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Feed Rate-Adaptive Predictive PH Interpolation for Real-Time NURBS Toolpath Execution in High-Speed CNC Machining Cover

Feed Rate-Adaptive Predictive PH Interpolation for Real-Time NURBS Toolpath Execution in High-Speed CNC Machining

By: Guosheng TianORCID  
Open Access
|Oct 2025

References

  1. Farouki, R. T. (2008). Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable. Springer. https://doi.org/10.1007/978-3-540-73398-0
  2. Hu, Z., Xu, J., Li, J. (2024). Trajectory tracking control of CNC system based on RBF neural network composite learning control. In 2024 3rd Conference on Fully Actuated System Theory and Applications (FASTA). IEEE. https://doi.org/10.1109/FASTA61401.2024.10595107
  3. Guo, H., Li, Y., Zhao, B., Guo, Y., Xie, Z., Morina, A., Lu, X. (2024). Transient lubrication of floating bush coupled with dynamics and kinematics of cam-roller in fuel supply mechanism of diesel engine. Physics of Fluids, 36 (12), 123103. https://doi.org/10.1063/5.0232226
  4. Hajrasouliha, A., Ghahfarokhi, B. S. (2021). Dynamic geo-based resource selection in LTE-V2V communications using vehicle trajectory prediction. Computer Communications, 177, 239–254. https://doi.org/10.1016/j.comcom.2021.08.006
  5. Sarvi, A., Mobayen, S., Fekih, A., Mohammadi, S., Asad, J. H. (2022). Design of a global discrete-time sliding mode control scheme for a class of nonlinear systems with state delays and uncertainties. Asian Journal of Control, 24 (5), 2761–2770. https://doi.org/10.1002/asjc.2639
  6. Lotfabadi, A. K., Ghorbansarvi, A., Ossareh, H., Marshall, J. S. (2025). Cost-optimized energy storage operation for a grid-connected solar PV system at community and individual scales. Journal of Energy Storage, 132, 117734. https://doi.org/10.1016/j.est.2025.117734
  7. Ghorbansarvi, A. (2025). Transformer temperature management and voltage control in electric distribution systems with high solar PV penetration. Thesis, University of Vermont, Burlington, US.
  8. Li, J., Wu, X., Wu, L. (2024). A computationally-efficient analytical model for SPM machines considering PM shaping and property distribution. IEEE Transactions on Energy Conversion, 39 (2), 1034–1046. https://doi.org/10.1109/TEC.2024.3352577
  9. Farahani, M. R., Khodaygan, S. (2021). Minimization of non-repeatable runout (NRRO) in high-speed spindle bearings. SAE Technical Paper 2021-01-5023. https://doi.org/10.4271/2021-01-5023
  10. Ghamari, S. M., Molaee, H., Ghahramani, M., Habibi, D., Aziz, A. (2025). Design of an improved robust fractional-order PID controller for buck–boost converter using snake optimization algorithm. IET Control Theory & Applications, 19 (1), e70008. https://doi.org/10.1049/cth2.70008
  11. Liu, X., Ahmad, F., Yamazaki, K., Mori, M. (2025). Adaptive interpolation scheme for NURBS curves with the integration of machining dynamics. International Journal of Machine Tools and Manufacture, 45 (4-5), 433–444. https://doi.org/10.1016/j.ijmachtools.2004.09.009
  12. Jia, Z., Song, D., Ma, J., Hu, G., Su, W. (2017). A NURBS interpolator with constant speed at feedrate-sensitive regions under drive and contour-error constraints. International Journal of Machine Tools and Manufacture, 116, 1–17. https://doi.org/10.1016/j.ijmachtools.2016.12.007
