Skip to main content
SpringerLink
Log in

Optimal trajectory generation method to find a smooth robot joint trajectory based on multiquadric radial basis functions

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

Abstract

A new technique to generate smooth motion trajectories for robot manipulators using multiquadric radial basis functions (MQ-RBFs) is presented in this paper. In order to get the optimal trajectory, two objective functions are minimized that are proportional to the execution time, the integral of the squared jerk (which denotes the time derivative of the acceleration) along the whole trajectory. Also, the proposed interpolation technique is introduced for solving the trajectory planning problem in the joint space, where the interpolation of via-points takes into account boundary conditions and also satisfies kinematics limits of velocity, acceleration, and jerk. Then, the proposed approach is compared with a set of classical interpolation techniques based on radial basis function models and cubic splines. Finally, the proposed technique has been tested for the six-joint PUMA 560 manipulator in two cases (minimum time and minimum time-jerk) and results are compared with those proposed of other important trajectory planning techniques. Numerical results show the competent performances of the proposed methodology to generate trajectories in short total transfer time and with high smooth profile.

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

Similar content being viewed by others

Availability of data and material

All data generated or analyzed during this study are included in this published article.

References

  1. Angeles J (2014) Fundamentals of robotic mechanical systems theory methods and algorithms. Springer International Publishing

  2. Gasparetto A, Boscariol P, Lanzutti A, Vidoni R (2015) Path planning and trajectory planning algorithms: a general overview. Motion and operation planning of robotic systems. Springer, pp 3–27

  3. Kahn ME, Roth B (1971) The near-minimum-time control of open-loop articulated kinematic chains. J Dyn Syst Meas Control Trans ASME 164–172

  4. Kim BK, Shin KG (1985) Minimum-time path planning for robot arms and their dynamics. IEEE Trans Syst Man Cybern 213–223

  5. Bobrow JE, Dubowsky S, Gibson JS (1985) Time-optimal control of robotic manipulators along specified paths. Int J Rob Res 4(3):3–17

    Article  Google Scholar 

  6. Shin KG, Mckay ND (1985) Minimum-time control of robotic manipulators with geometric path constraints. IEEE Trans Auto Cont

  7. Constantinescu D, Croft EA (2000) Smooth and time-optimal trajectory planning for industrial manipulators along specified paths. J Robot Syst

  8. Xiao Y, Du Z, Dong W (2012) Smooth and near time‐optimal trajectory planning of industrial robots for online applications. Ind Rob

  9. Bendali N, Ouali M (2018) Optimization the trajectories of robot manipulators along specified task. Int J Intell Eng Sys

  10. Bendali N, Ouali M, Gellal K, (2015) Optimal trajectory planning under kino-dynamics constraints for a 6-DOF PUMA 560. In: Proceedings of the The International Conference on Engineering. Turkey

  11. Abu-Dakka FJ, Valero FJ, Suñer JL, Mata V (2015) A direct approach to solving trajectory planning problems using genetic algorithms with dynamics considerations in complex environments. Robotica 33(3):669–683

    Article  Google Scholar 

  12. Bendali N, Ouali M, Kifouche A (2014) A new method for time-jerk optimal trajectory planning under kino-dynamic constraint of robot manipulators in pick-and-place operations. IAES Int J Robot Autom

  13. Kyriakopoulos KJ, Saridis GN (1988) Minimum jerk path generation

  14. Piazzi A, Visioli A (2000) Global minimum-jerk trajectory planning of robot manipulators. IEEE Tran Indu Elec

  15. Piazzi A, Visioli A (1997) Interval algorithm for minimum-jerk trajectory planning of robot manipulators. In: Proceedings of the IEEE Conference on Decision and Control

  16. Liny HI, Liu YC (2011) Minimum-jerk robot joint trajectory using particle swarm optimization. In: Proceedings - 1st International Conference on Robot, Vision and Signal Processing, RVSP

