{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,22]],"date-time":"2025-12-22T05:24:00Z","timestamp":1766381040779,"version":"build-2065373602"},"reference-count":62,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,1]],"date-time":"2025-04-01T00:00:00Z","timestamp":1743465600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Robotics"],"abstract":"<jats:p>Autonomy of underwater vehicles has become an imperative feature due to increasingly challenging deep sea mission scenarios. In particular, for trajectory-tracking problems of Autonomous Underwater Vehicles (AUVs), the use of Lyapunov theory tools in state-of-the-art methods is common practice. These often require special assumptions, according to the application considered, and \u2018intuition\u2019 for the choice of a control law, which often leads to conservative results. This article suggests a systematic analysis for the horizontal motion of an AUV which ensures global asymptotic stability for the closed loop system. A nonlinear underactuated AUV system is considered with linear and angular velocity constraints. The Takagi\u2013Sugeno (TS) framework design is adopted for the representation of the original nonlinear system. The control law is synthesised using the standard parallel distributed compensation (PDC) control law structure and stability is guaranteed for the closed loop system. The design criteria are posed as linear matrix inequalities (LMIs) where sufficient conditions for the design of the control law are shown. The proposed approach can be easily adopted for different types of autonomous vehicles with minor modifications.<\/jats:p>","DOI":"10.3390\/robotics14040045","type":"journal-article","created":{"date-parts":[[2025,4,1]],"date-time":"2025-04-01T11:17:57Z","timestamp":1743506277000},"page":"45","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Optimal Trajectory Tracking for Underactuated Systems via the Takagi\u2013Sugeno Framework: An Autonomous Underwater Vehicle Mission Case Study"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7569-3302","authenticated-orcid":false,"given":"Georgios P.","family":"Kladis","sequence":"first","affiliation":[{"name":"Department of Aeronautical Science, Automatic Control, Aerospace Technology, Defence Systems and Operations Academic Section, Hellenic Air Force Academy, 13671 Dekeleia, Acharnes Attikis, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7701-4820","authenticated-orcid":false,"given":"Lefteris","family":"Doitsidis","sequence":"additional","affiliation":[{"name":"School of Production Engineering and Management, Technical University of Crete, 73100 Chania, Crete, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nikos C.","family":"Tsourveloudis","sequence":"additional","affiliation":[{"name":"School of Production Engineering and Management, Technical University of Crete, 73100 Chania, Crete, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,1]]},"reference":[{"key":"ref_1","unstructured":"Larkin, K.E., Donaldson, K., and McDonough, N. (2015). Delving Deeper: Critical Challenges for 21st Century Deep-Sea Research, European Marine Board. Position Paper 22."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"159","DOI":"10.31181\/smeor11202414","article-title":"Exploring the Role of Robotics in Maritime Technology: Innovations, Challenges, and Future Prospects","volume":"1","author":"Sahoo","year":"2024","journal-title":"Spectr. Mech. Eng. Oper. Res."},{"key":"ref_3","unstructured":"Fossen, T.I. (1994). Guidance and Control of Ocean Vehicles, Wiley."},{"key":"ref_4","unstructured":"Do, K.D., and Pan, J. (2009). Control of Ships and Underwater Vehicles: Design for Underactuated and Nonlinear Marine Systems, Springer."},{"key":"ref_5","unstructured":"Khalil, H.K. (2002). Nonlinear Systems, Prentice Hall, Inc.. [3rd ed.]."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1109\/9.362901","article-title":"Exponential stabilization of nonholonomic chained systems","volume":"40","author":"Sordalen","year":"1995","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_7","unstructured":"Pettersen, K., and Egeland, O. (1996, January 11\u201313). Exponential stabilization of an underactuated autonomous surface vessel. Proceedings of the 35th IEEE Conference on Decision and Control, Kobe, Japan."},{"key":"ref_8","first-page":"219","article-title":"Control of nonholonomic systems using chained forms","volume":"1","author":"Murray","year":"1993","journal-title":"Fields Inst. Commun."},{"key":"ref_9","unstructured":"Nakamura, Y., and Savant, S. (1992, January 12\u201314). Nonlinear tracking control of autonomous underwater vehicles. Proceedings of the 1992 IEEE International Conference on Robotics and Automation, Nice, France."