  13. Wang, T.-Y., Zhang, Y.-B., Dong, J.-C., Ke, R.-J., Ding, Y.-Y. (2020). NURBS interpolator with adaptive smooth feedrate scheduling and minimal feedrate fluctuation. International Journal of Precision Engineering and Manufacturing, 21, 273–290. https://doi.org/10.1007/s12541-019-00288-6
  14. Yang, S., Wang, J., Wang, K. (2025). NURBS-OT: An advanced model for generative curve modeling. Journal of Mechanical Design, 147 (3), 031703. https://doi.org/10.1115/1.4066549
  15. Sharafkhani, F., Corns, S., Holmes, R. (2024). Multi-step ahead water level forecasting using deep neural networks. Water, 16 (21), 3153. https://doi.org/10.3390/w16213153
  16. Tsai, M.-S., Nien, H.-W., Yau, H.-T. (2008). Development of an integrated look-ahead dynamics-based NURBS interpolator for high precision machinery. Computer-Aided Design, 40 (5), 554–566. https://doi.org/10.1016/j.cad.2008.01.015
  17. Liu, M., Huang, Y., Yin, L., Guo, J., Shao, X., Zhang, G. (2014). Development and implementation of a NURBS interpolator with smooth feedrate scheduling for CNC machine tools. International Journal of Machine Tools and Manufacture, 87, 1–15. https://doi.org/10.1016/j.ijmachtools.2014.07.002
  18. Chen, M., Sun, Y. (2019). Contour error–bounded parametric interpolator with minimum feedrate fluctuation for five-axis CNC machine tools. The International Journal of Advanced Manufacturing Technology, 103, 567–584. https://doi.org/10.1007/s00170-019-03586-5
  19. Tai, C.-H., Tsai, Y.-T., Li, K.-M. (2023). Establishment of real-time adaptive control strategy for milling parameters. IEEE Access, 11, 125972–125983. https://doi.org/10.1109/ACCESS.2023.3330587
  20. Ghasemi-Falavarjani, N., Moallem, P., Rahimi, A. (2025). High performance frame selection algorithm for gray-level frames within the framework of multi-frame super-resolution. Digital Signal Processing, 164, 105217. https://doi.org/10.1016/j.dsp.2025.105217
  21. Lashaki, R. A., Raeisi, Z., Makki, M., Zare, S. (2025). Dendrite neural network scheme for estimating output power and efficiency for a class of solar free-piston Stirling engine. International Journal of Modelling and Simulation. https://doi.org/10.1080/02286203.2025.2459989
  22. Jiang, T., Tang, Y., Xu, C., Liu, W. (2025). A calibration and error evaluation method of a combined tracking-based vision measurement system for meter-scale components. IEEE Transactions on Industrial Informatics, 21 (6), 4958–4967. https://doi.org/10.1109/TII.2025.3547351
  23. Mollaee, H., Ghamari, S. M., Khavari, F. (2022). Self-tuning regulator adaptive controller design for DC-DC boost converter with a novel robust improved identification method. IET Power Electronics, 15 (13), 1365–1379. https://doi.org/10.1049/pel2.12310
  24. Farouki, R. T., Tsai, Y.-F. (2001). Exact Taylor series coefficients for variable-feedrate CNC curve interpolators. Computer-Aided Design, 33 (2), 155–165. https://doi.org/10.1016/S0010-4485(00)00085-3
  25. Hayasaka, T., Minoura, K., Ishizaki, K., Shamoto, E., Sence, B. (2019). A lightweight interpolation algorithm for short-segmented machining tool paths to realize vibration avoidance, high accuracy, and short machining time. Precision Engineering, 59, 1–17. https://doi.org/10.1016/j.precisioneng.2019.05.006
  26. Du, D., Liu, Y., Guo, X., Yamazaki, K., Fujishima, M. (2010). An accurate adaptive NURBS curve interpolator with real-time flexible acceleration/deceleration control. Robotics and Computer-Integrated Manufacturing, 26 (4), 273–281. https://doi.org/10.1016/j.rcim.2009.09.001
