Skip to main content
SpringerLink
Log in

Smooth trajectory generation based on contour error constraint and parameter correction b-spline

  • ORIGINAL ARTICLE
  • Published:
The International Journal of Advanced Manufacturing Technology Aims and scope Submit manuscript

Abstract

In terms of CNC machining task, high feedrate and high accuracy are of great significance. In this paper, an integrated jerk-limited method of minimal time trajectory planning with confined contour error (IJLMTPC) is proposed. Firstly, an optimal control method is utilized to obtain the minimal time feedrate profile. To compensate the contour error, it is added into the optimal control framework as a constraint. Based on the high-order transfer functions of every axis servo model, the contour errors constraint is formulated as the function of tracking errors. However, the optimal control problem (OCP) is difficult to be solved. Hence, the OCP is transformed into two convex subproblems. Since the axial jerk elements are included in the contour error constraint, they are no longer considered as dependent constraints in the two subproblems specially. Afterwards, two subproblems are discretized with control vector parameterization. Consequently, two subproblems are more efficient to be solved with nonlinear programming, and the minimal time feedrate is obtained. Secondly, considering that the OCP is solved on the domain of curve parameter, it is inconvenient for real-time interpolation. The resulting feedrate from solving the OCP is transformed into the corresponding profile on time domain, while the limited tangent jerk is imposed on. Hence, the feedrate is smoothed. Finally, to alleviate the fluctuation of feedrate, the real-time interpolation is performed based on the parameter correction b-spline. Two benchmark tool paths are adopted to test the proposed scheme, and the effectiveness is verified.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Song D, Zhong Y, Ma JW (2020) Look-ahead-window-based interval adaptive feedrate scheduling for long five-axis spline toolpaths under axial drive constraints. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 234(13):1656–1670

    Article  Google Scholar 

  2. Tajima S, Sencer B, Shamoto E (2018) Accurate interpolation of machining tool-paths based on FIR filtering. Precis Eng 52:332–344

    Article  Google Scholar 

  3. Lin M, Lin W, Lin Y, Lee C, Lee J (2019) Smooth Feedrate Planning and Trajectory Generation for Five-axis Tool Path. In: 2019 IEEE 15th International Conference on Control and Automation (ICCA), Edinburgh, UK, vol 2019, pp 405–410

  4. Yang J, Yuen A (2017) An analytical local corner smoothing algorithm for five-axis CNC machining. Int J Mach Tools Manuf 123:22–35

    Article  Google Scholar 

  5. Dong J, Stori JA (2006) A generalized time-optimal bidirectional scan algorithm for constrained feed-rate optimization. J Dynam Syst Measure Cont 128(2):379–390

    Article  Google Scholar 

  6. Bharathi A, Dong J (2016) Feedrate optimization for smooth minimum-time trajectory generation with higher order constraints. Int J Adv Manufact Technol 82(5):1029–1040

    Article  Google Scholar 

  7. Sencer B, Dumanli A, Yamada Y (2018) Spline interpolation with optimal frequency spectrum for vibration avoidance. CIRP Ann 67(1):377–380

    Article  Google Scholar 

  8. Farouki RT, Pelosi F, Sampoli ML (2019) Optimization of corner blending curves. Comput Aided Des 117:102739

    Article  MathSciNet  Google Scholar 

  9. Yang Y, Liao L, Yang H, Li S (2020) An optimal control strategy for multi-UAVs target tracking and cooperative competition. IEEE/CAA J Automatica Sinica 99:1–15

    Google Scholar 

  10. Elbanhawi M, Simic M (2014) Sampling-Based Robot motion planning: a review. IEEE Access 2(1):56–77

    Article  Google Scholar 

  11. Tang X, Chen J (2016) Direct trajectory optimization and costate estimation of infinite-horizon optimal control problems using collocation at the flipped legendre-gauss-radau points. IEEE/CAA J Automatica Sinica 3(2):174–183

    Article  MathSciNet  Google Scholar 

  12. Ji C, Kong M, Li R (2019) Time-Energy Optimal trajectory planning for variable stiffness actuated robot. IEEE Access 7:14366–14377

    Article  Google Scholar 

  13. Ye P, Zhang Y, Xiao J, Zhao M, Zhang H (2019) A novel feedrate planning and interpolating method for parametric toolpath in Frenet-Serret frame. Int J Adv Manufact Technol 101(5-8):1915–1925

    Article  Google Scholar 

  14. Erkorkmaz K, Chen QG, Zhao MY, Beudaert X, Gao XS (2017) Linear programming and windowing based feedrate optimization for spline toolpaths. CIRP Ann 66(1):393–396

    Article  Google Scholar 

  15. Zhang Q, Li SR, Guo JX, Gao XS (2016) Time-optimal path tracking for robots under dynamics constraints based on convex optimization. Robotica 34(9):2116–2139

    Article  Google Scholar 

  16. Zhang Q, Li SR, Guo JX (2014) Minimum time trajectory optimization of CNC machining with tracking error constraints. Abstract Appl Anal 2014(5):835098

    MathSciNet  MATH  Google Scholar 

  17. Saleem O, Rizwan M (2020) ILC-Adapted parameter optimization of cross-coupled single-input fuzzy tracking controllers for an X-Y positioning table. J Chin Inst Eng 43(6):519–531

    Article  Google Scholar 

  18. El Khalick MA, Uchiyama N (2011) Discrete-time model predictive contouring control for biaxial feed drive systems and experimental verification. Mechatronics 21(6):918–926

