Screw-Theoretic PDE-Based Synthesis for Unified Representation of Multibody Flexible Manipulator Dynamics

Screw-Theoretic PDE-Based Synthesis for Unified Representation of Multibody Flexible Manipulator Dynamics | 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 Screw-Theoretic PDE-Based Synthesis for Unified Representation of Multibody Flexible Manipulator Dynamics Sadeq Yaqubi, Alexander Kitzinger, Andreas Müller, Hubert Gattringer, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9336591/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 4 You are reading this latest preprint version Abstract This paper addresses the open problem of partial differential equation (PDE)-based dynamic modeling for flexible multibody robotic systems by presenting a screw-theoretic synthesis methodology-developed within a unified Lie-algebraic framework-for serial flexible manipulators with an arbitrary number of links in three-dimensional motion. The proposed approach expresses all dynamic states — rigid-body motion, elastic deformation, and inter-link interaction forces — uniformly within the screw theoretic structure as body-fixed twists and their dual wrenches, treating all physical components consistently within the same geometric framework. Hence, the PDE structure of the deformation field is retained exactly, while the se(3) representation admits the linear-algebraic formulation required for multibody assembly and formal well-posedness analysis. Building on a previously developed single-link screw-theoretic PDE model, joint constraints are enforced as screw-compatibility equations connecting the twist and wrench fields of adjacent links at their connection points, and the per-link models are assembled via a closed-form linear-algebraic stacking procedure. The resulting system matrix exhibits near-tridiagonal block structure, interaction wrenches appear explicitly as algebraic variables, and adding a link requires only appending block rows — making the synthesis automatable for arbitrary $n$. The assembled system is formulated as a semi-explicit index-1 differential-algebraic equation, and well-posedness is established by recasting it in abstract Cauchy form through modal projection. Presented solutions are validated experimentally on a two-link flexible manipulator in three-dimensional motion, confirming implementability and physical consistency. Mathematics Subject Classification (2020) 74Kxx · 74H45 · 70E60 Screw Theory Synthesis Flexible Manipulator PDE Dynamic Modeling Nonlinear Dynamics Multibody Systems Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 07 Apr, 2026 Editor assigned by journal 07 Apr, 2026 Submission checks completed at journal 07 Apr, 2026 First submitted to journal 06 Apr, 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. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-9336591","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":619109129,"identity":"de749f41-01fb-428d-8cdb-a7e5c134af7b","order_by":0,"name":"Sadeq Yaqubi","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5UlEQVRIiWNgGAWjYHACAyjNw3DgA5BiY2B8cIBoLQdnJIC0MBsQr4WZJwHEYDbArRgI5NsPb3zwhuGwPN+N3IOHbX8czuNjYGbEa4vBmbRiwzkMhw1n3shLOJyTcLgY6DAG/FoYcsykeRgOM264kWMA0pLYxsB/AK8W+f435r+BWuzBWizAWgjYwnAjx4wZqCURrIWBGC0GN54VS84xSE+eeeZdwsGetPTENmYCWuT7kzd+eFNhbdt3PPfwhx821onz25uZP+B1GAjwGDQzIExmJqgepIWhjoGAl0fBKBgFo2AkAwBFOFAvptRF+AAAAABJRU5ErkJggg==","orcid":"","institution":"Tampere University","correspondingAuthor":true,"prefix":"","firstName":"Sadeq","middleName":"","lastName":"Yaqubi","suffix":""},{"id":619109130,"identity":"8586f4e5-ab81-4697-a1e3-b45936416d73","order_by":1,"name":"Alexander Kitzinger","email":"","orcid":"","institution":"Johannes Kepler University of Linz","correspondingAuthor":false,"prefix":"","firstName":"Alexander","middleName":"","lastName":"Kitzinger","suffix":""},{"id":619109131,"identity":"b44dbcb9-cbf3-4709-906c-44f3f0690808","order_by":2,"name":"Andreas Müller","email":"","orcid":"","institution":"Johannes Kepler University of Linz","correspondingAuthor":false,"prefix":"","firstName":"Andreas","middleName":"","lastName":"Müller","suffix":""},{"id":619109132,"identity":"703ff67d-3bea-4bb0-a5aa-665972693e49","order_by":3,"name":"Hubert Gattringer","email":"","orcid":"","institution":"Johannes Kepler University of Linz","correspondingAuthor":false,"prefix":"","firstName":"Hubert","middleName":"","lastName":"Gattringer","suffix":""},{"id":619109133,"identity":"29b08bed-0e55-4fa5-a84a-a0e6d2b57184","order_by":4,"name":"Jouni Mattila","email":"","orcid":"","institution":"Tampere University","correspondingAuthor":false,"prefix":"","firstName":"Jouni","middleName":"","lastName":"Mattila","suffix":""}],"badges":[],"createdAt":"2026-04-06 18:38:32","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9336591/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9336591/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":107488443,"identity":"2d89d344-c43f-4b50-ab47-610173f72a6a","added_by":"auto","created_at":"2026-04-22 02:44:49","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":10373878,"visible":true,"origin":"","legend":"","description":"","filename":"ManuscriptNLSubmission.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9336591/v1_covered_0d9633a3-8d91-4aed-ae7a-4e35ed431f79.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Screw-Theoretic PDE-Based Synthesis for Unified Representation of Multibody Flexible Manipulator Dynamics","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"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":"Screw Theory, Synthesis, Flexible Manipulator, PDE, Dynamic Modeling, Nonlinear Dynamics, Multibody Systems","lastPublishedDoi":"10.21203/rs.3.rs-9336591/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9336591/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper addresses the open problem of partial differential equation (PDE)-based dynamic modeling for flexible multibody robotic systems by presenting a screw-theoretic synthesis methodology-developed within a unified Lie-algebraic framework-for serial flexible manipulators with an arbitrary number of links in three-dimensional motion. The proposed approach expresses all dynamic states — rigid-body motion, elastic deformation, and inter-link interaction forces — uniformly within the screw theoretic structure as body-fixed twists and their dual wrenches, treating all physical components consistently within the same geometric framework. Hence, the PDE structure of the deformation field is retained exactly, while the se(3) representation admits the linear-algebraic formulation required for multibody assembly and formal well-posedness analysis. Building on a previously developed single-link screw-theoretic PDE model, joint constraints are enforced as screw-compatibility equations connecting the twist and wrench fields of adjacent links at their connection points, and the per-link models are assembled via a closed-form linear-algebraic stacking procedure. The resulting system matrix exhibits near-tridiagonal block structure, interaction wrenches appear explicitly as algebraic variables, and adding a link requires only appending block rows — making the synthesis automatable for arbitrary $n$. The assembled system is formulated as a semi-explicit index-1 differential-algebraic equation, and well-posedness is established by recasting it in abstract Cauchy form through modal projection. 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