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/S0005-1098(01)00199-6","article-title":"Global tracking control of underactuated ships by Lyapunov\u2019s direct method","volume":"38","author":"Jiang","year":"2002","journal-title":"Automatica"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1109\/9.739086","article-title":"Time-Varying Exponential Stabilization of the Position and Attitude of an Underactuated Autonomous Underwater Vehicle","volume":"44","author":"Pettersen","year":"1999","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Pettersen, K., and Nijmeijer, H. (1998, January 12\u201319). Global practical stabilization and tracking for an underactuated ship-a combined averaging and backstepping approach. Proceedings of the IFAC Conference on Systems Structure Control, Tampa, FL, USA.","DOI":"10.1016\/S1474-6670(17)41967-7"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1109\/TCST.2002.806465","article-title":"Tracking Control of an Underactuated Ship","volume":"11","author":"Lefeber","year":"2003","journal-title":"IEEE Trans. Control Syst. Technol."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Li, D., and Du, L. (2021). AUV Trajectory Tracking Models and Control Strategies: A Review. J. Mar. Sci. Eng., 9.","DOI":"10.3390\/jmse9091020"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Wang, Y., and Du, Z. (2024). Trajectory Tracking Control for an Underactuated AUV via Nonsingular Fast Terminal Sliding Mode Approach. J. Mar. Sci. Eng., 12.","DOI":"10.3390\/jmse12081442"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/j.oceaneng.2020.107131","article-title":"Robust trajectory tracking of autonomous underwater vehicles using back-stepping control and time delay estimation","volume":"201","author":"Cho","year":"2020","journal-title":"Ocean Eng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1007\/s11071-015-2551-x","article-title":"Trajectory tracking sliding mode control of underactuated AUVs","volume":"86","author":"Elmokadem","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Ma, C., Jia, J., Zhang, T., Wu, S., and Jiang, D. (2023). Horizontal trajectory tracking control for underactuated autonomous underwater vehicles based on contraction theory. J. Mar. Sci. Eng., 11.","DOI":"10.3390\/jmse11040805"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1054","DOI":"10.1109\/TSMC.2019.2894171","article-title":"Trajectory tracking control of autonomous underwater vehicle with unknown parameters and external disturbances","volume":"51","author":"Yang","year":"2021","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_20","first-page":"447","article-title":"Cooperative Impedance Control for Multiple Underwater Vehicle Manipulator Systems Under Lean Communication","volume":"46","author":"Bechlioulis","year":"2020","journal-title":"IEEE J. Ocean. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1002\/(SICI)1099-1115(199812)12:8<649::AID-ACS515>3.0.CO;2-P","article-title":"Nonlinear and adaptive backstepping designs for tracking control of ships","volume":"12","author":"Godhavn","year":"1998","journal-title":"Int. J. Adapt. Control Signal Process."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"111493","DOI":"10.1016\/j.oceaneng.2022.111493","article-title":"Robust yaw control of autonomous underwater vehicle based on fractional-order PID controller","volume":"257","author":"Liu","year":"2022","journal-title":"Ocean Eng."},{"key":"ref_23","unstructured":"Olfati-Saber, R. (2001). Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles. [Ph.D. Thesis, Massachusetts Institute of Technology]."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"81","DOI":"10.14257\/ijca.2016.9.1.08","article-title":"Trajectory tracking control of underactuated AUV in the presence of ocean currents","volume":"9","author":"Yang","year":"2016","journal-title":"Int. J. Control Autom."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Yan, Z., Zhang, M., Zhou, J., and Yue, L. (2024). Distributed Lyapunov-Based Model Predictive Control for AUV Formation Systems with Multiple Constraints. J. Mar. Sci. Eng., 12.","DOI":"10.3390\/jmse12030363"},{"key":"ref_26","unstructured":"Repoulias, F., and Papadopoulos, E. (2005, January 18\u201322). Trajectory planning and tracking control design of underactuated AUVs. Proceedings of the IEEE International Conference on Robotics and Automation, ICRA 05, Barcelona, Spain."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1650","DOI":"10.1016\/j.oceaneng.2006.11.007","article-title":"Planar trajectory planning and tracking control design for underactuated AUVs","volume":"34","author":"Repoulias","year":"2007","journal-title":"Ocean Eng."