{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T15:16:17Z","timestamp":1761664577886,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,4,7]],"date-time":"2023-04-07T00:00:00Z","timestamp":1680825600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Natural Science Foundation of Hunan Province","award":["2021JJ3057","21B0424","202250015428"],"award-info":[{"award-number":["2021JJ3057","21B0424","202250015428"]}]},{"name":"the Research Foundation of Education Bureau of Hunan Province","award":["2021JJ3057","21B0424","202250015428"],"award-info":[{"award-number":["2021JJ3057","21B0424","202250015428"]}]},{"name":"Hengyang Science and Technology Major Project","award":["2021JJ3057","21B0424","202250015428"],"award-info":[{"award-number":["2021JJ3057","21B0424","202250015428"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Feedrate plays a crucial role in determining the machining quality, tool life, and machining time. Thus, this research aimed to improve the accuracy of NURBS interpolator systems by minimizing feedrate fluctuations during CNC machining. Previous studies have proposed various methods to minimize these fluctuations. However, these methods often require complex calculations and are not suitable for real-time and high-precision machining applications. Given the sensitivity of the curvature-sensitive region to feedrate variations, this paper proposed a two-level parameter compensation method to eliminate the feedrate fluctuation. First, in order to address federate fluctuations in non-curvature sensitive areas with low computational costs, we employed the first-level parameter compensation (FLPC) using the Taylor series expansion method. This compensation allows us to achieve a chord trajectory for the new interpolation point that matches the original arc trajectory. Second, even in curvature-sensitive areas, feedrate fluctuations can still occur because of truncation errors in the first-level parameter compensation. To address this, we employed the Secant-based method for second-level parameter compensation (SLPC), which does not require derivative calculations and can regulate feedrate fluctuation within the fluctuation tolerance. Finally, we applied the proposed method to the simulation of butterfly-shaped NURBS curves. These simulations demonstrated that our method achieved maximum feedrate fluctuation rates below 0.01% with an average computational time of 360 us, which is sufficient for high-precision and real-time machining. Additionally, our method outperformed four other feedrate fluctuation elimination methods, highlighting its feasibility and effectiveness.<\/jats:p>","DOI":"10.3390\/s23083789","type":"journal-article","created":{"date-parts":[[2023,4,7]],"date-time":"2023-04-07T04:04:16Z","timestamp":1680840256000},"page":"3789","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["NURBS Interpolator with Minimum Feedrate Fluctuation Based on Two-Level Parameter Compensation"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2720-0470","authenticated-orcid":false,"given":"Mingxing","family":"Nie","sequence":"first","affiliation":[{"name":"School of Computer Science, University of South China, Hengyang 421001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5879-5980","authenticated-orcid":false,"given":"Tao","family":"Zhu","sequence":"additional","affiliation":[{"name":"School of Computer Science, University of South China, Hengyang 421001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4785-5095","authenticated-orcid":false,"given":"Yue","family":"Li","sequence":"additional","affiliation":[{"name":"School of Computer Science, University of South China, Hengyang 421001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"7492762","DOI":"10.1155\/2022\/7492762","article-title":"Real-Time NURBS Interpolation under Multiple Constraints","volume":"2022","author":"Nie","year":"2022","journal-title":"Comput. Intell. Neurosci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3192","DOI":"10.1109\/ACCESS.2021.3138049","article-title":"Novel Feedrate Optimization Method for NURBS Tool Paths Under Various Constraints","volume":"10","author":"Lu","year":"2021","journal-title":"IEEE Access."