    Article  Google Scholar 

  19. Kornmaneesang W, Chen SL (2020) MPC-Based robust contouring control for a robotic machining system. Asian J Control 2020:1–13

    Google Scholar 

  20. El Khalick MA, Uchiyama N, Sano S (2013) Sliding mode contouring control design using nonlinear sliding surface for three-dimensional machining. Int J Mach Tools Manuf 65:8–14

    Article  Google Scholar 

  21. Kim H, Okwudire CE (2020) Simultaneous servo error pre-compensation and feedrate optimization with tolerance constraints using linear programming. Int J Adv Manufact Technol 109(2):809–821

    Article  Google Scholar 

  22. Du D, Liu Y, Guo X, Yamazaki K, Fujishima M (2010) An accurate adaptive NURBS curve interpolator with Real-Time flexible Acceleration/Deceleration control. Robot Comput Integr Manuf 26(4):273–281

    Article  Google Scholar 

  23. Tsai MC, Cheng CW (2003) A Real-Time predictor corrector interpolator for CNC machining. J Manufact Sci Eng-Trans ASME 125(3):449–460

    Article  Google Scholar 

  24. Jahanpour J, Alizadeh MR (2015) A novel acc-jerk-limited nurbs interpolation enhanced with an optimized s-shaped quintic feedrate scheduling scheme. Int J Adv Manufact Technol 77(9-12):1889–1905

    Article  Google Scholar 

  25. 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 Manufact Technol 90(9):2751–2759

    Article  Google Scholar 

  26. Lu L, Zhang L, Gu Y, Zhao J (2017) A parametric interpolation method with minimal feedrate fluctuation by nonuniform rational basis spline. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 231(18):3301–3317

    Google Scholar 

  27. 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 

  28. Lei WT, Sung MP, Lin LY, Huang JJ (2007) Fast real-time NURBS path interpolation for CNC machine tools. Int J Mach Tools Manuf 47(10):1530–1541

    Article  Google Scholar 

  29. Zhang Q, Li SR, Guo JX (2012) Smooth time-optimal tool trajectory generation for CNC manufacturing systems. J Manuf Syst 31(3):280–287

    Article  MathSciNet  Google Scholar 

  30. Zhang Q, Li SR (2013) Efficient computation of smooth minimum time trajectory for CNC machining. Int J Adv Manufact Technol 68(1-4):683–692

    Article  Google Scholar 

  31. Zhang Q, Li S (2011) Minimum time feed-rate optimization along predefined toolpath with acceleration constraints on each axis. Proceedings of the 30th Chinese Control Conference, (2011):5462–5467

  32. Zhao K, Kang ZJ, Guo XB (2020) Smooth trajectory generation for predefined path with pseudo spectral method. IEEE Access 8:158735–158744

    Article  Google Scholar 

  33. Zhang Q, Gao XS (2016) Practical feedrate optimization for planar high precision contouring. In: 2016 IEEE International Conference on Information and Automation (ICIA), Ningbo, China, vol 2016, pp 1872–1877

  34. Koren Y, Lo Ch-Ch (1991) Variable-Gain Cross-Coupling Controller for Contouring. CIRP Annals 40(1):371–374

    Article  Google Scholar 

  35. Zhao K, Li SR, Kang ZJ (2019) Smooth minimum time trajectory planning with minimal feed fluctuation. Int J Adv Manufact Technol 105(9-12):1099–1111

    Article  Google Scholar 

  36. Wu JC, Zhou HC, Tang XQ, Chen JH (2012) A NURBS interpolation algorithm with continuous feedrate. Int J Adv Manufact Technol 59(5-8):623–632

    Article  Google Scholar 

  37. Dong J, Wang T, Li B, Ding Y (2014) Smooth feedrate planning for continuous short line tool path with contour error constraint. Int J Mach Tools Manuf 76:1–12

    Article  Google Scholar 

  38. Tsai MS, Nien HW, Yau HT (2008) Development of an integrated look-ahead dynamics-based nurbs interpolator for high precision machinery. Comput Aided Des 40(5):554–566

    Article  Google Scholar 

  39. Yang M, Yang JX, Ding H (2018) A high accuracy on-line estimation algorithm of five-axis contouring errors based on three-point arc approximation. Int J Mach Tools Manuf 130-131:73–84

    Article  Google Scholar 

Download references

Funding

This work is supported by the National Natural Science Foundation of China under Grant 61573378.

Author information

Authors and Affiliations

  1. College of Control Science and Engineering, China University of Petroleum (East China), Qingdao, 266580, China

    Kai Zhao

  2. College of Computer Science and Information Engineering, Anyang Institute of Technology, Anyang, 455000, China

    Kai Zhao

  3. School of Artificial Intelligence, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Shurong Li & Zhe Liu

  4. College of New Energy, China University of Petroleum (East China), Qingdao, 266580, China

    Zhongjian Kang

Authors
  1. Kai Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shurong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhongjian Kang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhe Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shurong Li.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Availability of data and material

The manuscript has no associated data and materials.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, K., Li, S., Kang, Z. et al. Smooth trajectory generation based on contour error constraint and parameter correction b-spline. Int J Adv Manuf Technol 119, 4359–4373 (2022). https://doi.org/10.1007/s00170-021-08367-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00170-021-08367-7

Keywords

Search

Navigation