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1734","DOI":"10.1016\/j.oceaneng.2006.10.019","article-title":"Nonlinear Path-following Control of an AUV","volume":"34","author":"Lapierre","year":"2007","journal-title":"Ocean Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1109\/37.642974","article-title":"Control architectures for autonomous underwater vehicles","volume":"17","author":"Valavanis","year":"1997","journal-title":"IEEE Control Syst. Mag."},{"key":"ref_30","unstructured":"Lefeber, E. (2000). Tracking Control of Nonlinear Mechanical Systems. [Ph.D. Thesis, University of Twente]."},{"key":"ref_31","unstructured":"Do, K.D., Pan, J., and Jiang, Z.P. (2000, January 8\u201310). Global Exponential Tracking Control of Underactuated Surface Ships in the Body Frame. Proceedings of the American Control Conference, Anchorage, AK, USA."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1529","DOI":"10.1109\/TAC.2002.802755","article-title":"Underactuated Ship Global Tracking under Relaxed Conditions","volume":"47","author":"Do","year":"2000","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_33","unstructured":"Millman, R.S., Brockett, R.W., and Sussmann, H.J. (1983). Asymptotic stability and feedback stabilization. Differential Geometric Control Theory, Birkauser."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Soltan, R., Ashrafiuon, H., and Muske, K. (2009, January 10\u201312). State-dependent trajectory planning and tracking control of unmanned surface vessels. Proceedings of the American Control Conference, St. Louis, MO, USA.","DOI":"10.1109\/ACC.2009.5160010"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"336","DOI":"10.5772\/56671","article-title":"Global asymptotic trajectory tracking and point stabilization of asymmetric underactuated ships with non-diagonal inertia\/damping matrices","volume":"10","author":"Ma","year":"2013","journal-title":"Int. J. Adv. Robot. Syst."},{"key":"ref_36","first-page":"2035","article-title":"Global tracking for underactuated ships with bounded feedback controllers","volume":"87","author":"Harmouche","year":"2014","journal-title":"Int. J. Control"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"540","DOI":"10.1007\/s11771-014-1972-x","article-title":"Trajectory planning and tracking control for underactuated unmanned surface vessels","volume":"21","author":"Liao","year":"2014","journal-title":"J. Cent. South Univ."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1515\/ijnaoe-2015-0058","article-title":"Trajectory tracking control of underactuated USV based on modified backstepping approach","volume":"7","author":"Dong","year":"2015","journal-title":"Int. J. Nav. Archit. Ocean Eng."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Tanaka, K., and Wang, H.O. (2001). Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach, John Wiley and Sons, Inc.","DOI":"10.1002\/0471224596"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"155","DOI":"10.51400\/2709-6998.2276","article-title":"Model-based fuzzy modeling and control for autonomous underwater vehicles in the horizontal plane","volume":"11","author":"Chang","year":"2003","journal-title":"J. Mar. Sci. Technol."},{"key":"ref_41","unstructured":"Jun, S., Kim, D., and Lee, H. (2011, January 26\u201329). Design of TS fuzzy model-based controller for depth control of autonomous underwater vehicles with parametric uncertainties. Proceedings of the 11th International Conference of Control, Automation and Systems (ICCAS), Gyeonggi-do, Republic of Korea."},{"key":"ref_42","unstructured":"Wang, H.O., Tanaka, K., and Griffin, M.F. (1995, January 20\u201324). Parallel Distributed Compensation of Nonlinear Systems by Takagi-Sugeno Fuzzy Model. Proceedings of the 1995 IEEE International Conference on Fuzzy Systems, Yokohama, Japan."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Kladis, G.P., Menon, P.P., and Edwards, C. (2011, January 12\u201315). Cooperative tracking for a swarm of Unmanned Aerial Vehicles: A distributed Tagaki-Sugeno fuzzy framework design. Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Orlando, FL, USA.","DOI":"10.1109\/CDC.2011.6161123"},{"key":"ref_44","unstructured":"Shanmugavel, M. (2007). Path Planning of Multiple Autonomous Vehicles. [Ph.D. Thesis, Cranfield University]."},{"key":"ref_45","unstructured":"Tanaka, K., and Wang, H.O. (1997, January 12). Fuzzy Regulators and Fuzzy Observers: A Linear Matrix Inequality Approach. Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, CA, USA."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1109\/91.669023","article-title":"Fuzzy Regulators and Fuzzy Observers","volume":"6","author":"Tanaka","year":"1998","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Delmotte, F., Lauber, J., and Guerra, T.M. (2008, January 1\u20136). A multi-model controller. Proceedings of the 2008 IEEE International Conference on Fuzzy Systems (IEEE World Congress on Computational Intelligence), Hong Kong, China.","DOI":"10.1109\/FUZZY.2008.4630574"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1248","DOI":"10.1016\/j.fss.2005.12.006","article-title":"Conditions of output stabilization for nonlinear models in the Takagi\u2013Sugeno form","volume":"157","author":"Guerra","year":"2006","journal-title":"Fuzzy Sets Syst."