{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,28]],"date-time":"2026-03-28T20:00:59Z","timestamp":1774728059484,"version":"3.50.1"},"reference-count":57,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,6,12]],"date-time":"2021-06-12T00:00:00Z","timestamp":1623456000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["JMSE"],"abstract":"<jats:p>A modified path-following control system using the vector field method for an underactuated autonomous surface ship model is proposed in the presence of static obstacles. With this integrated system, autonomous ships are capable of following the predefined path, while avoiding the obstacles automatically. It is different from the methods in most published papers, which usually study path-following and obstacle collision avoidance, separately. This paper considers the coupled path following and collision avoidance task as a whole. Meanwhile, the paper also shows the heading control design method in the presence of static obstacles. To obtain a strong stability property, a nonlinear autopilot is designed based on the manoeuvring tests of the free-running ship model. The equilibrium point of the controller is globally exponentially stable. For the guidance system, a novel vector field method was proposed, and the proof shows the coupled guidance and control system is uniform semi-global exponentially stable (USGES). To prevent the obstacles near the predefined path, the proposed guidance law is augmented by integrating the repelling field of obstacles so that it can control the ship travel toward the predefined path through the obstacles safely. The repelling field function is given considering the obstacle shape and collision risk using the velocity obstacle (VO) algorithm. The simulations and ship model test were performed to validate the integrated system of autonomous ships.<\/jats:p>","DOI":"10.3390\/jmse9060652","type":"journal-article","created":{"date-parts":[[2021,6,14]],"date-time":"2021-06-14T22:25:46Z","timestamp":1623709546000},"page":"652","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":29,"title":["Modified Vector Field Path-Following Control System for an Underactuated Autonomous Surface Ship Model in the Presence of Static Obstacles"],"prefix":"10.3390","volume":"9","author":[{"given":"Haitong","family":"Xu","sequence":"first","affiliation":[{"name":"Centre for Marine Technology and Ocean Engineering (CENTEC), Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal"}]},{"given":"Miguel A.","family":"Hinostroza","sequence":"additional","affiliation":[{"name":"Centre for Marine Technology and Ocean Engineering (CENTEC), Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8570-4263","authenticated-orcid":false,"given":"C.","family":"Guedes Soares","sequence":"additional","affiliation":[{"name":"Centre for Marine Technology and Ocean Engineering (CENTEC), Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2021,6,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/S0951-8320(01)00104-1","article-title":"Risk Assessment in Maritime Transportation","volume":"74","author":"Teixeira","year":"2001","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_2","unstructured":"Allianz Global Corporate and Speciality (2018). Safety and Shipping Review 2018, Allianz Global Corporate & Specialty."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2282","DOI":"10.1109\/OCEANS.2004.1406507","article-title":"Path Following for Marine Surface Vessels","volume":"Volume 4","author":"Breivik","year":"2004","journal-title":"Oceans \u201904 MTS\/IEEE Techno-Ocean \u201904 (IEEE Cat. No.04CH37600)"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"131","DOI":"10.4173\/mic.2008.4.2","article-title":"Straight-Line Target Tracking for Unmanned Surface Vehicles","volume":"29","author":"Breivik","year":"2008","journal-title":"Model. Identif. Control A Nor. Res. Bull."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Yanushevsky, R. (2011). Guidance of Unmanned Aerial Vehicles, Taylor & Francis.","DOI":"10.1201\/b10807"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Fossen, T.I. (2011). Handbook of Marine Craft Hydrodynamics and Motion Control, John Wiley & Sons, Ltd.","DOI":"10.1002\/9781119994138"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"820","DOI":"10.1109\/TCST.2014.2338354","article-title":"Line-of-Sight Path Following for Dubins Paths with Adaptive Sideslip Compensation of Drift Forces","volume":"23","author":"Fossen","year":"2015","journal-title":"IEEE Trans. Control Syst. Technol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"99","DOI":"10.5957\/jsr.2015.59.2.99","article-title":"Integrated Line of Sight and Model Predictive Control for Path Following and Roll Motion Control Using Rudder","volume":"59","author":"Liu","year":"2015","journal-title":"J. Sh. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"153","DOI":"10.4173\/mic.1996.2.7","article-title":"Identification of Dynamically Positioned Ships","volume":"17","author":"Fossen","year":"1996","journal-title":"Model. Identif. Control"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Moe, S., Pettersen, K.Y., Fossen, T.I., and Gravdahl, J.T. (2016, January 21\u201324). Line-of-Sight Curved Path Following for Underactuated USVs and AUVs in the Horizontal Plane under the Influence of Ocean Currents. Proceedings of the 24th Mediterranean Conference on Control and Automation, MED 2016, Athens, Greece.","DOI":"10.1109\/MED.2016.7536018"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Moe, S., and Pettersen, K.Y. (2016, January 21\u201324). Set-Based Line-of-Sight (LOS) Path Following with Collision Avoidance for Underactuated Unmanned Surface Vessel. Proceedings of the 24th Mediterranean Conference on Control and Automation, MED 2016, Athens, Greece.","DOI":"10.1109\/MED.2016.7535964"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"610","DOI":"10.1109\/TRO.2017.2651119","article-title":"Integral Line-of-Sight Guidance for Path Following Control of Underwater Snake Robots: Theory and Experiments","volume":"33","author":"Kelasidi","year":"2017","journal-title":"IEEE Trans. Robot."},{"key":"ref_13","first-page":"A-411","article-title":"An Optimized Energy-Efficient Path Following Algorithm for Underactuated Marine Surface Ship Model","volume":"160","author":"Xu","year":"2018","journal-title":"Int. J. Marit. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"398","DOI":"10.3182\/20120919-3-IT-2046.00068","article-title":"A Time-Varying Lookahead Distance Guidance Law for Path Following","volume":"9","author":"Lekkas","year":"2012","journal-title":"IFAC Proc. Vol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2074","DOI":"10.1016\/j.oceaneng.2007.02.005","article-title":"Path Following Control System for a Tanker Ship Model","volume":"34","author":"Moreira","year":"2007","journal-title":"Ocean Eng."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Vu, M.T., Le, T.-H., Thanh, H.L.N.N., Huynh, T.-T., Van, M., Hoang, Q.-D., and Do, T.D. (2021). Robust Position Control of an Over-actuated Underwater Vehicle under Model Uncertainties and Ocean Current Effects Using Dynamic Sliding Mode Surface and Optimal Allocation Control. Sensors, 21.","DOI":"10.3390\/s21030747"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"6855","DOI":"10.1109\/ACCESS.2020.3048706","article-title":"Station-Keeping Control of a Hovering Over-Actuated Autonomous Underwater Vehicle under Ocean Current Effects and Model Uncertainties in Horizontal Plane","volume":"9","author":"Vu","year":"2021","journal-title":"IEEE Access"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Borhaug, E., Pavlov, A., and Pettersen, K.Y. (2008, January 9\u201311). Integral LOS Control for Path Following of Underactuated Marine Surface Vessels in the Presence of Constant Ocean Currents. Proceedings of the 2008 47th IEEE Conference on Decision and Control, Cancun, Mexico.","DOI":"10.1109\/CDC.2008.4739352"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"380","DOI":"10.3182\/20120919-3-IT-2046.00065","article-title":"Relative Velocity Control and Integral Los for Path Following of Underactuated Surface Vessels","volume":"9","author":"Caharija","year":"2012","journal-title":"IFAC Proc. Vol."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2287","DOI":"10.1109\/TCST.2014.2306774","article-title":"Integral LOS Path Following for Curved Paths Based on a Monotone Cubic Hermite Spline Parametrization","volume":"22","author":"Lekkas","year":"2014","journal-title":"IEEE Trans. Control Syst. Technol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1623","DOI":"10.1109\/TCST.2015.2504838","article-title":"Integral Line-of-Sight Guidance and Control of Underactuated Marine Vehicles: Theory, Simulations, and Experiments","volume":"24","author":"Caharija","year":"2016","journal-title":"IEEE Trans. Control Syst. Technol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1002\/acs.2550","article-title":"Direct and Indirect Adaptive Integral Line-of-Sight Path-Following Controllers for Marine Craft Exposed to Ocean Currents","volume":"31","author":"Fossen","year":"2015","journal-title":"Int. J. Adapt. Control Signal Process."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Nelson, D.R., Barber, D.B., McLain, T.W., and Beard, R.W. (2006, January 14\u201316). Vector Field Path Following for Small Unmanned Air Vehicles. Proceedings of the 2006 American Control Conference, Minneapolis, MN, USA.","DOI":"10.1109\/ACC.2006.1657648"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1109\/TRO.2007.898976","article-title":"Vector Field Path Following for Miniature Air Vehicles","volume":"23","author":"Nelson","year":"2007","journal-title":"IEEE Trans. Robot."