  27. Zare, S., Raeisi, Z., Lashaki, R. A., Makki, M., Ghasemi, N. (2025). Evaluation of a thermoacoustic Stirling oscillator using a describing function and a genetic algorithm. Applied Thermal Engineering, 278, 127125. https://doi.org/10.1016/j.applthermaleng.2025.127125
  28. Yan, X., Hu, J., Zhang, X., Xu, W. (2022). Obtaining superior low-temperature wear resistance in Q&P-processed medium Mn steel with a low initial hardness. Tribology International, 175, 107803. https://doi.org/10.1016/j.triboint.2022.107803
  29. Peng, C., Ghamari, S. M., Mollaee, H., Rezaei, O. (2025). Design of a novel robust adaptive fractional-order model predictive controller for boost converter using grey wolf optimization algorithm. Scientific Reports, 15, 27670. https://doi.org/10.1038/s41598-025-10125-8
  30. Nagao, S., Kawai, Y., Yokokura, Y., Ohishi, K., Miyazaki, T. (2023). Load-side acceleration control based on single inertialization compensator and jerk observer for industrial robots. IEEJ Journal of Industry Applications, 12 (6), 1034–1045. https://doi.org/10.1541/ieejjia.22011983
  31. Kitayoshi, R., Yoshiura, Y., Kaku, Y. (2023). Σ-X series: AC Servo drive for achievement of digital solution. IEEJ Journal of Industry Applications, 12 (5), 859–867. https://doi.org/10.1541/ieejjia.22010390
  32. Takeuchi, M., Katsura, S. (2023). Robust velocity control for electromagnetic friction brake based on disturbance observer. IEEJ Journal of Industry Applications, 12 (5), 876–884. https://doi.org/10.1541/ieejjia.22004484
  33. Takeuchi, K., Sakaino, S., Tsuji, T. (2023). Motion generation based on contact state estimation using two-stage clustering. IEEJ Journal of Industry Applications, 12 (5), 1000–1007. https://doi.org/10.1541/ieejjia.22012635
  34. Kato, Y., Sakaino, S., Tsuji, T. (2023). Motion planning for cutting flexible objects based on contact state recognition. IEEJ Journal of Industry Applications, 12 (4), 786–792. https://doi.org/10.1541/ieejjia.22004392
  35. Zhang, Z., Liu, S., Zhang, Y., Wang, C., Zhang, S., Yang, Z., Xu, W. (2022). Optimization of low-power femtosecond laser trepan drilling by machine learning and a high-throughput multi-objective genetic algorithm. Optics & Laser Technology, 148, 107688. https://doi.org/10.1016/j.optlastec.2021.107688
  36. Liu, Y., Li, X., Zhang, Y., Ge, L., Guan, Y., Zhang, Z. (2024). Ultra-large scale stitchless AFM: Advancing nanoscale characterization and manipulation with zero stitching error and high throughput. Small, 20 (1), 2303838. https://doi.org/10.1002/smll.202303838
  37. Liu, Y., Li, X., Ge, L., Zhang, Z. (2024). Ultralarge-area stitchless scanning probe lithography and in situ characterization system using a compliant nanomanipulator. IEEE/ASME Transactions on Mechatronics, 29 (2), 924–935. https://doi.org/10.1109/TMECH.2023.3323385
  38. Tao, Z., Li, W., Guo, Z., Chen, Y., Song, L., Li, J. (2024). Aerothermal optimization of a turbine rotor tip configuration based on free-form deformation approach. International Journal of Heat and Fluid Flow, 110, 109644. https://doi.org/10.1016/j.ijheatfluidflow.2024.109644
  39. Farouki, R. T., Sakkalis, T. (1990). Pythagorean hodographs. IBM Journal of Research and Development, 34 (5), 736–752. https://doi.org/10.1147/rd.345.0736
  40. Tsai, Y.-F., Farouki, R. T., Feldman, B. (2001). Performance analysis of CNC interpolators for time-dependent feedrates along PH curves. Computer Aided Geometric Design, 18 (3), 245–265. https://doi.org/10.1016/S0167-8396(01)00029-2
