Early detection of unstable car-and-driver motion – a Floquet theory approach

Early detection of unstable car-and-driver motion – a Floquet theory approach | 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 Early detection of unstable car-and-driver motion – a Floquet theory approach Samuele Giacintucci, Fabio Della Rossa, Gianpiero Mastinu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4673548/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 20 Feb, 2025 Read the published version in Nonlinear Dynamics → Version 1 posted 10 You are reading this latest preprint version Abstract The global stability of car-and-driver is studied. The aim is to distinguish stable versus unstable trajectories as early as possible, after a disturbance has acted. First, a simple car-and-driver model is introduced to simulate the response of the system to severe perturbations, e.g. wind gusts or evasive maneuvers. Both straight and curved motions are analysed, considering an oversteering vehicle. The motion of the system is influenced by the existence of unstable limit cycles, generated from a Hopf bifurcation that occurs at relatively high vehicle forward velocity. Resorting to Floquet theory, we demonstrate that unstable limit cycles are saddle-type cycles with an $N-1$-dimensional stable manifold, being $N$ the dimension of the system. Such stable manifold divides the phase space into two regions, delimiting the stability region of the vehicle. Initial states outside this region cause an uncontrolled motion. By exploiting the properties of the manifolds and by resorting to Floquet theory, we derive a \emph{Degree of Stability (DoS) criterion} valid for motions close to the saddle limit cycle. The criterion serves as a strategy to promptly detect unstable car-and-driver motion in real time during a maneuver, also offering a quantitative indication of the severity of the instability. Two examples show that the DoS criterion can distinguish between a controlled and an uncontrolled maneuver when the corresponding trajectories are still almost equivalent. Global stability Vehicle and driver model Nonlinear tyre character- istic Hopf bifurcation Saddle-type limit cycle Floquet theory Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 20 Feb, 2025 Read the published version in Nonlinear Dynamics → Version 1 posted Editorial decision: Revision requested 09 Aug, 2024 Reviews received at journal 26 Jul, 2024 Reviews received at journal 15 Jul, 2024 Reviewers agreed at journal 10 Jul, 2024 Reviewers agreed at journal 08 Jul, 2024 Reviewers agreed at journal 08 Jul, 2024 Reviewers invited by journal 08 Jul, 2024 Editor assigned by journal 03 Jul, 2024 Submission checks completed at journal 03 Jul, 2024 First submitted to journal 02 Jul, 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. 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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-4673548","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":331423782,"identity":"c841b3f4-50fa-4e47-b8b2-1b28821d6214","order_by":0,"name":"Samuele Giacintucci","email":"","orcid":"","institution":"Politecnico di Milano","correspondingAuthor":false,"prefix":"","firstName":"Samuele","middleName":"","lastName":"Giacintucci","suffix":""},{"id":331423783,"identity":"0c037254-3918-4bb2-8a3c-7b926bf2969d","order_by":1,"name":"Fabio Della Rossa","email":"","orcid":"","institution":"Politecnico di Milano","correspondingAuthor":false,"prefix":"","firstName":"Fabio","middleName":"Della","lastName":"Rossa","suffix":""},{"id":331423784,"identity":"9b4debc2-bf27-4557-b489-743c06f9c05e","order_by":2,"name":"Gianpiero Mastinu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABE0lEQVRIiWNgGAWjYDACZhBhAOdKMDCwNzAwPDBg4IFIEdSSANTCcwBIg7Xg0oMCEkAWJUBoBhzWyLszP3tcUGCTz8/Anfjx5w+LPIObbw8+SCjYJsPAzn8AmxbDw2zmxjMM0ixnNvBuluZJkCg2uJ2XbJBgcBunwwybGcykeQwOGxgc4N0gDfRL4obbOWYS+LWwfwNq+Q/SsvnnD5CWm2fwa5Fn5gHZcgCkZZsED0jLDR78WgyYecqkZxgkG0g2826z5kmTSJx5BuoXNmZmA2xa5PuPb5Mu+GNnwM/eu/nmD5u6xL7jZw8++PDntj0//8EHWG05AAt+hCt4IBQbVmcBbWnAjDEeHGpHwSgYBaNgpAIA+oNT36T2xD0AAAAASUVORK5CYII=","orcid":"","institution":"Politecnico di Milano","correspondingAuthor":true,"prefix":"","firstName":"Gianpiero","middleName":"","lastName":"Mastinu","suffix":""}],"badges":[],"createdAt":"2024-07-02 10:36:18","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4673548/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4673548/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11071-024-10601-2","type":"published","date":"2025-02-20T15:58:05+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":77058341,"identity":"6c2402ff-8521-46d8-abc9-e48da1d436c5","added_by":"auto","created_at":"2025-02-24 16:58:51","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3523333,"visible":true,"origin":"","legend":"","description":"","filename":"submitted.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4673548/v1_covered_9eea5f0c-8efd-4191-892f-04d77d2c6eda.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Early detection of unstable car-and-driver motion – a Floquet theory approach","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":"Global stability, Vehicle and driver model, Nonlinear tyre character- istic, Hopf bifurcation, Saddle-type limit cycle, Floquet theory","lastPublishedDoi":"10.21203/rs.3.rs-4673548/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4673548/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"The global stability of car-and-driver is studied. 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