{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T13:00:36Z","timestamp":1780491636059,"version":"3.54.1"},"reference-count":52,"publisher":"MDPI AG","issue":"16","license":[{"start":{"date-parts":[[2021,8,6]],"date-time":"2021-08-06T00:00:00Z","timestamp":1628208000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The determination of a ship\u2019s safe trajectory in collision situations at sea is one of the basic functions in autonomous navigation of ships. While planning a collision avoiding manoeuvre in open waters, the navigator has to take into account the ships manoeuvrability and hydrometeorological conditions. To this end, the ship\u2019s state vector is predicted\u2014position coordinates, speed, heading, and other movement parameters\u2014at fixed time intervals for different steering scenarios. One possible way to solve this problem is a method using the interpolation of the ship\u2019s state vector based on the data from measurements conducted during the sea trials of the ship. This article presents the interpolating function within any convex quadrilateral with the nodes being its vertices. The proposed function interpolates the parameters of the ship\u2019s state vector for the specified point of a plane, where the values in the interpolation nodes are data obtained from measurements performed during a series of turning circle tests, conducted for different starting conditions and various rudder settings. The proposed method of interpolation was used in the process of determining the anti-collision manoeuvre trajectory. The mechanism is based on the principles of a modified Dijkstra algorithm, in which the graph takes the form of a regular network of points. The transition between the graph vertices depends on the safe passing level of other objects and the degree of departure from the planned route. The determined shortest path between the starting vertex and the target vertex is the optimal solution for the discrete space of solutions. The algorithm for determining the trajectory of the anti-collision manoeuvre was implemented in autonomous sea-going vessel technology. This article presents the results of laboratory tests and tests conducted under quasi-real conditions using physical ship models. The experiments confirmed the effective operation of the developed algorithm of the determination of the anti-collision manoeuvre trajectory in the technological framework of autonomous ship navigation.<\/jats:p>","DOI":"10.3390\/s21165332","type":"journal-article","created":{"date-parts":[[2021,8,8]],"date-time":"2021-08-08T21:35:40Z","timestamp":1628458540000},"page":"5332","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["The Algorithm of Determining an Anti-Collision Manoeuvre Trajectory Based on the Interpolation of Ship\u2019s State Vector"],"prefix":"10.3390","volume":"21","author":[{"given":"Piotr","family":"Borkowski","sequence":"first","affiliation":[{"name":"Faculty of Computer Science and Telecommunications, Maritime University of Szczecin, Wa\u0142y Chrobrego 1, 70500 Szczecin, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zbigniew","family":"Pietrzykowski","sequence":"additional","affiliation":[{"name":"Faculty of Computer Science and Telecommunications, Maritime University of Szczecin, Wa\u0142y Chrobrego 1, 70500 Szczecin, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Janusz","family":"Magaj","sequence":"additional","affiliation":[{"name":"Faculty of Computer Science and Telecommunications, Maritime University of Szczecin, Wa\u0142y Chrobrego 1, 70500 Szczecin, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"202111","DOI":"10.1109\/ACCESS.2020.3034948","article-title":"Development of an Automated Camera-Based Drone Landing System","volume":"8","author":"Demirhan","year":"2020","journal-title":"IEEE Access"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2758","DOI":"10.1109\/JSEN.2018.2888909","article-title":"Speed-up automatic quadcopter position detection by sensing propeller rotation","volume":"19","author":"Premachandra","year":"2019","journal-title":"IEEE Sens. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"253","DOI":"10.7225\/toms.v08.n02.011","article-title":"Review of Autonomous and Remotely Controlled Ships in Maritime Sector","volume":"8","year":"2019","journal-title":"Trans. Marit. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Dudoit, A., and Skorupski, J. (2019, January 9\u201312). A Simulation-Based Approach for the Conflict Resolution Method Optimization in a Distributed Air Traffic Control System. Proceedings of the International Scientific Conference Transport of the 21st Century, Ryn, Poland.","DOI":"10.1007\/978-3-030-27687-4_11"},{"key":"ref_5","unstructured":"(2021, June 03). Taking Maritime Transport into the Digital Age. Available online: https:\/\/www.seatrafficmanagement.info\/projects\/monalisa-2."},{"key":"ref_6","unstructured":"(2021, June 03). Maritime Unmanned Navigation through Intelligence in Networks. Available online: http:\/\/www.unmanned-ship.org\/munin."},{"key":"ref_7","unstructured":"(2021, June 03). STM Validation\u2014Seeing Is Believing. Available online: https:\/\/www.seatrafficmanagement.info\/projects\/stm-validation."},{"key":"ref_8","unstructured":"(2021, June 03). Advanced Autonomous Waterborne Applications Initiative. Available online: https:\/\/www.rolls-royce.com\/~\/media\/Files\/R\/Rolls-Royce\/documents\/customers\/marine\/ship-intel\/aawa-whitepaper-210616.pdf."