An Embedded High-Order Runge–Kutta 6(5) Pair with Adaptive Step Control for Low- Latency Real-Time Signal Processing on Resource-Constrained Platforms

An Embedded High-Order Runge–Kutta 6(5) Pair with Adaptive Step Control for Low- Latency Real-Time Signal Processing on Resource-Constrained Platforms | 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 An Embedded High-Order Runge–Kutta 6(5) Pair with Adaptive Step Control for Low- Latency Real-Time Signal Processing on Resource-Constrained Platforms Chao Zhao, Yucheng Shen This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8887021/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 6 You are reading this latest preprint version Abstract Real-time signal processing on small embedded devices often depends on solving ordinary differential equations to model dynamic behavior in audio effects, sensor data fusion, and control loops. Fixed-step low-order Runge-Kutta solvers like RK4 give predictable timing but require very small steps on problems with mixed fast and slow dynamics, which wastes cycles and raises power use. Higher-order adaptive methods can adjust the step size automatically for better efficiency, but their extra stages and variable timing make them hard to run on low-power microcontrollers or small FPGAs without breaking latency guarantees. This work describes a practical 6(5) embedded Runge-Kutta pair with bounded step-size adaptation designed specifically for such constrained hardware. The implementation uses a compact eight-stage Verner coefficient set that supports FSAL and dense output, fixed-point arithmetic with careful stage reuse, and a simple controller that limits step changes while adding hysteresis to keep execution time within known bounds. Tests on an STM32F407 microcontroller and an Artix-7 FPGA covered a stiff Van der Pol oscillator, a nonlinear diode ladder filter, and an extended Kalman filter for inertial sensors. Compared to fixed RK4 and a bounded Dormand-Prince 5(4) variant, the new solver cut global errors by more than an order of magnitude for similar average latency, kept worst-case latency suitable for sampled loops, and often used less energy on varying workloads. Step rejections stayed below 8% in all cases. The results show that a tuned high-order adaptive solver can run reliably on severely limited platforms, allowing richer dynamic models in battery-powered edge applications without losing real-time predictability. Adaptive Step Control Fixed-point Implementation Low-latency Real-time Signal Processing Runge-Kutta 6(5) Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviews received at journal 09 Apr, 2026 Reviewers agreed at journal 29 Mar, 2026 Reviewers invited by journal 24 Feb, 2026 Editor assigned by journal 16 Feb, 2026 Submission checks completed at journal 16 Feb, 2026 First submitted to journal 15 Feb, 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-8887021","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":596121627,"identity":"0450370a-22eb-4e0a-8ff6-6674b5e37bf2","order_by":0,"name":"Chao Zhao","email":"","orcid":"","institution":"Southwest Jiaotong University Hope College","correspondingAuthor":false,"prefix":"","firstName":"Chao","middleName":"","lastName":"Zhao","suffix":""},{"id":596121628,"identity":"68edf2b0-2b8a-4bfb-bb9c-f95cdef68e89","order_by":1,"name":"Yucheng Shen","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+0lEQVRIiWNgGAWjYDACCQiVAMSMDyoY5EAcA6K1MBucYTAmTQubBFFa5Gc3H3vM22aXxy/dfq3iwB+DxAb25m0SDDV3cGoxuHMs3Zi3LblYcs6ZshsHeIBaeI6VSTAce4Zbi0SOmTRv24HEDTdy0m5/kPiT2AAUkWBsOIzbYTPyv4G17AdqKThgALRF/g1+LQw3ctggtkikH2M4kADUIsGDX4vBjTQzyTnnkhNn3MhhljhwwMC4jSet2CLhGD6HJT+TeFNml9g/I/3hB2CIyfazH95440MNHocBARMPmOKBRAcbiEjAqwGYUH6AKfYHBNSNglEwCkbBSAUAgWxYoNssNMkAAAAASUVORK5CYII=","orcid":"","institution":"Fuzhou University of International Studies and Trad","correspondingAuthor":true,"prefix":"","firstName":"Yucheng","middleName":"","lastName":"Shen","suffix":""}],"badges":[],"createdAt":"2026-02-15 15:38:12","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8887021/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8887021/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":103507729,"identity":"767e241c-1ab9-46f2-86e8-505546bc0d14","added_by":"auto","created_at":"2026-02-26 13:44:04","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":641632,"visible":true,"origin":"","legend":"","description":"","filename":"AnEmbeddedHighOrderRungeKutta65Pairanoy.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8887021/v1_covered_eb9319e9-5171-4e16-84a0-82608ac8c7a5.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"An Embedded High-Order Runge–Kutta 6(5) Pair with Adaptive Step Control for Low- Latency Real-Time Signal Processing on Resource-Constrained Platforms","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":"journal-of-king-saud-university-engineering-sciences","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [Journal of King Saud University – Engineering Sciences](https://link.springer.com/journal/44444)","snPcode":"44444","submissionUrl":"https://submission.springernature.com/new-submission/44444/3","title":"Journal of King Saud University – Engineering Sciences","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Open","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Adaptive Step Control, Fixed-point Implementation, Low-latency, Real-time Signal Processing, Runge-Kutta 6(5)","lastPublishedDoi":"10.21203/rs.3.rs-8887021/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8887021/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eReal-time signal processing on small embedded devices often depends on solving ordinary differential equations to model dynamic behavior in audio effects, sensor data fusion, and control loops. Fixed-step low-order Runge-Kutta solvers like RK4 give predictable timing but require very small steps on problems with mixed fast and slow dynamics, which wastes cycles and raises power use. Higher-order adaptive methods can adjust the step size automatically for better efficiency, but their extra stages and variable timing make them hard to run on low-power microcontrollers or small FPGAs without breaking latency guarantees.\u003c/p\u003e \u003cp\u003eThis work describes a practical 6(5) embedded Runge-Kutta pair with bounded step-size adaptation designed specifically for such constrained hardware. The implementation uses a compact eight-stage Verner coefficient set that supports FSAL and dense output, fixed-point arithmetic with careful stage reuse, and a simple controller that limits step changes while adding hysteresis to keep execution time within known bounds.\u003c/p\u003e \u003cp\u003eTests on an STM32F407 microcontroller and an Artix-7 FPGA covered a stiff Van der Pol oscillator, a nonlinear diode ladder filter, and an extended Kalman filter for inertial sensors. Compared to fixed RK4 and a bounded Dormand-Prince 5(4) variant, the new solver cut global errors by more than an order of magnitude for similar average latency, kept worst-case latency suitable for sampled loops, and often used less energy on varying workloads. 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