Skip to main content
SpringerLink
Log in

Cubic spline trajectory generation with axis jerk and tracking error constraints

  • Published:
International Journal of Precision Engineering and Manufacturing Aims and scope Submit manuscript

Abstract

This paper presents a cubic spline trajectory generation algorithm that produces continuous position, velocity, and acceleration profiles for CNC machines with confined axis jerk and tracking error. A series of reference knots are connected using cubic spline functions in time t for constructing axis trajectories. They are generated in such a way that continuity up to the second derivative is preserved along the overall composite curve. For CNC machines governed by a typical PID controller, the tracking error is also constrained as well as the velocity, acceleration, and jerk for each axis. The time intervals between each pair of adjacent knots are scheduled such that the machining time is minimized subject to all the constraints. Experiment results are presented to illustrate the application of the algorithm.

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

Similar content being viewed by others

References

  1. Bobrow, J. E., Dubowsky, S., and Gibson, J. S., “Time-optimal control of robotic manipulators along specified paths,” International Journal of Robotics Research, Vol. 4, No. 3, pp. 3–17, 1985.

    Article  Google Scholar 

  2. Timar, S. D., Farouki, R. T., Smith, T. S., and Boyadjieff, C. L., “Algorithms for time-optimal control of CNC machines along curved tool paths,” Robotics and Computer-Integrated Manufacturing, Vol. 21, No. 1, pp. 37–53, 2005.

    Article  Google Scholar 

  3. Lin, C. S., Chang, P. R., and Luh, J. Y. S., “Formulation and optimization of cubic polynomial joint trajectories for industrial robots,” IEEE Transactions on Automatic Control, Vol. 28, No. 12, pp. 1066–1074, 1983.

    Article  MATH  Google Scholar 

  4. Altintas, Y. and Erkorkmaz, K., “Feedrate optimization for spline interpolation in high speed machine tools,” CIRP Annals-Manufacturing Technology, Vol. 52, No. 1, pp. 297–302, 2003.

    Article  Google Scholar 

  5. Dong, J. Y., Ferreira, P. M., and Stori, J. A., “Feedrate optimization with jerk constraints for generating minimum-time trajectories,” International Journal of Machine Tools and Manufacture, Vol. 47, No. 12–13, pp. 1941–1955, 2007.

    Article  Google Scholar 

  6. Gasparetto, A. and Zanotto, V., “A technique for time-jerk optimal planning of robot trajectories,” Robotics and Computer-Integrated Manufacturing, Vol. 24, No. 3, pp. 415–426, 2008.

    Article  Google Scholar 

  7. Sencer, B., Altintas, Y., and Croft, E., “Feed optimization for fiveaxis CNC machine tools with drive constraints,” International Journal of Machine Tools and Manufacture, Vol. 48, No. 7–8, pp. 733–745, 2008.

    Article  Google Scholar 

  8. Beudaert, X., Lavernhe, S., and Tournier, C., “Feedrate interpolation with axis jerk constraints on 5-axis NURBS and G1 tool path,” International Journal of Machine Tools and Manufacture, Vol. 57, pp. 73–82, 2012.

    Article  Google Scholar 

  9. Zhang, K., Yuan, C. M., Gao, X. S., and Li, H. B., “A greedy algorithm for feedrate planning of CNC machines along curved tool paths with confined jerk,” Robotics and Computer-Integrated Manufacturing, Vol. 28, No. 4, pp. 472–483, 2012.

    Article  Google Scholar 

  10. Fan, W., Gao, X. S., Yan, W., and Yuan, C. M., “Interpolation of parametric CNC machining path under confined jounce,” International Journal of Advanced Manufacturing Technology, Vol. 62, No. 5–8, pp. 719–739, 2012.

    Article  Google Scholar 

  11. Lai, J. Y., Lin, K. Y., Tseng, S. J., and Ueng, W. D., “On the development of a parametric interpolator with confined chord error, feedrate, acceleration and jerk,” International Journal of Advanced Manufacturing Technology, Vol. 37, No. 1–2, pp. 104–121, 2008.

    Article  Google Scholar 

  12. Zhang, K., Yuan, C. M., and Gao, X. S., “Efficient algorithm for time-optimal feedrate planning and smoothing with confined chord error and acceleration,” International Journal of Advanced Manufacturing Technology, Vol. 66, No. 9–12, pp. 1685–1697, 2013.

    Article  Google Scholar 

  13. Renton, D. and Elbestawi, M. A., “High speed servo control of multi-axis machine tools,” International Journal of Machine Tools and Manufacture, Vol. 40, No. 4, pp. 539–559, 2000.

    Article  Google Scholar 

  14. Erkorkmaz, K. and Altintas, Y., “High speed CNC system design. Part III: high speed tracking and contouring control of feed drives,” International Journal of Machine Tools and Manufacture, Vol. 41, No. 11, pp. 1637–1658, 2001.

    Article  Google Scholar 

  15. Ernesto, C. A. and Farouki, R. T., “Solution of inverse dynamics problems for contour error minimization in CNC machines,” International Journal of Advanced Manufacturing Technology, Vol. 49, No. 5–8, pp. 589–604, 2010.

    Article  Google Scholar 

  16. Altintas, Y., “Manufacturing automation: metal cutting mechanics, machine tool vibrations, and CNC design,” Cambridge University Press, Cambridge, 2000.

    Google Scholar 

  17. Pan, Y. R., Shih, Y. T., Horng, R. H., and Lee, A. C., “Advanced parameter identification for a linear-motor-driven motion system using disturbance observer,” Int. J. Precis. Eng. Manuf., Vol. 10, No. 4, pp. 35–47, 2009.

    Article  Google Scholar 

  18. Nocedal, J. and Wright, S. J., “Numerical optimization,” Springer, 2nd Edition, 2006.

    MATH  Google Scholar 

  19. Yang, W. Y., Cao, W. W., Chung, T. S., and Morris, J., “Applied numerical methods using MATLAB,” John Wiley and Sons, 2005.

    Google Scholar 

  20. Yuan, H. L. and Shim, T., “Model based real-time collision-free motion planning for nonholonomic mobile robots in unknown dynamic environments,” Int. J. Precis. Eng. Manuf., Vol. 14, No. 3, pp. 359–365, 2013.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Key Laboratory of Mathematics Mechanization, Institute of Systems Science, Chinese Academy of Sciences, Beijing, China, 100190

    Ke Zhang, Jian-Xin Guo & Xiao-Shan Gao

Authors
  1. Ke Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jian-Xin Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiao-Shan Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiao-Shan Gao.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, K., Guo, JX. & Gao, XS. Cubic spline trajectory generation with axis jerk and tracking error constraints. Int. J. Precis. Eng. Manuf. 14, 1141–1146 (2013). https://doi.org/10.1007/s12541-013-0155-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12541-013-0155-2

Keywords

Search

Navigation