  17. Lin HI (2014) A fast and unified method to find a minimum-jerk robot joint trajectory using particle swarm optimization. J Intell Robot Syst: Theo Appl

  18. Simon D, Isik C (1993) A trigonometric trajectory generator for robotic arms. Int J Control

  19. Simon D (2004) Neural networks for optimal robot trajectory planning, In: Handbook of Neural Computation

  20. Simon D (1993) The application of neural networks to optimal robot trajectory planning. Rob Auton Syst

  21. Lu S, Zhao J, Jiang L, Liu H (2017) Solving the time-jerk optimal trajectory planning problem of a robot using augmented lagrange constrained particle swarm optimization. Math Probl Eng

  22. Lu S, Zhao J, Jiang L, Liu H (2017) Time-jerk optimal trajectory planning of a 7-DOF redundant robot. Turk J Electric Eng Comput Sci

  23. Jiang L, Lu S, Gu Y, Zhao J (2017) Time-jerk optimal trajectory planning for a 7-DOF redundant robot using the sequential quadratic programming method. In: Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

  24. Rout A, Dileep M, Mohanta GB, Deepak BBVL, Biswal BB (2018) Optimal time-jerk trajectory planning of 6 axis welding robot using TLBO method. Procedia Comput Sci 133:537–544

    Article  Google Scholar 

  25. Cong M, Xu X, Xu P (2010) Time-jerk synthetic optimal trajectory planning of robot based on fuzzy genetic algorithm. I J Inte Syst Tech Appl

  26. Gasparetto A, Lanzutti A, Vidoni R, Zanotto V (2012) Experimental validation and comparative analysis of optimal time-jerk algorithms for trajectory planning. Robot Comput Integr Manuf 28(2):164–181

    Article  Google Scholar 

  27. Gasparetto A, Zanotto V (2007) A new method for smooth trajectory planning of robot manipulators. Mech Mach Theory

  28. Lombai F, Szederkenyi G (2008) Trajectory tracking control of a 6-degreeof-freedom robot arm using nonlinear optimization. In: International Workshop on Advanced Motion Control AMC

  29. Lombai F, Szederkenyi G (2009) Throwing motion generation using nonlinear optimization on a 6-degree-of-freedom robot manipulator. In: IEEE 2009 International Conference on Mechatronics ICM

  30. Lin CS, Chang PR, Luh JYS (1983) Formulation and optimization of cubic polynomial joint trajectories for industrial robots. IEEE Trans Automat Contr

  31. Thompson SE, Patel RV (1987) Formulation of joint trajectories for industrial robots using B-splines. IEEE Trans Ind Electron

  32. Liu H, Lai X, Wu W (2013) Time-optimal and jerk-continuous trajectory planning for robot manipulators with kinematic constraints. Robot Comput Integr Manuf

  33. Ramabalan S, Saravanan R, Balamurugan C (2009) Multi-objective dynamic optimal trajectory planning of robot manipulators in the presence of obstacles. Int J Adv Manuf Technol

  34. Gasparetto A, Zanotto V (2008) A technique for time-jerk optimal planning of robot trajectories. Robot Comput Integr Manuf

  35. Huang J, Hu P, Wu K, Zeng M (2018) Optimal time-jerk trajectory planning for industrial robots. Mech Mach Theory

  36. Xu Z, Wei S, Wang N, Zhang X (2014) Trajectory planning with Bezier curve in Cartesian space for industrial gluing robot. In: Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

  37. Fang S, Ma X, Zhao Y, Zhang Q, Li Y (2019) Trajectory planning for Seven-DOF robotic arm based on quintic polynormial. In: Proceedings - 2019 11th International Conference on Intelligent Human-Machine Systems and Cybernetics, IHMSC 2019

  38. Xie Z, Wu P, Ren P (2016) A comparative study on the pick-andplace trajectories for a delta robot. In: Proceedings of the ASME Design Engineering Technical Conference

