Galerkin-based time integration approaches to rigid body dynamics in terms of unit quaternions

Galerkin-based time integration approaches to rigid body dynamics in terms of unit quaternions | 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 Systematic Review Galerkin-based time integration approaches to rigid body dynamics in terms of unit quaternions Marvin May, Peter Betsch This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6354821/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 21 Jul, 2025 Read the published version in Multibody System Dynamics → Version 1 posted 9 You are reading this latest preprint version Abstract Galerkin-based time integration approaches are investigated for discrete mechanical systems subject to holonomic constraints. In this connection, rigid body rotations are described by using unit quaternions, resulting in a configuration-dependent mass matrix. Two different Galerkin methods are developed: (i) a Petrov-Galerkin approach leading to the common time-stepping format of numerical integrators for initial value problem and (ii) a Bubnov-Galerkin method leading to a global solution procedure. The numerical performance of the alternative schemes is compared, including their algorithmic conservation properties, accuracy, and satisfaction of the constraints on configuration and velocity level. The global Galerkin-based approach is deemed to be advantageous for the solution of inverse dynamics and optimal control problems, in which a flow of information backwards in time is required. Numerical examples involving large rotations are investigated to assess the numerical performance of the alternative Galerkin approaches. In addition to that, a comparison of the quaternion-based rigid body formulation with a director-based formulation is included. Galerkin methods Rotation parameterization Rigid Body dynamics Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 21 Jul, 2025 Read the published version in Multibody System Dynamics → Version 1 posted Editorial decision: Revision requested 22 May, 2025 Reviews received at journal 20 May, 2025 Reviewers agreed at journal 21 Apr, 2025 Reviews received at journal 16 Apr, 2025 Reviewers agreed at journal 16 Apr, 2025 Reviewers invited by journal 15 Apr, 2025 Editor assigned by journal 07 Apr, 2025 Submission checks completed at journal 02 Apr, 2025 First submitted to journal 01 Apr, 2025 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. 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