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1002\/j.2161-4296.2001.tb00238.x","article-title":"A Computational Method for the Solution of Optimal Control Problems in Ship Routing","volume":"48","author":"Bijlsma","year":"2001","journal-title":"Navigation"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"28","DOI":"10.2478\/pomr-2013-0032","article-title":"An Ant Colony Algorithm for efficient ship routing","volume":"20","author":"Tsau","year":"2013","journal-title":"Pol. Marit. Res."},{"key":"ref_11","first-page":"395","article-title":"Ship Ocean Route Programming","volume":"74","year":"2004","journal-title":"Sci. J. Marit. Univ. Szczec."},{"key":"ref_12","first-page":"777","article-title":"Collision avoidance path planning for ships by particle swarm optimization","volume":"26","author":"Kang","year":"2018","journal-title":"J. Mar. Sci. Tech."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Karbowska-Chilinska, J., Koszelew, J., Ostrowski, K., Kuczynski, P., Kulbiej, E., and Wolejsza, P. (2019). Beam Search Algorithm for Ship Anti-Collision Trajectory Planning. Sensors, 19.","DOI":"10.3390\/s19245338"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Koszelew, J., Karbowska-Chilinska, J., Ostrowski, K., Kuczy\u0144ski, P., Kulbiej, E., and Wo\u0142ejsza, P. (2020). Beam Search Algorithm for Anti-Collision Trajectory Planning for Many-to-Many Encounter Situations with Autonomous Surface Vehicles. Sensors, 20.","DOI":"10.3390\/s20154115"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.oceaneng.2012.09.003","article-title":"Cooperative Path Planning Algorithm for Marine Surface Vessels","volume":"57","author":"Tam","year":"2013","journal-title":"Ocean Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1016\/j.ssci.2019.09.018","article-title":"Ship Collision Avoidance Methods: State-of-the-art","volume":"121","author":"Huang","year":"2020","journal-title":"Saf. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Lisowski, J. (2020). Game Control Methods Comparison when Avoiding Collisions with Multiple Objects Using Radar Remote Sensing. Remote Sens., 12.","DOI":"10.3390\/rs12101573"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Lisowski, J. (2021). Synthesis of a Path-Planning Algorithm for Autonomous Robots Moving in a Game Environment during Collision Avoidance. Electronics, 10.","DOI":"10.3390\/electronics10060675"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10957-006-9051-6","article-title":"Optimal trajectories and guidance schemes for ship collision avoidance","volume":"129","author":"Miele","year":"2006","journal-title":"J. Optim. Theory Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"31","DOI":"10.12716\/1001.11.01.02","article-title":"Multi-criteria ACO-based algorithm for the ship\u2019s trajectory planning","volume":"11","author":"Lazarowska","year":"2017","journal-title":"TransNav J."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1109\/4235.873234","article-title":"Modeling of a ship trajectory in collision situations at sea by evolutionary algorithm","volume":"4","author":"Michalewicz","year":"2000","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/s10489-011-0319-7","article-title":"On evolutionary computing in multi-ship trajectory planning","volume":"37","year":"2012","journal-title":"Appl. Intell."},{"key":"ref_23","first-page":"746","article-title":"The study of ship collision avoidance route planning by ant colony algorithm","volume":"18","author":"Tsau","year":"2010","journal-title":"J. Mar. Sci. Tech."},{"key":"ref_24","first-page":"914689","article-title":"Multiobjective Optimization Based Vessel Collision Avoidance Strategy Optimization","volume":"2014","author":"Xu","year":"2014","journal-title":"Math. Probl. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1016\/j.apor.2012.05.008","article-title":"A study on the collision avoidance of a ship using neural networks and fuzzy logic","volume":"37","author":"Ahn","year":"2012","journal-title":"Appl. Ocean Res."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Borkowski, P. (2018). The Ship Movement Trajectory Prediction Algorithm Using Navigational Data Fusion. Sensors, 17.","DOI":"10.3390\/s17061432"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Lisowski, J. (2020). Multistage Dynamic Optimization with Different Forms of Neural-State Constraints to Avoid Many Object Collisions Based on Radar Remote Sensing. Remote Sens., 12.","DOI":"10.3390\/rs12061020"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"294","DOI":"10.5391\/IJFIS.2017.17.4.294","article-title":"Study on the Construction of Stage Discrimination Model and Consecutive Waypoints Generation Method for Ship\u2019s Automatic Avoiding Action","volume":"17","author":"Dinh","year":"2017","journal-title":"Int. J. Fuzzy Log. Intell. Syst."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Lisowski, J., and Mohamed-Seghir, M. (2019). Comparison of Computational Intelligence Methods Based on Fuzzy Sets and Game Theory in the Synthesis of Safe Ship Control Based on Information from a Radar ARPA System. Remote Sens., 11.","DOI":"10.3390\/rs11010082"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1016\/j.procs.2014.08.115","article-title":"The branch-and-bound method, genetic algorithm, and dynamic programming to determine a safe ship trajectory in fuzzy, knowledge-based and intelligent information and engineering systems","volume":"35","year":"2014","journal-title":"Procedia Comput. Sci."},{"key":"ref_31","unstructured":"Perera, L.P., Carvalho, J.P., and Soares, C.G. (2010, January 15\u201317). Bayesian Network based sequential collision avoidance action execution for an Ocean Navigational System. Proceedings of the 8th IFAC Conference on Control Applications in Marine Systems, Rostock, Germany."