  39. Petrinec K, Kovacic Z (2007) Trajectory planning algorithm based on the continuity of jerk. In: Mediterranean Conference on Control and Automation

  40. Kucuk S (2017) Optimal trajectory generation algorithm for serial and parallel manipulators. Robot Comput Integr Manuf

  41. Liu H, Lai X, Wu W (2013) Time-optimal and jerk-continuous trajectory planning for robot manipulators with kinematic constraints. Robot Comput Integr Manuf 29(2):309–317

    Article  Google Scholar 

  42. Buhmann MD, Levesley J (2004) Radial basis functions: theory and implementations. Math Comput

  43. Chettibi T (2019) Smooth point-to-point trajectory planning for robot manipulators by using radial basis functions. Robotica

  44. Mirinejad H, Inanc T (2016) An RBF collocation method for solving optimal control problems. Rob Auton Syst

  45. Mirinejad H, Inanc T, Zurada JM (2021) Radial basis function interpolation and Galerkin projection for direct trajectory optimization and costate estimation. IEEE/CAA J Autom Sinica 8(8):1380–1388

    Article  MathSciNet  Google Scholar 

  46. Zlajpah L, Petric T (2020) Generation of smooth Cartesian paths using radial basis functions. In: In International Conference on Robotics in Alpe-Adria Danube Region

  47. Mishra PK, Nath SK, Sen MK, Fasshauer GE (2018) Hybrid Gaussian cubic radial basis functions for scattered data interpolation. Comput Geosci

  48. Hardy RL (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res

  49. Hardy RL (1990) Theory and applications of the multiquadric-biharmonic method. Comput Math App

  50. Franke R (1982) Scattered data interpolation: Tests of some methods. Math Comput 38(157):181–200

    MathSciNet  MATH  Google Scholar 

  51. Mirinejad H, Inanc T (2015) A radial basis function method for direct trajectory optimization. In: Proceedings of the American Control Conference

  52. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2007) Numerical recipes 3rd Edition: the Art of Scientific Computing

  53. Abu-Dakka FJ, Assad IF, Alkhdour RM, Abderahim M (2017) Statistical evaluation of an evolutionary algorithm for minimum time trajectory planning problem for industrial robots. Int J Adv Manuf Technol 89(1–4):389–406

    Article  Google Scholar 

  54. Wang CH, Horng JG (1990) Constrained minimum-time path planning for robot manipulators via virtual knots of the cubic B-spline functions. IEEE Trans Automat Contr

Download references

Funding

This work was supported by “La Direction Générale de la Recherche Scientifique et du Développement Technologique (DGRSDT)” of Algeria to whom we are grateful.

Author information

Authors and Affiliations

  1. Faculty of Sciences and Technology, University of Khemis Miliana, Road of Theniet El Had, Khemis Miliana, 44225, Algeria

    Bendali Nadir & Said Abderrezak

  2. Structural Mechanics Research Laboratory, Department of Mechanical Engineering, University Blida-1, Blida, Algeria

    Bendali Nadir & Ouali Mohammed

  3. Hung Yen University of Technology and Education, Hung Yen, Vietnam

    Nguyen Minh-Tuan

Authors
  1. Bendali Nadir
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ouali Mohammed
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nguyen Minh-Tuan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Said Abderrezak
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Bendali Nadir: performed the algorithm design, code writing, and was a major contributor in writing the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Bendali Nadir.

Ethics declarations

Ethics approval

Not applicable.

Consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.

Additional information

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

Nadir, B., Mohammed, O., Minh-Tuan, N. et al. Optimal trajectory generation method to find a smooth robot joint trajectory based on multiquadric radial basis functions. Int J Adv Manuf Technol 120, 297–312 (2022). https://doi.org/10.1007/s00170-022-08696-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00170-022-08696-1

Keywords

Search

Navigation