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1220","DOI":"10.2514\/1.34896","article-title":"Lyapunov Vector Fields for Autonomous Unmanned Aircraft Flight Control","volume":"31","author":"Lawrence","year":"2008","journal-title":"J. Guid. Control. Dyn."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1007\/s11424-018-8006-y","article-title":"Vector Field Based Sliding Mode Control of Curved Path Following for Miniature Unmanned Aerial Vehicles in Winds","volume":"31","author":"Wang","year":"2018","journal-title":"J. Syst. Sci. Complex."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/j.oceaneng.2015.12.037","article-title":"Vector Field Path Following for Surface Marine Vessel and Parameter Identification Based on LS-SVM","volume":"113","author":"Xu","year":"2016","journal-title":"Ocean Eng."},{"key":"ref_28","first-page":"409","article-title":"Waypoint-Following for a Marine Surface Ship Model Based on Vector Field Guidance Law","volume":"Volume 1","author":"Santos","year":"2016","journal-title":"Maritime Technology and Engineering 3"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/j.ifacol.2015.10.263","article-title":"A Comparison between the ILOS Guidance and the Vector Field Guidance","volume":"28","author":"Caharija","year":"2015","journal-title":"IFAC-PapersOnLine"},{"key":"ref_30","unstructured":"Khalil, H.K. (2002). Nonlinear Systems, Prentice Hall. [3rd ed.]."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Lamnabhi-Lagarrigue, F., Lor\u00eda, A., and Panteley, E. (2005). Cascaded Nonlinear Time-Varying Systems: Analysis and Design. Advanced Topics in Control Systems Theory: Lecture Notes from FAP 2004, Springer.","DOI":"10.1007\/b101671"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2912","DOI":"10.1016\/j.automatica.2014.10.018","article-title":"On Uniform Semiglobal Exponential Stability (USGES) of Proportional Line-of-Sight Guidance Laws","volume":"50","author":"Fossen","year":"2014","journal-title":"Automatica"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1109\/CDC.2005.1582222","article-title":"Cross-Track Control for Underactuated Autonomous Vehicles","volume":"Volume 2005","author":"Borhaug","year":"2005","journal-title":"Proceedings of the 44th IEEE Conference on Decision and Control"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1016\/j.automatica.2005.12.020","article-title":"Global \u03ba-Exponential Way-Point Maneuvering of Ships: Theory and Experiments","volume":"42","author":"Fredriksen","year":"2006","journal-title":"Automatica"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Beard, R.W., and McLain, T.W. (2013). Small Unmanned Aircraft: Theory and Practice, Princeton University Press.","DOI":"10.1515\/9781400840601"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Zhang, J., Zhang, D., Yan, X., Haugen, S., and Guedes Soares, C. (2015). A Distributed Anti-Collision Decision Support Formulation in Multi-Ship Encounter Situations under COLREGs. Ocean Eng.","DOI":"10.1016\/j.oceaneng.2015.06.054"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"465","DOI":"10.3182\/20120919-3-IT-2046.00079","article-title":"A Navigation and Control Platform for Real-Time Manoeuvring of Autonomous Ship Models","volume":"9","author":"Perera","year":"2012","journal-title":"IFAC Proc. Vol."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1007\/s00773-010-0106-x","article-title":"Fuzzy Logic Based Decision Making System for Collision Avoidance of Ocean Navigation under Critical Collision Conditions","volume":"16","author":"Perera","year":"2011","journal-title":"J. Mar. Sci. Technol."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"374","DOI":"10.1109\/JOE.2014.2304793","article-title":"Experimental Evaluations on Ship Autonomous Navigation and Collision Avoidance by Intelligent Guidance","volume":"40","author":"Perera","year":"2015","journal-title":"IEEE J. Ocean. Eng."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1017\/S037346330700447X","article-title":"Autonomous Ship Collision Avoidance Navigation Concepts, Technologies and Techniques","volume":"61","author":"Statheros","year":"2008","journal-title":"J. Navig."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1016\/j.oceaneng.2018.01.001","article-title":"Velocity Obstacle Algorithms for Collision Prevention at Sea","volume":"151","author":"Huang","year":"2018","journal-title":"Ocean Eng."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1016\/j.oceaneng.2018.12.053","article-title":"Generalized Velocity Obstacle Algorithm for Preventing Ship Collisions at Sea","volume":"173","author":"Huang","year":"2019","journal-title":"Ocean Eng."