  41. Bawazeer, S. A., Baakeem, S. S., Mohamad, A. A. (2021). New approach for radial basis function based on partition of unity of Taylor series expansion with respect to shape parameter. Algorithms, 14 (1). https://doi.org/10.3390/a14010001
  42. Yeh, S.-S., Hsu, P.-L. (1999). The speed-controlled interpolator for machining parametric curves. Computer-Aided Design, 31 (5), 349–357. https://doi.org/10.1016/S0010-4485(99)00035-4
  43. Du, X., Huang, J., Zhu, L.-M., Ding, H. (2020). An error-bounded B-spline curve approximation scheme using dominant points for CNC interpolation of micro-line toolpath. Robotics and Computer-Integrated Manufacturing, 64, 101930. https://doi.org/10.1016/j.rcim.2019.101930
  44. Ni, H., Yuan, J., Ji, S., Zhang, C., Hu, T. (2018). Feedrate scheduling of NURBS interpolation based on a novel jerk-continuous ACC/DEC algorithm. IEEE Access, 6, 66403–66417. https://doi.org/10.1109/ACCESS.2018.2813334
  45. Novák, L., Novák, D. (2020). On taylor series expansion for statistical moments of functions of correlated random variables. Symmetry, 12 (8), 1379. https://doi.org/10.3390/sym12081379
  46. Paudel, A., Gupta, S., Thapa, M., Mulani, S. B., Walters, R. W. (2022). Higher-order Taylor series expansion for uncertainty quantification with efficient local sensitivity. Aerospace Science and Technology, 126, 107574. https://doi.org/10.1016/j.ast.2022.107574
  47. Yeh, S.-S., Hsu, P.-L. (2002). Adaptive-feedrate interpolation for parametric curves with a confined chord error. Computer-Aided Design, 34 (3), 229–237. https://doi.org/10.1016/S0010-4485(01)00082-3
  48. Zhiming, X., Jincheng, C., Zhengjin, F. (2002). Performance evaluation of a real-time interpolation algorithm for NURBS curves. The International Journal of Advanced Manufacturing Technology, 20, 270–276. https://doi.org/10.1007/s001700200152
  49. Sarguroh, S. S., Rane, A. B. (2018). Using GRBL-Arduino-based controller to run a two-axis computerized numerical control machine. In 2018 International Conference on Smart City and Emerging Technology (ICSCET). IEEE. https://doi.org/10.1109/ICSCET.2018.8537315
  50. Wu, G., Zhao, M., Cong, Y., Hu, Z., Li, G. (2021). Algorithm of berthing and maneuvering for catamaran unmanned surface vehicle based on ship maneuverability. Journal of Marine Science and Engineering, 9 (3), 289. https://doi.org/10.3390/jmse9030289
  51. Liu, D., Luo, M., Pelayo, G. U., Trejo, D. O., Zhang, D. (2021). Position-oriented process monitoring in milling of thin-walled parts. Journal of Manufacturing Systems, 60, 360–372. https://doi.org/10.1016/j.jmsy.2021.06.010
  52. Lv, S., Liu, Z. (2022). Step motor control based on PLC. In 2022 5th World Conference on Mechanical Engineering and Intelligent Manufacturing (WCMEIM). IEEE. https://doi.org/10.1109/WCMEIM56910.2022.10021545
  53. Hu, Y., Jiang, X., Huo, G., Su, C., Wang, B., Li, H., Zheng, Z. (2021). A novel S-shape based NURBS interpolation with acc-jerk-Continuity and round-off error elimination. arXiv:2103.14433. https://doi.org/10.48550/arXiv.2103.14433
  54. Liu, X., Tan, J., Long, S. (2024). Multi-axis fatigue load spectrum editing for automotive components using generalized S-transform. International Journal of Fatigue, 188, 108503. https://doi.org/10.1016/j.ijfatigue.2024.108503
