controller is designed separately for each subsystem, namely, (i) the position controller synthesis is derived from the Maximum Principle, Lyapunov, and linear quadratic regulator (LQR) theory, ensuring the global exponential stability and steady-state optimality of the controller within the linear region, and global asymptotic stability is guaranteed for the saturation region when coupled with any local exponential stable attitude controller, and (ii) the attitude system, with the quaternion angles and the angular velocity as the controlled variables, is designed in the error space through the backstepping technique, which renders the overall system, position, and attitude, with desirable closed-loop properties that are almost global. The overall stability of the system is achieved through the propagation of the position interconnection term to the attitude system. To enhance the robustness of the tracking system, integrative action is devised for both position and attitude, with emphasis on the modular approach for the integrative action on the position controller. The proposed method is experimentally validated on board an off-the-shelf quadrotor to assess the resulting performance.<\/jats:p>","DOI":"10.3390\/drones8040163","type":"journal-article","created":{"date-parts":[[2024,4,19]],"date-time":"2024-04-19T08:44:31Z","timestamp":1713516271000},"page":"163","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Saturated Trajectory Tracking Controller in the Body-Frame for Quadrotors"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2752-6398","authenticated-orcid":false,"given":"Jo\u00e3o","family":"Madeiras","sequence":"first","affiliation":[{"name":"IDMEC \u2014Institute of Mechanical Engineering, Instituto Superior T\u00e9cnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7966-4648","authenticated-orcid":false,"given":"Carlos","family":"Cardeira","sequence":"additional","affiliation":[{"name":"IDMEC \u2014Institute of Mechanical Engineering, Instituto Superior T\u00e9cnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5799-390X","authenticated-orcid":false,"given":"Paulo","family":"Oliveira","sequence":"additional","affiliation":[{"name":"IDMEC \u2014Institute of Mechanical Engineering, Instituto Superior T\u00e9cnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal"},{"name":"ISR\u2014Institute for Systems and Robotics, Instituto Superior T\u00e9cnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6079-0436","authenticated-orcid":false,"given":"Pedro","family":"Batista","sequence":"additional","affiliation":[{"name":"ISR\u2014Institute for Systems and Robotics, Instituto Superior T\u00e9cnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5096-5527","authenticated-orcid":false,"given":"Carlos","family":"Silvestre","sequence":"additional","affiliation":[{"name":"ISR\u2014Institute for Systems and Robotics, Instituto Superior T\u00e9cnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal"},{"name":"Department of Electrical and Computer Engineering, Faculty of Science and Technology, University of Macau, Macao 999078, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1016\/j.conengprac.2010.02.008","article-title":"Modelling and control of a large quadrotor robot","volume":"18","author":"Pounds","year":"2010","journal-title":"Control Eng. 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