{"paper_id":"2ba08795-8a4b-4dbe-af8c-afd803709aed","body_text":"A Universal Approach For Solving The Ultra-Revolution Lambert's Problem | 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 A Universal Approach For Solving The Ultra-Revolution Lambert's Problem James William McElreath, Ian Down, Manoranjan Majji This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6279141/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 27 Jun, 2025 Read the published version in Celestial Mechanics and Dynamical Astronomy → Version 1 posted 9 You are reading this latest preprint version Abstract Lambert's problem, a cornerstone of orbital mechanics, is crucial in the design and planning of space missions, from satellite rendezvous to deep-space trajectory optimization. This paper presents a universal approach (valid across all conics) to solving Lambert’s Problem, specializing in the ultra-revolution problem. By employing matrix exponential solutions to the Sundman transform, the transfer time equation--constructed of universal functions--is evaluated through exponential functions, resulting in two key advantages. The first advantage is the removal of all trigonometric functions and associated inverses when evaluating the transfer time function, reducing computation cost. The second is a unified transfer time formulation across elliptic and hyperbolic regimes, reducing algorithm complexity. Furthermore, the multi-revolution term is removed from the transfer time equation, reducing the complexity of transfer time derivatives and improving the numerical stability of multi-revolution solutions. A simple-but-accurate initial guess scheme that leverages the transfer angle is employed, resulting in accurate initial guesses and solutions even at 100,000 revolutions. Simulations indicate computation rates similar to state-of-the-art methods. Tests across a wide range of representative cases confirm that the solver achieves 10-digit velocity accuracy within two iterations for most transfer geometries and one iteration for near-circular cases, regardless of revolution count. Lambert’s Problem Orbital Rendezvous Lambert Solver Sundman Transform Gooding’s Method Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 27 Jun, 2025 Read the published version in Celestial Mechanics and Dynamical Astronomy → Version 1 posted Editorial decision: Revision requested 18 May, 2025 Reviews received at journal 15 May, 2025 Reviews received at journal 12 May, 2025 Reviewers agreed at journal 28 Mar, 2025 Reviewers agreed at journal 23 Mar, 2025 Reviewers invited by journal 23 Mar, 2025 Editor assigned by journal 22 Mar, 2025 Submission checks completed at journal 22 Mar, 2025 First submitted to journal 21 Mar, 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. 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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-6279141\",\"acceptedTermsAndConditions\":true,\"allowDirectSubmit\":false,\"archivedVersions\":[],\"articleType\":\"Research Article\",\"associatedPublications\":[],\"authors\":[{\"id\":435283278,\"identity\":\"34f4f15d-7c47-44b6-a96f-40e3a145306c\",\"order_by\":0,\"name\":\"James William McElreath\",\"email\":\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3UlEQVRIiWNgGAWjYBACxgYQWQAieBgfQAUNiNACUsPGw2xwgBgtCDVsPGwSRGlhbuBO/PDB4J4c//zeY9UfarYlNrA3b5PA7zDezZIzDIqNJY7xpd04cOx2YgPPsTJCWrYx8xgkJDYc4zG7cYANqEUix4ywlj8GCfXzgVoKDvwDapF/Q4QWBoOEBAOgFoaDbSBbeAhoaQb6pccgwXDjsRxjibN9t43beNKKLfBpMWzv3fjhR0WCvNzhM4YfKr7dlu1nP7zxBl4tzegibPiUg4A8IQWjYBSMglEwChgAoM5Ja1H0Uw8AAAAASUVORK5CYII=\",\"orcid\":\"\",\"institution\":\"Texas A\\u0026M University\",\"correspondingAuthor\":true,\"prefix\":\"\",\"firstName\":\"James\",\"middleName\":\"William\",\"lastName\":\"McElreath\",\"suffix\":\"\"},{\"id\":435283279,\"identity\":\"11cf8d19-b278-4851-8031-49e830499025\",\"order_by\":1,\"name\":\"Ian Down\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"Texas A\\u0026M University\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Ian\",\"middleName\":\"\",\"lastName\":\"Down\",\"suffix\":\"\"},{\"id\":435283280,\"identity\":\"ff99dd0e-a57c-490d-829d-7112d90dd413\",\"order_by\":2,\"name\":\"Manoranjan Majji\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"Texas A\\u0026M University\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Manoranjan\",\"middleName\":\"\",\"lastName\":\"Majji\",\"suffix\":\"\"}],\"badges\":[],\"createdAt\":\"2025-03-21 16:08:24\",\"currentVersionCode\":1,\"declarations\":\"\",\"doi\":\"10.21203/rs.3.rs-6279141/v1\",\"doiUrl\":\"https://doi.org/10.21203/rs.3.rs-6279141/v1\",\"draftVersion\":[],\"editorialEvents\":[{\"content\":\"https://doi.org/10.1007/s10569-025-10251-5\",\"type\":\"published\",\"date\":\"2025-06-27T15:57:35+00:00\"}],\"editorialNote\":\"\",\"failedWorkflow\":false,\"files\":[{\"id\":85686202,\"identity\":\"b6a370f0-f2ec-408d-9a93-0a6e99bfef05\",\"added_by\":\"auto\",\"created_at\":\"2025-06-30 16:04:50\",\"extension\":\"pdf\",\"order_by\":1,\"title\":\"\",\"display\":\"\",\"copyAsset\":false,\"role\":\"manuscript-pdf\",\"size\":21055873,\"visible\":true,\"origin\":\"\",\"legend\":\"\",\"description\":\"\",\"filename\":\"JCMDAUniversalLambertSolver.pdf\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-6279141/v1_covered_697fd877-b334-4990-8f1c-197e070062a6.pdf\"}],\"financialInterests\":\"No competing interests reported.\",\"formattedTitle\":\"A Universal Approach For Solving The Ultra-Revolution Lambert's Problem\",\"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\":\"info@researchsquare.com\",\"identity\":\"celestial-mechanics-and-dynamical-astronomy\",\"isNatureJournal\":false,\"hasQc\":true,\"allowDirectSubmit\":false,\"externalIdentity\":\"cele\",\"sideBox\":\"Learn more about [Celestial Mechanics and Dynamical Astronomy](http://link.springer.com/journal/10569)\",\"snPcode\":\"10569\",\"submissionUrl\":\"https://submission.nature.com/new-submission/10569/3\",\"title\":\"Celestial Mechanics and Dynamical Astronomy\",\"twitterHandle\":\"\",\"acdcEnabled\":true,\"dfaEnabled\":true,\"editorialSystem\":\"em\",\"reportingPortfolio\":\"Springer Hybrid\",\"inReviewEnabled\":true,\"inReviewRevisionsEnabled\":false},\"keywords\":\"Lambert’s Problem, Orbital Rendezvous, Lambert Solver, Sundman Transform, Gooding’s Method\",\"lastPublishedDoi\":\"10.21203/rs.3.rs-6279141/v1\",\"lastPublishedDoiUrl\":\"https://doi.org/10.21203/rs.3.rs-6279141/v1\",\"license\":{\"name\":\"CC BY 4.0\",\"url\":\"https://creativecommons.org/licenses/by/4.0/\"},\"manuscriptAbstract\":\"Lambert's problem, a cornerstone of orbital mechanics, is crucial in the design and planning of space missions, from satellite rendezvous to deep-space trajectory optimization. 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