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1109\/JOE.2013.2254214","article-title":"Safe Maritime Autonomous Navigation with COLREGS, Using Velocity Obstacles","volume":"39","author":"Kuwata","year":"2014","journal-title":"IEEE J. Ocean. Eng."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1016\/j.oceaneng.2010.01.012","article-title":"Study on Collision Avoidance in Busy Waterways by Using AIS Data","volume":"37","author":"Mou","year":"2010","journal-title":"Ocean Eng."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Vu, M.T., Van, M., Bui, D.H.P., Do, Q.T., Huynh, T.-T., Lee, S.D., and Choi, H.S. (2020). Study on Dynamic Behavior of Unmanned Surface Vehicle-Linked Unmanned Underwater Vehicle System for Underwater Exploration. Sensors, 20.","DOI":"10.3390\/s20051329"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Guedes Soares, C., Garbatov, Y., Fonseca, N., and Teixeira, A.P. (2011). Mathematical Models for Simulation of Manoeuvring Performance of Ships. Maritime Engineering and Technology, Taylor & Francis Group.","DOI":"10.1201\/b12726"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Xu, H., Hassani, V., Hinostroza, M.A., and Guedes Soares, C. (2018, January 17\u201322). Real-Time Parameter Estimation of Nonlinear Vessel Steering Model Using Support Vector Machine. Proceedings of the ASME 2018 37th International Conference on Ocean, Offshore and Arctic Engineering, Madrid, Spain.","DOI":"10.1115\/OMAE2018-78234"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/j.ejcon.2019.09.007","article-title":"Uniformly Semiglobally Exponential Stability of Vector Field Guidance Law and Autopilot for Path-Following","volume":"53","author":"Xu","year":"2020","journal-title":"Eur. J. Control"},{"key":"ref_49","unstructured":"ITTC (2002, January 8\u201314). Recommended Procedures and Guidelines: Free Running Model Tests. Proceedings of the 23rd International Towing Tank Conference, Venice, Italy."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"106488","DOI":"10.1016\/j.oceaneng.2019.106488","article-title":"Hydrodynamic Coefficient Estimation for Ship Manoeuvring in Shallow Water Using an Optimal Truncated LS-SVM","volume":"191","author":"Xu","year":"2019","journal-title":"Ocean Eng."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"106676","DOI":"10.1016\/j.oceaneng.2019.106676","article-title":"Manoeuvring Modelling of a Containership in Shallow Water Based on Optimal Truncated Nonlinear Kernel-Based Least Square Support Vector Machine and Quantum-Inspired Evolutionary Algorithm","volume":"195","author":"Xu","year":"2020","journal-title":"Ocean Eng."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1016\/j.ejcon.2020.08.003","article-title":"L1 adaptive backstepping control for path-following of underactuated marine surface ship","volume":"58","author":"Xu","year":"2021","journal-title":"Eur. J. Control"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1016\/j.oceaneng.2017.09.020","article-title":"Review of Ship Safety Domains: Models and Applications","volume":"145","author":"Szlapczynski","year":"2017","journal-title":"Ocean Eng."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Guy, S.J., Chhugani, J., Kim, C., Satish, N., Lin, M., Manocha, D., and Dubey, P. (2015, January 7\u20139). ClearPath: Highly Parallel Collision Avoidance for Multi-Agent Simulation. Proceedings of the 2009 ACM SIGGRAPH\/Eurographics Symposium on Computer Animation-SCA \u201909, Los Angeles, CA, USA.","DOI":"10.1145\/1599470.1599494"},{"key":"ref_55","first-page":"A-213","article-title":"Estimation of Hydrodynamic Coefficients of a Nonlinear Manoeuvring Mathematical Model with Free-Running Ship Model Tests","volume":"160","author":"Xu","year":"2018","journal-title":"Int. J. Marit. Eng."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Silveira, P.A.M., Teixeira, A.P., and Guedes Soares, C. (2013). Use of AIS Data to Characterise Marine Traffic Patterns and Ship Collision Risk off the Coast of Portugal. J. Navig.","DOI":"10.1017\/S0373463313000519"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"031302","DOI":"10.1115\/1.4029826","article-title":"System Identification of Nonlinear Vessel Steering","volume":"137","author":"Perera","year":"2015","journal-title":"J. Offshore Mech. Arct. Eng."}],"container-title":["Journal of Marine Science and Engineering"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2077-1312\/9\/6\/652\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:13:47Z","timestamp":1760163227000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2077-1312\/9\/6\/652"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,12]]},"references-count":57,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2021,6]]}},"alternative-id":["jmse9060652"],"URL":"https:\/\/doi.org\/10.3390\/jmse9060652","relation":{},"ISSN":["2077-1312"],"issn-type":[{"value":"2077-1312","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,12]]}}}