Conserving integration of multibody systems with singular and non-constant mass matrix including quaternion-based rigid body dynamics

Conserving integration of multibody systems with singular and non-constant mass matrix including quaternion-based rigid body 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 Conserving integration of multibody systems with singular and non-constant mass matrix including quaternion-based rigid body dynamics Philipp L. Kinon, Peter Betsch This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3999974/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 9 You are reading this latest preprint version Abstract Mechanical systems with singular and/or configuration-dependent mass matrix can pose difficulties to Hamiltonian formulations, which are the standard choice for the design of energy-momentum conserving time integrators. In this work, we derive a structure-preserving time integrator for constrained mechanical systems, based on a mixed variational approach. Livens’ principle (or sometimes called Hamilton-Pontryagin principle) features independent velocity and momentum quantities and circumvents the need to invert the mass matrix. In particular, we take up the description of rigid body rotations using unit quaternions. Using Livens’ principle, a new and comparatively easy approach to the simulation of these problems is presented. The equations of motion are approximated by using (partitioned) midpoint discrete gradients, thus generating a new energy-momentum conserving integration scheme for mechanical systems with singular and/or configuration-dependent mass matrix. The derived method is second-order accurate and algorithmically preserves a generalized energy function as well as the holonomic constraints and momentum maps corresponding to symmetries of the system. We study the numerical performance of the newly devised scheme in representative examples for multibody dynamics and rigid body rotations. Holonomic constraints Singular mass matrix Rigid body rotations Unit quaternions Euler parameters Livens’ principle Structure-preserving integration Energy-momentum methods Discrete gradients Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 12 Apr, 2024 Reviews received at journal 12 Apr, 2024 Reviews received at journal 19 Mar, 2024 Reviewers agreed at journal 04 Mar, 2024 Reviewers agreed at journal 03 Mar, 2024 Reviewers invited by journal 01 Mar, 2024 Editor assigned by journal 01 Mar, 2024 Submission checks completed at journal 01 Mar, 2024 First submitted to journal 29 Feb, 2024 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-3999974","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":276163318,"identity":"f01ce351-c4d3-48c3-bb5c-1c75838edf08","order_by":0,"name":"Philipp L. Kinon","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3UlEQVRIiWNgGAWjYDACCcbGAwkgmhnEq0jgAQsS0NKApOVMAg8PYS0MDAfgBjO2JTAQ1GIu3dxw4OGOOwyS7bwHHxfOS5OxZ2A+eJsHjxbLOQcbDiSeecYgzcyXbDxzWw7QYWzJ1vi0GNxIBGppO8wgx8xjJs27rQKoBcggQcsckBb+b8RpkQZraQA5jIcNrxbLGYlgv/BINvMYG/McS+PhOcxmbDkHjxZzifSHD3/uuCMncf6M4WOemmR79vbmhzfe4HMYiADGJpJLmPEoR9ZCQNkoGAWjYBSMaAAATlNG14xM9F8AAAAASUVORK5CYII=","orcid":"","institution":"Karlsruhe Institute of Technology","correspondingAuthor":true,"prefix":"","firstName":"Philipp","middleName":"L.","lastName":"Kinon","suffix":""},{"id":276163319,"identity":"9dba901d-0122-4583-834b-40eb07597a1d","order_by":1,"name":"Peter Betsch","email":"","orcid":"","institution":"Karlsruhe Institute of Technology","correspondingAuthor":false,"prefix":"","firstName":"Peter","middleName":"","lastName":"Betsch","suffix":""}],"badges":[],"createdAt":"2024-02-29 13:59:29","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3999974/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3999974/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":52030353,"identity":"b680bbb9-ca7e-46c6-9c2b-40e3d0bf4491","added_by":"auto","created_at":"2024-03-05 16:21:15","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":835945,"visible":true,"origin":"","legend":"","description":"","filename":"2024MBSDkinonbetschsubmitted.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3999974/v1_covered_b1b1b6e0-bb62-4274-97e4-baf93748a291.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Conserving integration of multibody systems with singular and non-constant mass matrix including quaternion-based rigid body dynamics","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":"multibody-system-dynamics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"mubo","sideBox":"Learn more about [Multibody System Dynamics](http://link.springer.com/journal/11044)","snPcode":"11044","submissionUrl":"https://submission.nature.com/new-submission/11044/3","title":"Multibody System Dynamics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Holonomic constraints, Singular mass matrix, Rigid body rotations, Unit quaternions, Euler parameters, Livens’ principle, Structure-preserving integration, Energy-momentum methods, Discrete gradients","lastPublishedDoi":"10.21203/rs.3.rs-3999974/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3999974/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eMechanical systems with singular and/or configuration-dependent mass matrix can pose difficulties to Hamiltonian formulations, which are the standard choice for the design of energy-momentum conserving time integrators. In this work, we derive a structure-preserving time integrator for constrained mechanical systems, based on a mixed variational approach. Livens’ principle (or sometimes called Hamilton-Pontryagin principle) features independent velocity and momentum quantities and circumvents the need to invert the mass matrix. In particular, we take up the description of rigid body rotations using unit quaternions. Using Livens’ principle, a new and comparatively easy approach to the simulation of these problems is presented. The equations of motion are approximated by using (partitioned) midpoint discrete gradients, thus generating a new energy-momentum conserving integration scheme for mechanical systems with singular and/or configuration-dependent mass matrix. The derived method is second-order accurate and algorithmically preserves a generalized energy function as well as the holonomic constraints and momentum maps corresponding to symmetries of the system. We study the numerical performance of the newly devised scheme in representative examples for multibody dynamics and rigid body rotations.\u003c/p\u003e","manuscriptTitle":"Conserving integration of multibody systems with singular and non-constant mass matrix including quaternion-based rigid body dynamics","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-03-05 16:12:39","doi":"10.21203/rs.3.rs-3999974/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2024-04-12T14:11:25+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-04-12T13:56:25+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-03-19T10:07:32+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"e51353e5-bc25-4a51-9ce5-c6180be5d424","date":"2024-03-04T08:08:18+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"ad60fff2-83a0-478b-b1cb-b5d759527f62","date":"2024-03-03T21:01:31+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-03-01T23:30:07+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-03-01T14:59:35+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-03-01T06:18:40+00:00","index":"","fulltext":""},{"type":"submitted","content":"Multibody System Dynamics","date":"2024-02-29T12:59:57+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"multibody-system-dynamics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"mubo","sideBox":"Learn more about [Multibody System Dynamics](http://link.springer.com/journal/11044)","snPcode":"11044","submissionUrl":"https://submission.nature.com/new-submission/11044/3","title":"Multibody System Dynamics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"4a4f0a2d-fe38-4bd3-bca7-e74a117eeb15","owner":[],"postedDate":"March 5th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2024-05-31T16:37:56+00:00","versionOfRecord":[],"versionCreatedAt":"2024-03-05 16:12:39","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3999974","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3999974","identity":"rs-3999974","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}