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Bernal, M., Sala, A., Lendek, Z., and Guerra, T.M. (2022). Analysis and Synthesis of Nonlinear Control Systems, Springer.","DOI":"10.1007\/978-3-030-90773-0"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1007\/s10846-013-9968-6","article-title":"Generation of bezier curve-based flyable trajectories for multi-uav systems with parallel genetic algorithm","volume":"74","author":"Sahingoz","year":"2014","journal-title":"J. Intell. Robot. Syst."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Shivgan, R., and Dong, Z. (2020, January 11\u201314). Energy-efficient drone coverage path planning using genetic algorithm. Proceedings of the 2020 IEEE 21st International Conference on High Performance Switching and Routing (HPSR), Newark, NJ, USA.","DOI":"10.1109\/HPSR48589.2020.9098989"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/s10846-023-01945-9","article-title":"A fuzzy logic approach of pareto optimality for multi-objective path planning in case of unmanned surface vehicle","volume":"109","author":"Ntakolia","year":"2023","journal-title":"J. Intell. Robot. Syst."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/j.robot.2016.03.011","article-title":"A comparison of optimization techniques for AUV path planning in environments with ocean currents","volume":"82","author":"Zeng","year":"2016","journal-title":"Robot. Auton. Syst."},{"key":"ref_54","first-page":"153","article-title":"Path planning for cooperating unmanned vehicles over 3-D terrain","volume":"24","author":"Nikolos","year":"2009","journal-title":"Inf. Control Autom. Robot."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1016\/j.engappai.2010.10.013","article-title":"Energy Conservation Based Fuzzy Tracking For Unmanned Aerial Vehicle Missions Under Priori Known Wind Information","volume":"24","author":"Kladis","year":"2011","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_56","unstructured":"Kawamoto, S., Tada, K., Ishigame, A., and Taniguchi, T. (1992, January 8\u201312). An Approach to Stability Analysis of Second Order Fuzzy Systems. Proceedings of the First IEEE International Conference on Fuzzy Systems, San Diego, CA, USA."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1571","DOI":"10.1016\/S0005-1098(03)00172-9","article-title":"New approaches to controller designs based on fuzzy observers for Takagi\u2013Sugeno fuzzy systems via LMI","volume":"39","author":"Liu","year":"2003","journal-title":"Automatica"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1046","DOI":"10.1109\/TFUZZ.2012.2191790","article-title":"Observer-based piecewise affine output feedback controller synthesis of continuous-time T-S fuzzy affine dynamic systems using quantized measurements","volume":"20","author":"Qiu","year":"2012","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"919","DOI":"10.1109\/TFUZZ.2010.2052259","article-title":"Fuzzy-model-based piecewise Hinf static-output-feedback controller design for networked nonlinear systems","volume":"18","author":"Qiu","year":"2010","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"3803","DOI":"10.1080\/00207721.2015.1126380","article-title":"Fuzzy distributed cooperative tracking for a swarm of unmanned aerial vehicles with heterogeneous goals","volume":"47","author":"Kladis","year":"2015","journal-title":"Int. J. Syst. Sci."},{"key":"ref_61","unstructured":"Lofberg, J. (2004, January 2\u20134). YALMIP: A toolbox for modeling and optimization in MATLAB. Proceedings of the 2004 IEEE International Conference on Robotics and Automation, Taipei, Taiwan."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"2043","DOI":"10.1016\/j.fss.2010.03.010","article-title":"Adaptive observers for TS fuzzy systems with unknown polynomial inputs","volume":"161","author":"Lendek","year":"2010","journal-title":"Fuzzy Sets Syst."}],"container-title":["Robotics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2218-6581\/14\/4\/45\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:07:43Z","timestamp":1760029663000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2218-6581\/14\/4\/45"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,1]]},"references-count":62,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["robotics14040045"],"URL":"https:\/\/doi.org\/10.3390\/robotics14040045","relation":{},"ISSN":["2218-6581"],"issn-type":[{"type":"electronic","value":"2218-6581"}],"subject":[],"published":{"date-parts":[[2025,4,1]]}}}