  55. Li, X., Wang, S., Li, Z., Yang, R., Li, Z. (2024). Measurement of bolt axial stress using a combination of trailing wave and shear wave ultrasound. NDT & E International, 143, 103056. https://doi.org/10.1016/j.ndteint.2024.103056
  56. Hao, D., Wang, W., Liu, Z., Yun, C. (2020). Experimental study of stability prediction for high-speed robotic milling of aluminum. Journal of Vibration and Control, 26 (7-8), 387–398. https://doi.org/10.1177/1077546319880376
  57. Chi, Y., Dong, Z., Cui, M., Shan, C., Xiong, Y., Zhang, D., Luo, M. (2024). Comparative study on machinability and surface integrity of γ-TiAl alloy in laser assisted milling. Journal of Materials Research and Technology, 33, 3743–3755. https://doi.org/10.1016/j.jmrt.2024.10.028
  58. Zou, Y., Tang, S., Guo, S., Song, X. (2024). Tool wear analysis in turning inconel-657 using various tool materials. Materials and Manufacturing Processes, 39 (10), 1363–1368. https://doi.org/10.1080/10426914.2024.2323434
  59. Wang, C., Zhang, Z., Jing, X., Yang, Z., Xu, W. (2022). Optimization of multistage femtosecond laser drilling process using machine learning coupled with molecular dynamics. Optics & Laser Technology, 156, 108442. https://doi.org/10.1016/j.optlastec.2022.108442
  60. Zhang, X., Liu, Y., Chen, X., Li, Z., Su, C.-Y. (2023). Adaptive pseudoinverse control for constrained hysteretic nonlinear systems and its application on dielectric elastomer actuator. IEEE/ASME Transactions on Mechatronics, 28 (4), 2142–2154. https://doi.org/10.1109/TMECH.2022.3231263
  61. Ramezani, M. K., Mohseni Shakib, S., Soltani, H. (2012). Numerical analysis of delamination growth in laminated composites under buckling behavior. Advanced Materials Research, 433-440, 379–384. https://doi.org/10.4028/www.scientific.net/AMR.433-440.379
  62. Wang, F., Chen, K., Zhen, S., Chen, X., Zheng, H., Wang, Z. (2024). Prescribed performance adaptive robust control for robotic manipulators with fuzzy uncertainty. IEEE Transactions on Fuzzy Systems, 32 (3), 1318–1330. https://doi.org/10.1109/TFUZZ.2023.3323090
  63. Huang, J., Kang, R., Dong, Z., Gao, S. (2025). Prediction model for surface shape of YAG wafers in wafer rotational grinding. International Journal of Mechanical Sciences, 287, 109982. https://doi.org/10.1016/j.ijmecsci.2025.109982
  64. Ramezani, M. K., Ramesh, S., Purbolaksono, J., Das, R. (2022). Closed-form solutions of stress intensity factors for semi-elliptical surface cracks in a cylindrical bar under pure tension. Acta Mechanica Solida Sinica, 35 (2), 344–356. https://doi.org/10.1007/s10338-021-00286-0
  65. Barrinaya, M. A., Alfiyurandaa, M. N., Ramezani, M., Putra, I. S., Ramesh, S., Kadarnoa, P., Hastutya, S., Purbolaksonoa, J. (2022). Modes I-II-III stress intensity factors of a semi-elliptical surface crack at a round bar under torsion loading by FEM and DBEM. Engineering Solid Mechanics, 10 (4), 399–406. http://dx.doi.org/10.5267/j.esm.2022.6.099
  66. Liu, H., Zhang, D., Geng, D. (2025). Design of a self-excited vibration tool bar for cutting difficult-to-machine alloys. International Journal of Mechanical Sciences, 300, 110456. https://doi.org/10.1016/j.ijmecsci.2025.110456
  67. Chang, H., Feng, S., Qiu, X., Meng, H., Guo, G., He, X., He, Q., Yang, X., Ma, W., Kan, R., Fittschen, C., Li, C. (2020). Implementation of the toroidal absorption cell with multi-layer patterns by a single ring surface. Optics Letters, 45 (21), 5897–5900. https://doi.org/10.1364/OL.404198
