A Semi-Infinte Optimization Approach to Optimal Spline Trajectory Planning of Mechanical Manipulators

  • Corrado Guarino Lo Bianco
  • Aurelio Piazzi
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 57)


The paper deals with the problem of optimal trajectory planning for rigid links industrial manipulators. According with actual industrial requirements, a technique for planning minimum-time spline trajectories under dynamics and kinematics constraints is proposed. More precisely, the evaluated trajectories, parametrized by means of cubic splines, have to satisfy joint torques and end-effector Cartesian velocities within given bounds. The problem solution is obtained by means of an hybrid genetic/interval algorithm for semi-infinite optimization. This algorithm provides an estimated global minimizer whose feasibility is guaranteed by the use of a deterministic interval procedure; i.e., a routine based on concepts of interval analysis. The proposed approach is tested by planning a 10 via points movement for a two link manipulator.


Interval Procedure Joint Torque Interval Analysis Trajectory Planning Redundant Manipulator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R.E. Bellman and S.E. Dreyfus. Applied Dynamic Programmming, Priceton University Press, 1962.Google Scholar
  2. [2]
    J.E. Bobrow, S. Dubowsky, and J.S. Gibson. Time-optimal control of robotic manipulators along specified paths, International Journal of Robotics Research, 4:554–561, 1985.Google Scholar
  3. [3]
    J.J. Craig. Introduction to Robotics: Mechanics and Control, AddisonWesley, 1989.MATHGoogle Scholar
  4. [4]
    A. De Luca, L. Lanari, and G. Oriolo. A sensitivity approach to optimal spline robot trajectories, Automatica, 27:535–539, 1991.CrossRefGoogle Scholar
  5. [5]
    D.E. Goldberg. Genetic Algorithms in Search, Optimization, and Machine Learning, Addison–Wesley, 1989.MATHGoogle Scholar
  6. [6]
    C. Guarino Lo Bianco and A. Piazzi. A hybrid genetic/interval algorithm for semi-infinite optimization. In Proceedings of the 35th Conference on Decision and Control, pages 2136–2138, Kobe, Japan, 1996.CrossRefGoogle Scholar
  7. [7]
    C. Guarino Lo Bianco and A. Piazzi. Mixed H2/H∞ fixed-structure control via semi-infinite optimization. In L. Boullart, editor, In Proceedings of the 7th IFAC Symposium on Computer Aided Control Systems Design, pages 329–334, Pergamon, 1997.Google Scholar
  8. [8]
    C. Guarino Lo Bianco and A. Piazzi. A worst-case approach to SISO mixed H2 /H control. In Proceedings of the 1998 IEEE International Conference on Control Applications, pages 684–688, Trieste, Italy, 1998.Google Scholar
  9. [9]
    C. Guarino Lo Bianco and A. Piazzi. A global optimization approach to scalar H2/H control, European Journal of Control, 6:358–367, 2000.MathSciNetCrossRefGoogle Scholar
  10. [10]
    C. Guarino Lo Bianco and A. Piazzi. A hybrid algorithm for infinitely constrained optimization, International Journal of Systems Science, 32:91–102, 2001.MATHGoogle Scholar
  11. [11]
    E. Haaren-Retagne. A Semmi-Infinite Programming Algorithm for Robotic Trajectory Planning, Phd thesis, Universität Trier, Trier, Germany, 1992.Google Scholar
  12. [12]
    E. Hansen. Global Optimization Using Interval Analysis, Marcel Dekker, 1992.MATHGoogle Scholar
  13. [13]
    R. Hettich and K.O. Kortanek. Semi-infinite programming: theory, methods, and applications, SIAM Review, 35:380–429, 1993.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    B. Hu, K.L. Teo, and H. P. Lee. Local optimization of weighted joint torques for redundant robotic manipulators, IEEE Transactions on Robotics and Automation, 11:422–425, 1995.CrossRefGoogle Scholar
  15. [15]
    L.S. Jennings and K.L. Teo. A computational algorithm for functional inequality constrained optimization problems, Automatica, 26:371–375, 1990.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    O. Knüppel. PROFIL programmer’s runtime optimized fast interval library, Technical Report 93.4, Technische Universität Hamburg-Harburg (Germany), 1993.Google Scholar