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1315","DOI":"10.1007\/s00170-022-09385-9","article-title":"NURBS interpolator with pre-compensation based on discrete inverse transfer function for CNC high-precision machining","volume":"121","author":"Wang","year":"2022","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Cao, L., Wang, T., Jia, S., Dong, J., Tian, C., Tian, Y., and Zheng, M. (Res. Sq., 2022). Innovation of NURBS feedrate direct interpolator in five-axis for CNC high-precision and high-efficiency machining, Res. Sq., Preprint.","DOI":"10.21203\/rs.3.rs-1678627\/v1"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1531","DOI":"10.1007\/s00170-021-08092-1","article-title":"A novel feedrate scheduling method based on Sigmoid function with chord error and kinematic constraints","volume":"119","author":"Li","year":"2022","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Guo, P., Wu, Y., Yang, G., Shen, Z., Zhang, H., Zhang, P., Lou, F., and Li, H. (2021). A feedrate planning method for the NURBS curve in CNC machining based on the critical constraint curve. Appl. Sci., 11.","DOI":"10.3390\/app11114959"},{"key":"ref_7","unstructured":"Hu, Y., Jiang, X., Huo, G., Su, C., Wang, B., Li, H., and Zheng, Z. (2021). A novel S-shape based NURBS interpolation with acc-jerk-Continuity and round-off error elimination. arXiv."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1007\/s12541-019-00288-6","article-title":"NURBS interpolator with adaptive smooth feedrate scheduling and minimal feedrate fluctuation","volume":"21","author":"Wang","year":"2020","journal-title":"Int. J. Precis. Eng. Manuf."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.precisioneng.2020.03.012","article-title":"An improved feedrate scheduling method for NURBS interpolation in five-axis machining","volume":"64","author":"Sang","year":"2020","journal-title":"Precis. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2095","DOI":"10.1007\/s00170-020-06258-x","article-title":"A NURBS curve interpolator with small feedrate fluctuation based on arc length prediction and correction","volume":"111","author":"Ji","year":"2020","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"112953","DOI":"10.1109\/ACCESS.2019.2934977","article-title":"A real-time feedrate planning method and efficient interpolator with minimal feedrate fluctuation for parametric toolpath","volume":"7","author":"Jiang","year":"2019","journal-title":"IEEE Access."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1099","DOI":"10.1007\/s00170-019-04308-7","article-title":"Smooth minimum time trajectory planning with minimal feed fluctuation","volume":"105","author":"Zhao","year":"2019","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/j.ymssp.2018.10.028","article-title":"Dynamic feedrate optimization for parametric toolpath with data-based tracking error prediction","volume":"120","author":"Zhang","year":"2019","journal-title":"Mech. Syst. Sig. Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"66403","DOI":"10.1109\/ACCESS.2018.2813334","article-title":"Feedrate scheduling of NURBS interpolation based on a novel jerk-continuous ACC\/DEC algorithm","volume":"6","author":"Ni","year":"2018","journal-title":"IEEE Access."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"63794","DOI":"10.1109\/ACCESS.2018.2875403","article-title":"A bidirectional adaptive feedrate scheduling method of NURBS interpolation based on S-shaped ACC\/DEC algorithm","volume":"6","author":"Ni","year":"2018","journal-title":"IEEE Access."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Cho, C.N., and Kim, H.J. (2017, January 11\u201314). In Feedrate fluctuation compensating NURBS interpolator for CNC machining. Proceedings of the 2017 20th International Conference on Electrical Machines and Systems (ICEMS), Sydney, Australia.","DOI":"10.1109\/ICEMS.2017.8056034"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"8718760","DOI":"10.1155\/2017\/8718760","article-title":"A novel approach for NURBS interpolation with minimal feed rate fluctuation based on improved adams-moulton method","volume":"2017","author":"Peng","year":"2017","journal-title":"Math. Probl. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2359","DOI":"10.1177\/0954405416629588","article-title":"Feedrate scheduling for interpolation of parametric tool path using the sine series representation of jerk profile","volume":"231","author":"Huang","year":"2017","journal-title":"Proc. Inst. Mech. Eng. Part B J. Eng. Manuf."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"911","DOI":"10.3901\/CJME.2016.0407.047","article-title":"High accurate interpolation of NURBS tool path for CNC machine tools","volume":"29","author":"Liu","year":"2016","journal-title":"Chin. J. Mech. Eng."},{"key":"ref_20","first-page":"329","article-title":"Advanced interpolation techniques for NC machines","volume":"3","author":"Bedi","year":"1993","journal-title":"J. Manuf. Sci. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/j.ijmachtools.2003.10.020","article-title":"NURBS interpolator for constant material removal rate in open NC machine tools","volume":"44","author":"Tikhon","year":"2004","journal-title":"Int. J. Mach. Tools Manuf."},{"key":"ref_22","first-page":"393","article-title":"In A generalized interpolator for command generation of parametric curves in computer-controlled machines, ASME Japan\/USA Symp","volume":"1","author":"Huang","year":"1992","journal-title":"Flex. Autom."