  68. Cao, Y., Zhang, Z. (2025). Enhanced contour tracking: A time-varying internal model principle-based approach. IEEE/ASME Transactions on Mechatronics, 30 (4), 3188–3196. https://doi.org/10.1109/TMECH.2025.3572743
  69. Liu, L., Jiang, X., Ying, E., Sun, Z., Geng, D., Zhang, D. (2025). High-performance milling of Ti-6Al-4V through rotary ultrasonic elliptical milling with anticlockwise elliptical vibration. Journal of Zhejiang University-SCIENCE A, 26, 707–722. https://doi.org/10.1631/jzus.A2500007
  70. Tang, J., Zhang, T., Liu, T., Zhang, Z., Cao, L., Yao, Q. (2025). A non-intrusive interval analysis method for chatter stability of uncertain milling systems. Journal of Manufacturing Processes, 145, 142–157. https://doi.org/10.1016/j.jmapro.2025.04.047
  71. Liu, H., Zhen, S., Liu, X., Zheng, H., Gao, L., Chen, Y.-H. (2024). Robust approximate constraint following control design for collaborative robots system and experimental validation. Robotica, 42 (11), 3957–3975. https://doi.org/10.1017/S0263574724001760
  72. Piegl, L. A., Tiller, W. (1999). Computing offsets of NURBS curves and surfaces. Computer-Aided Design, 31 (2), 147–156. https://doi.org/10.1016/S0010-4485(98)00066-9
  73. Lv, P., Pan, S. (2024). Research on NURBS curve interpolation algorithm based on Mline-Gear. International Journal of Computer Science and Information Technology, 2 (1), 84–94. https://doi.org/10.62051/ijcsit.v2n1.10
  74. Moetakef Imani, B., Ghandehariun, A. (2011). Real-time PH-based interpolation algorithm for high speed CNC machining. The International Journal of Advanced Manufacturing Technology, 56, 619–629. https://doi.org/10.1007/s00170-011-3200-2
  75. Jacobs, E. N., Ward, K. E., Pinkerton, R. M. (1933). The characteristics of 78 related airfoil section from tests in the Variable-Density Wind Tunnel. NACA Technical Report NACA-TR-460.
  76. Moon, H. P., Farouki, R. T., Choi, H. I. (2001). Construction and shape analysis of PH quintic Hermite interpolants. Computer Aided Geometric Design, 18 (2), 93–115. https://doi.org/10.1016/S0167-8396(01)00016-4
  77. Liao, H., Vaitheeswaran, P. K., Song, T., Subbarayan, G. (2020). Algebraic point projection for immersed boundary analysis on low degree NURBS curves and surfaces. Algorithms, 13 (4), 82. https://doi.org/10.3390/a13040082
  78. Cheng, C.-W., Tsai, M.-C. (2004). Real-time variable feed rate NURBS curve interpolator for CNC machining. The International Journal of Advanced Manufacturing Technology, 23, 865–873. https://doi.org/10.1007/s00170-003-1732-9
  79. Li, J., Liu, Y., Li, Y., Zhong, G. (2019). S-model speed planning of NURBS curve based on uniaxial performance limitation. IEEE Access, 7, 60837–60849. https://doi.org/10.1109/ACCESS.2019.2914509
  80. Zhang, M., Liu, J., Li, N. (2013). Control of stepper motor based on VC++ and PCI-1240. In 2013 IEEE 11th International Conference on Electronic Measurement & Instruments. IEEE. https://doi.org/10.1109/ICEMI.2013.6743226
Language: English
Page range: 284 - 299
Submitted on: May 10, 2025
Accepted on: Sep 4, 2025
Published on: Oct 23, 2025
Published by: Slovak Academy of Sciences, Institute of Measurement Science
In partnership with: Paradigm Publishing Services
Publication frequency: Volume open

© 2025 Guosheng Tian, published by Slovak Academy of Sciences, Institute of Measurement Science
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.