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A NURBS curve interpolator with small feedrate fluctuation based on arc length prediction and correction

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Abstract

Non-uniform rational B-spline (NURBS) curve has been widely used in manufacturing systems. A good interpolator can help the system to improve the contour accuracy and get smooth dynamics performance, but it is hard to get a balance between the interpolation performance and computational load. As the derivative and curvature of NURBS curves used in manufacturing systems are high-order continuous, it is possible to predict a desired interpolation arc length based on the relationship of historical feed chord length and its corresponding arc length in one interpolation cycle. Therefore, this paper proposes a novel interpolation method, which consists three stages. Firstly, a NURBS curve is split into several high-order continuous segments based on its degrees and control points. Secondly, a prediction model based on Newton’s divided differences interpolation equation is derived from the relationship of interpolated chord length and its corresponding arc length, so that the target arc length of the next interpolation cycle can be predicted. Finally, target parameter u of every interpolation cycle is calculated with Taylor’s expansion, whose values are corrected by iteration, and then the position of the target interpolation point can be achieved. Performance of the proposed algorithm is tested and compared with other methods, and the simulation results show the proposed method can achieve smaller velocity fluctuation with low computational load.

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References

  1. Shamsuddin SM, Ahmed M, Smian Y (2006) NURBS skinning surface for ship hull design based on new parameterization method. Int J Adv Manuf Technol 28(9-10):936–941

    Article  Google Scholar 

  2. Lei W, Sung M, Lin L, Huang J (2007) Fast real-time NURBS path interpolation for CNC machine tools. Int J Mach Tools Manuf 47(10):1530–1541

    Article  Google Scholar 

  3. Koren Y, Lo C, Shpitalni M (1993) CNC interpolators: algorithms and analysis, vol 64. Asme Prod Eng Div Publ Ped, Asme, New York, pp 83–92

    Google Scholar 

  4. Chen M, Sun Y (2019) Contour error–bounded parametric interpolator with minimum feedrate fluctuation for five-axis CNC machine tools. Int J Adv Manuf Technol 103(1-4):567–584

    Article  Google Scholar 

  5. Zhao K, Li S, Kang Z (2019) Smooth minimum time trajectory planning with minimal feed fluctuation. Int J Adv Manuf Technol 105(9-12):1099–1111

  6. Cheng CW, Tsai MC (2004) Real-time variable feed rate NURBS curve interpolator for CNC machining. Int J Adv Manuf Technol 23(11-12):865–873

    Article  Google Scholar 

  7. Lin MT, Tsai MS, Yau HT (2007) Development of a dynamics-based NURBS interpolator with real-time look-ahead algorithm. Int J Mach Tool Manu 47(15):2246–2262

    Article  Google Scholar 

  8. Baek D-K, Ko T-J, Yang S-H (2012) Fast and precision NURBS interpolator for CNC systems. Int J Precis Eng Manuf 13(6):955–961

    Article  Google Scholar 

  9. Jia ZY, Song DN, Ma JW, Hu GQ, Su WW (2017) A NURBS interpolator with constant speed at feedrate-sensitive regions under drive and contour-error constraints. Int J Mach Tool Manu 116:1–17

    Article  Google Scholar 

  10. Zhang X-t (2012) An iterative feedrate optimization method for real-time NURBS interpolator. Int J Adv Manuf Technol 62(9-12):1273–1280

    Article  Google Scholar 

  11. Heng M, Erkorkmaz K (2010) Design of a NURBS interpolator with minimal feed fluctuation and continuous feed modulation capability. Int J Mach Tools Manuf 50(3):281–293

    Article  Google Scholar 

  12. Liu H, Liu Q, Zhou S, Li C, Yuan S (2015) A NURBS interpolation method with minimal feedrate fluctuation for CNC machine tools. Int J Adv Manuf Technol 78(5-8):1241–1250

    Article  Google Scholar 

  13. Lo CC (1997) Feedback Interpolators for CNC Machine Tools. J Manuf Sci Eng 119(4A):587–592

    Article  Google Scholar 

  14. Tsai M-C, Cheng C-W (2003) A Real-Time Predictor-Corrector Interpolator for CNC Machining. J Manuf Sci Eng 125(3):449–460