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"832","DOI":"10.1016\/0010-4485(94)90097-3","article-title":"Realtime curve interpolators","volume":"26","author":"Shpitalni","year":"1994","journal-title":"Comput. Aided Des."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1016\/S0010-4485(99)00035-4","article-title":"The speed-controlled interpolator for machining parametric curves","volume":"31","author":"Yeh","year":"1999","journal-title":"Comput. Aided Des."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1016\/S0890-6955(02)00015-9","article-title":"Real-time NURBS command generators for CNC servo controllers","volume":"42","author":"Cheng","year":"2002","journal-title":"Int. J. Mach. Tools Manuf."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1016\/j.ijmachtools.2009.01.007","article-title":"Robust real-time NURBS path interpolators","volume":"49","author":"Lei","year":"2009","journal-title":"Int. J. Mach. Tools Manuf."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1115\/1.1578670","article-title":"A real-time predictor-corrector interpolator for CNC machining","volume":"125","author":"Tsai","year":"2003","journal-title":"J. Manuf. Sci. Eng."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1855","DOI":"10.1243\/09544054JEM401","article-title":"Feedrate control for non-uniform rational B-spline motion command generation","volume":"220","author":"Cheng","year":"2006","journal-title":"Proc. Inst. Mech. Eng. Part B J. Eng. Manuf."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1530","DOI":"10.1016\/j.ijmachtools.2006.11.011","article-title":"Fast real-time NURBS path interpolation for CNC machine tools","volume":"47","author":"Lei","year":"2007","journal-title":"Int. J. Mach. Tools Manuf."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ijmachtools.2014.07.002","article-title":"Development and implementation of a NURBS interpolator with smooth feedrate scheduling for CNC machine tools","volume":"87","author":"Liu","year":"2014","journal-title":"Int. J. Mach. Tools Manuf."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1007\/s00170-011-3520-2","article-title":"A NURBS interpolation algorithm with continuous feedrate","volume":"59","author":"Wu","year":"2012","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1016\/j.ijmachtools.2009.11.005","article-title":"Design of a NURBS interpolator with minimal feed fluctuation and continuous feed modulation capability","volume":"50","author":"Heng","year":"2010","journal-title":"Int. J. Mach. Tools Manuf."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1007\/s00170-019-03586-5","article-title":"Contour error\u2013bounded parametric interpolator with minimum feedrate fluctuation for five-axis CNC machine tools","volume":"103","author":"Chen","year":"2019","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ijmachtools.2013.08.002","article-title":"A parametric interpolator with minimal feed fluctuation for CNC machine tools using arc-length compensation and feedback correction","volume":"75","author":"Zhao","year":"2013","journal-title":"Int. J. Mach. Tools Manuf."},{"key":"ref_35","first-page":"307","article-title":"Superresolved Raman spectroscopy Spectrosc","volume":"46","author":"Malka","year":"2013","journal-title":"Lett"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"75801","DOI":"10.1088\/2040-8986\/ab2625","article-title":"Improving Raman spectra of pure silicon using Super-Resolved method","volume":"21","author":"Malka","year":"2019","journal-title":"J. Opt."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1016\/S0010-4485(01)00082-3","article-title":"Adaptive-feedrate interpolation for parametric curves with a confined chord error","volume":"34","author":"Yeh","year":"2002","journal-title":"Comput. Aided Des."},{"key":"ref_38","first-page":"278","article-title":"A precise direct parametric interpolation method for NURBS-toolpath generation","volume":"31","author":"Song","year":"2019","journal-title":"Int. J. Ind. Syst. Eng."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1642","DOI":"10.1177\/0954405416673680","article-title":"Fast parametric curve interpolation with minimal feedrate fluctuation by cubic B-spline","volume":"232","author":"Lu","year":"2018","journal-title":"Proc. Inst. Mech. Eng. Part B J. Eng. Manuf."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/8\/3789\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:11:49Z","timestamp":1760123509000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/8\/3789"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,7]]},"references-count":39,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["s23083789"],"URL":"https:\/\/doi.org\/10.3390\/s23083789","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2023,4,7]]}}}