},{"key":"ref_32","first-page":"122","article-title":"The safe ships trajectory in a restricted area","volume":"111","author":"Pietrzykowski","year":"2014","journal-title":"Sci. J. Marit. Univ. Szczec."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1007\/BF01386390","article-title":"A note on two problems in connexion with graphs","volume":"1","author":"Dijkstra","year":"1959","journal-title":"Numer. Math."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Hofmann-Wellenhof, B., Lichtenegger, H., and Collins, J. (2001). Global Positioning System, Springer. [2rd ed.].","DOI":"10.1007\/978-3-7091-6199-9"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Borkowski, P. (2018). Adaptive system for steering a ship along the desired route. Mathematics, 6.","DOI":"10.3390\/math6100196"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2561383","DOI":"10.1155\/2017\/2561383","article-title":"Inference engine in an intelligent ship course-keeping system","volume":"2017","author":"Borkowski","year":"2017","journal-title":"Comput. Intell. Neurosci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1002\/j.2161-4296.1979.tb01389.x","article-title":"Optimal collision avoidance in unconfined waters","volume":"26","author":"Wit","year":"1984","journal-title":"J. Inst. Navig."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1017\/S0373463315000223","article-title":"Analysis of Collision Threat Parameters and Criteria","volume":"68","author":"Lenart","year":"2015","journal-title":"J. Navig."},{"key":"ref_39","first-page":"99","article-title":"Manoeuvring to required approach parameters\u2014CPA distance and time","volume":"1","author":"Lenart","year":"1999","journal-title":"Annu. Navig."},{"key":"ref_40","first-page":"109","article-title":"Manoeuvring to required approach parameters\u2014Distance and time on course","volume":"1","author":"Lenart","year":"1999","journal-title":"Annu. Navig."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Koszelew, J., Wolejsza, P., and Oldziej, D. (2018, January 21\u201323). Autonomous Vessel with an Air Look. Proceedings of the 2018 Baltic Geodetic Congress, Olsztyn, Poland.","DOI":"10.1109\/BGC-Geomatics.2018.00025"},{"key":"ref_42","first-page":"26","article-title":"Filtering and reconstruction of course parameters of the shiphandling training boat \u201cKo\u0142obrzeg\u201d","volume":"7","author":"Gierusz","year":"2000","journal-title":"Pol. Marit. Res."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.oceaneng.2016.06.024","article-title":"Simulation model of the LNG carrier with podded propulsion, Part II: Full model and experimental results","volume":"123","author":"Gierusz","year":"2016","journal-title":"Ocean Eng."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1016\/S1474-6670(17)35092-9","article-title":"Simulation Model of the Shiphandling Training Boat \u201cBlue Lady\u201d","volume":"34","author":"Gierusz","year":"2001","journal-title":"IFAC Proc. Vol."},{"key":"ref_45","unstructured":"(2021, June 03). Ship Handling Research and Training Centre at I\u0142awa. Available online: http:\/\/www.ilawashiphandling.com.pl."},{"key":"ref_46","first-page":"20","article-title":"Presentation algorithm of possible collision solutions in a navigational decision support system","volume":"38","author":"Borkowski","year":"2014","journal-title":"Sci. J. Marit. Univ. Szczec."},{"key":"ref_47","first-page":"88","article-title":"The area-dynamic approach to the assessment of the risks of ship collision in the restricted water","volume":"117","author":"Guze","year":"2016","journal-title":"Sci. J. Marit. Univ. Szczec."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Kazimierski, W., and Zaniewicz, G. (2021). Determination of Process Noise for Underwater Target Tracking with Forward Looking Sonar. Remote Sens., 13.","DOI":"10.3390\/rs13051014"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Stateczny, A., Kazimierski, W., Burdziakowski, P., Motyl, W., and Wisniewska, M. (2019). Shore Construction Detection by Automotive Radar for the Needs of Autonomous Surface Vehicle Navigation. ISPRS Int. J. Geo Inf., 8.","DOI":"10.3390\/ijgi8020080"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Stateczny, A., Kazimierski, W., and Kulpa, K. (2020). Radar and Sonar Imaging and Processing. Remote Sens., 12.","DOI":"10.3390\/rs12111811"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"5","DOI":"10.2478\/pomr-2018-0016","article-title":"Automatic watercraft recognition and identification on water areas covered by video monitoring as extension for sea and river traffic supervision systems","volume":"25","author":"Wawrzyniak","year":"2018","journal-title":"Pol. Marit. Res."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Wlodarczyk-Sielicka, M., Stateczny, A., and Lubczonek, J. (2019). The Reduction Method of Bathymetric Datasets that Preserves True Geodata. Remote Sens., 11.","DOI":"10.3390\/rs11131610"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/16\/5332\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:42:04Z","timestamp":1760164924000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/16\/5332"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,6]]},"references-count":52,"journal-issue":{"issue":"16","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["s21165332"],"URL":"https:\/\/doi.org\/10.3390\/s21165332","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,8,6]]}}}