    Article  Google Scholar 

  15. Zhao H, Zhu LM, Han D (2013) A parametric interpolator with minimal feed fluctuation for CNC machine tools using arc-length compensation and feedback correction. Int J Mach Tool Manu 75:12):1–12):8

    Article  Google Scholar 

  16. 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. Int J Precis Eng Manuf 21(2):273–290

    Article  Google Scholar 

  17. Liu H, Liu Q, Sun P, Liu Q, Yuan S (2017) A polynomial equation-based interpolation method of NURBS tool path with minimal feed fluctuation for high-quality machining. Int J Adv Manuf Technol 90(9):2751–2759

    Article  Google Scholar 

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Funding

This research is supported by National Science and Technology Major Project of China (Grant No. 2019ZX04004001), Natural Science Foundation of Shandong Province (Grant No. ZR2019QEE042), The Project of Innovative Application Experiencing Center of Industrial Internet Platform.

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Authors and Affiliations

  1. School of Mechanical and Electronic Engineering, Shandong Jianzhu University, Jinan, 250101, China

    Shuai Ji

  2. School of Mechanical Engineering, Shandong University, Jinan, 250061, China

    Tianliang Hu & Chengrui Zhang

  3. China National Machine Tool Quality Supervision Testing Center, Beijing, 100102, China

    Zuguang Huang

Authors
  1. Shuai Ji
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  2. Tianliang Hu
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  3. Zuguang Huang
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  4. Chengrui Zhang
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Correspondence to Tianliang Hu.

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Appendix A Parameters of the Butterfly shape NURBS curve

Appendix A Parameters of the Butterfly shape NURBS curve

Control point (mm):

(54.493, 52.139), (55.507, 52.139), (56.082,49.615), (56.780, 44.971), (69.575, 51.358), (77.786, 58.573),(90.526, 67.081), (105.973, 63.801), (100.400, 47.326), (94.567,39.913), (92.369, 30.485), (83.440, 33.757), (91.892, 28.509),(89.444, 20.393), (83.218, 15.446), (87.621, 4.830), (80.945,9.267), (79.834, 14.535), (76.074, 8.522), (70.183, 12.550), (64.171,16.865), (59.993, 22.122), (55.680, 36.359), (56.925, 24.995),(59.765, 19.828), (54.493, 14.940), (49.220, 19.828), (52.060,24.994), (53.305, 36.359), (48.992, 22.122), (44.814, 16.865),(38.802, 12.551), (32.911, 8.521), (29.152, 14.535), (28.040, 9.267),(21.364, 4.830), (25.768, 15.447), (19.539, 20.391), (17.097, 28.512),(25.537, 33.750), (16.602, 30.496), (14.199, 39.803), (8.668, 47.408), (3.000, 63.794), (18.465, 67.084), (31.197, 58.572), (39.411, 51.358), (52.204, 44.971), (52.904, 49.614), (53.478, 52.139), (54.492,52.139).

Knot vector:

[0, 0, 0, 0, 0.0083, 0.0150, 0.0361, 0.0855, 0.1293, 0.1509, 0.1931, 0.2273, 0.2435, 0.2561, 0.2692, 0.2889, 0.3170, 0.3316, 0.3482, 0.3553, 0.3649, 0.3837, 0.4005, 0.4269, 0.4510, 0.4660, 0.4891, 0.5000, 0.5109, 0.5340, 0.5489, 0.5731, 0.5994, 0.6163, 0.6351, 0.6447, 0.6518, 0.6683, 0.6830, 0.7111, 0.7307, 0.7439, 0.7565, 0.7729, 0.8069, 0.8491, 0.8707, 0.9145, 0.9639, 0.9850, 0.9917, 1, 1, 1, 1]

Weight vector:

[1.0000, 1.0000, 1.0000, 1.2000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 2.0000, 1.0000, 1.0000, 5.0000, 3.0000, 1.0000, 1.1000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.1000, 1.0000, 3.0000, 5.0000, 1.0000, 1.0000, 2.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.2000, 1.0000, 1.0000, 1.0000]

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Ji, S., Hu, T., Huang, Z. et al. A NURBS curve interpolator with small feedrate fluctuation based on arc length prediction and correction. Int J Adv Manuf Technol 111, 2095–2104 (2020). https://doi.org/10.1007/s00170-020-06258-x

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  • DOI: https://doi.org/10.1007/s00170-020-06258-x

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