Data-driven modeling and prediction of ship course keeping in shallow water waves using higher order dynamic mode decomposition incorporating control input

Data-driven modeling and prediction of ship course keeping in shallow water waves using higher order dynamic mode decomposition incorporating control input | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Data-driven modeling and prediction of ship course keeping in shallow water waves using higher order dynamic mode decomposition incorporating control input Chang-Zhe Chen, Tian-Ye Yu, Lu Zou, Zao-Jian Zou This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8865149/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 13 You are reading this latest preprint version Abstract The implementation of autonomous navigation of Maritime Autonomous Surface Ships (MASS) in complex environments critically depends on accurate and real-time prediction of ship motions. To address the combined challenges of wave excitations and shallow water effects, this paper proposes a novel data-driven modeling framework called Higher Order Dynamic Mode Decomposition incorporating Control Input (HODMD-CI). It extends the standard HODMD by explicitly integrating the time-delayed control inputs, including both actuator commands (i.e., rudder angle) and measurable environmental disturbances (i.e., wave elevation), into a state-space prediction model. The proposed method is validated using the free-running model test data of the Duisburg Test Case (DTC) container carrier performing course keeping in regular head waves under shallow water conditions. The prediction performance of HODMD-CI under three control-input configurations (rudder angle only, wave elevation only, and both) is evaluated and compared against the standard HODMD and several neural network models. Results demonstrate that HODMD-CI with combined rudder angle and wave elevation inputs achieves the highest overall accuracy and trend consistency, as evidenced by the lowest Average Relative Root Mean Square Error (ARRMSE) and the Pearson Correlation Coefficient (PCC) values closest to 1. Furthermore, HODMD-CI exhibits stronger noise resistance compared to neural network counterparts, particularly for roll motion prediction. This study confirms the efficacy of jointly modeling rudder angle and wave elevation for accurate ship motion prediction in complex, shallow water wave scenarios, offering a promising data-driven tool for intelligent ship navigation and control. Course keeping shallow water waves HODMD-CI data-driven modeling ship motion prediction control input Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 13 Apr, 2026 Reviews received at journal 03 Apr, 2026 Reviews received at journal 30 Mar, 2026 Reviews received at journal 27 Mar, 2026 Reviewers agreed at journal 21 Mar, 2026 Reviewers agreed at journal 18 Mar, 2026 Reviewers agreed at journal 05 Mar, 2026 Reviewers agreed at journal 24 Feb, 2026 Reviewers agreed at journal 23 Feb, 2026 Reviewers invited by journal 23 Feb, 2026 Editor assigned by journal 13 Feb, 2026 Submission checks completed at journal 13 Feb, 2026 First submitted to journal 12 Feb, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8865149","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":597200544,"identity":"dfeb5b39-b44a-4bad-8a35-b3f1e860269b","order_by":0,"name":"Chang-Zhe Chen","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABBElEQVRIiWNgGAWjYDACCSBmbACxeBgYPjCwwQWJ08I4g4FNgjQtzDxQ1Xi18M9uPvbg5w67PHn/swcf2/zhqzM4wHzwNg+DXR5OS+4cSzfsPZNcbHjgXLJxbhubhMEBtmRrHobkYlxaDCRyzKQZ25gTNzb2mEnnNoC08JhJ8zAcSGzAqSX/G1BLfeLGZh7z3xZ/QFr4vxHQksMG1HI4cT4bjxkzAxvYFja8WiRupJlJ9rYdT9zAw2MMZLBJzjzMZmw5xyAZpxb+GcnPJH62VSfO7z9j+OHHn2P8fMebH954U2GHUwvChQfA1DEGBmYwl5B6IJCHGFpDhNJRMApGwSgYaQAAxg5RclnVMPoAAAAASUVORK5CYII=","orcid":"","institution":"Shanghai Jiao Tong University","correspondingAuthor":true,"prefix":"","firstName":"Chang-Zhe","middleName":"","lastName":"Chen","suffix":""},{"id":597200545,"identity":"56ffb1a9-bac1-48bd-8051-04389119b1cd","order_by":1,"name":"Tian-Ye Yu","email":"","orcid":"","institution":"Shanghai Jiao Tong University","correspondingAuthor":false,"prefix":"","firstName":"Tian-Ye","middleName":"","lastName":"Yu","suffix":""},{"id":597200546,"identity":"a704c4be-181d-4c91-bb19-4e53f4366416","order_by":2,"name":"Lu Zou","email":"","orcid":"","institution":"Shanghai Jiao Tong University","correspondingAuthor":false,"prefix":"","firstName":"Lu","middleName":"","lastName":"Zou","suffix":""},{"id":597200547,"identity":"cfed407e-062b-43ed-900d-49cbeb6fa515","order_by":3,"name":"Zao-Jian Zou","email":"","orcid":"","institution":"Shanghai Jiao Tong University","correspondingAuthor":false,"prefix":"","firstName":"Zao-Jian","middleName":"","lastName":"Zou","suffix":""}],"badges":[],"createdAt":"2026-02-12 19:25:00","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8865149/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8865149/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":104397765,"identity":"b11e89c6-d196-4321-b908-755abe03a794","added_by":"auto","created_at":"2026-03-11 11:56:01","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":4741068,"visible":true,"origin":"","legend":"","description":"","filename":"DATADR1.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8865149/v1_covered_0a4f99c8-2e6a-4d21-b8df-36065298f431.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Data-driven modeling and prediction of ship course keeping in shallow water waves using higher order dynamic mode decomposition incorporating control input","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"nonlinear-dynamics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"nody","sideBox":"Learn more about [Nonlinear Dynamics](https://www.springer.com/journal/11071)","snPcode":"11071","submissionUrl":"https://submission.nature.com/new-submission/11071/3","title":"Nonlinear Dynamics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Course keeping, shallow water waves, HODMD-CI, data-driven modeling, ship motion prediction, control input","lastPublishedDoi":"10.21203/rs.3.rs-8865149/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8865149/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe implementation of autonomous navigation of Maritime Autonomous Surface Ships (MASS) in complex environments critically depends on accurate and real-time prediction of ship motions. To address the combined challenges of wave excitations and shallow water effects, this paper proposes a novel data-driven modeling framework called Higher Order Dynamic Mode Decomposition incorporating Control Input (HODMD-CI). It extends the standard HODMD by explicitly integrating the time-delayed control inputs, including both actuator commands (i.e., rudder angle) and measurable environmental disturbances (i.e., wave elevation), into a state-space prediction model. The proposed method is validated using the free-running model test data of the Duisburg Test Case (DTC) container carrier performing course keeping in regular head waves under shallow water conditions. The prediction performance of HODMD-CI under three control-input configurations (rudder angle only, wave elevation only, and both) is evaluated and compared against the standard HODMD and several neural network models. Results demonstrate that HODMD-CI with combined rudder angle and wave elevation inputs achieves the highest overall accuracy and trend consistency, as evidenced by the lowest Average Relative Root Mean Square Error (ARRMSE) and the Pearson Correlation Coefficient (PCC) values closest to 1. Furthermore, HODMD-CI exhibits stronger noise resistance compared to neural network counterparts, particularly for roll motion prediction. 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