  17. [17]
    D. Li, A. A. Goldenberg, and J. W. Zu. A new method of peak torque reduction with redundant manipulators, IEEE Transactions on Robotics and Automation, 13:845–853, 1997.CrossRefGoogle Scholar
  18. [18]
    C.-S. Lin, P.-R. Chang, and J.Y.S. Luh. Formulation and optimization of cubic polynomial joint trajectories for industrial robots, IEEE Transactions Automatic Control, AC-28:1066–1074, 1983.Google Scholar
  19. [19]
    D.G. Luenberger. Linear and Nonlinear Programming (2nd ed.). Addison—Wesley, 1989.Google Scholar
  20. [20]
    S.P. Marin. Optimal parametrization of curves for robotic trajectory design, IEEE Transactions on Automatic Control, AC-33:209–214, 1988.MATHCrossRefGoogle Scholar
  21. [21]
    ] R. Menozzi, A. Piazzi, and F. Contini. Small-signal modeling for microwave FET linear circuits based on a genetic algorithm, IEEE Transactions on Circuits and Systems, Part I: Fundamental Theory and Applications, 43:839–847, 1996.CrossRefGoogle Scholar
  22. [22]
    M. Mitchell. An Introduction to Genetic Algorithms, MIT Press, Cambridge, MA, 1996.Google Scholar
  23. [23]
    R.E. Moore. Interval Analysis, Prentice-Hall, Englewood Cliffs, 1966.MATHGoogle Scholar
  24. [24]
    R.E. Moore. Methods and Applications of Interval Analysis, SIAM Press, Philadelphia, 1979.MATHCrossRefGoogle Scholar
  25. [25]
    F. Pfeiffer and R. Johanni. A concept for manipulator trajectory planning, IEEE Transactions on Robotics and Automation, RA-3:115–123, 1987.CrossRefGoogle Scholar
  26. [26]
    A. Piazzi and G. Marro. Robust stability using interval analysis, International Journal of Systems Science, 27:1381–1390, 1996.MATHCrossRefGoogle Scholar
  27. [27]
    A. Piazzi and A. Visioli. Global minimum-time trajectory planning of mechanical manipulators using interval analysis, International Journal of Control, 71:631–652, 1998.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    R. P. Podhorodeski, A. A. Goldenberg, and R. G. Fenton. Resolving redundant manipulator joint rates and identifying special arm configurations using jacobian null-space bases, IEEE Transactions on Robotics and Automation, 7:607–618, 1991.CrossRefGoogle Scholar
  29. [29]
    E. Polak. Optimization: Algorithms and Consistent Approximations. Springer-Verlag, 1997.MATHGoogle Scholar
  30. [30]
    H. Ratschek and J. Rokne. New Computer Methods for Global Optimization, Ellis Horwood Limited, Chichester, UK, 1988.MATHGoogle Scholar
  31. [31]
    R. Reemtsen and S. Görner. Numerical methods for semi-infinite programming: a survey. In R. Reemtsen and J.-J. Rückmann, editors, Semi-Infinite Programming, pages 195–275, Kluwer, 1998.CrossRefGoogle Scholar
  32. [32]
    M. Schoenauer and S. Xanthakis. Constrained GA optimization. In Proceedings of the Fifth International Conference on Genetic Algorithms, pages 573–580, Urbana-Champaign, USA, 1993.Google Scholar
  33. [33]
    L. Sciavicco and B. Siciliano. Modelling and Control of Robot Manipulators, Advanced Textbooks in Control and Signal Processing, SpringerVerlag, 2000.MATHCrossRefGoogle Scholar
  34. [34]
    Z. Shiller and H.-H. Lu. Computation of path constrained time optimal motions with dynamic singularities, Journal of Dynamic Systems, Measurment and Control, 114:34–40, 1992.MATHCrossRefGoogle Scholar
  35. [35]
    K. G. Shin and N. D. McKay. Minimum-time control of robotic manipulators with geometric path constriants, IEEE Transactions on Automatic Control, AC-30:531–541, 1985.MATHCrossRefGoogle Scholar
  36. [36]
    K.L. Teo, C.J. Goh, and K.H. Wong. A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, Harlow, UK, 1991.MATHGoogle Scholar
  37. [37]
    K.L. Teo, V. Rehbock, and L.S. Jennings. A new computational algorithm for functional inequality constrained optimization problems, Automatica, 29:789–792, 1993.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Corrado Guarino Lo Bianco
    • 1
  • Aurelio Piazzi
    • 1
  1. 1.Dipartimento di Ingegneria dell’InformazioneUniversità di ParmaParmaItaly

Personalised recommendations