Multibody System Dynamics

, Volume 47, Issue 4, pp 347–362 | Cite as

A geometric optimization method for the trajectory planning of flexible manipulators

  • Arthur LismondeEmail author
  • Valentin Sonneville
  • Olivier Brüls


Lightweight and flexible robots offer an interesting answer to industrial needs for safety and efficiency. The control of such systems should be able to deal properly with the flexible behavior in the links and the joints. In this paper, a feedforward control action is computed by solving the inverse dynamics of the system. Flexibility in the system is modeled using finite elements formulated in the local frame. The inverse problem is then solved using a constrained optimization formulation. This local frame representation reduces the nonlinearity in the equations of motion and improves the convergence of the numerical scheme. To illustrate the method, numerical examples of a serial and a parallel 3D robot are shown.


Inverse dynamics Optimization Flexible robot Trajectory tracking 



The first author would like to acknowledge the Belgian Fund for Research training in Industry and Agriculture for its financial support (FRIA grant).


  1. 1.
    Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008) CrossRefGoogle Scholar
  2. 2.
    Bastos, G.J.: Contribution to the inverse dynamics of flexible manipulators. Ph.D. thesis, University of Liège (2013) Google Scholar
  3. 3.
    Bastos, G.J., Seifried, R., Brüls, O.: Inverse dynamics of serial and parallel underactuated multibody systems using a DAE optimal control approach. Multibody Syst. Dyn. 30, 359–376 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bastos, G.J., Seifried, R., Brüls, O.: Analysis of stable model inversion methods for constrained underactuated mechanical systems. Mech. Mach. Theory 111, 99–117 (2017) CrossRefGoogle Scholar
  5. 5.
    Bauchau, O.: Flexible Multibody Dynamics. Springer, Berlin (2011) CrossRefGoogle Scholar
  6. 6.
    Betts, J.T.: Practical Method for Optimal Control and Estimation Using Nonlinear Programming. Advances in Design and Control. SIAM, Philadelphia (2010) CrossRefGoogle Scholar
  7. 7.
    Blajer, W., Kolodziejczyk, K.: A geometric approach to solving problems of control constraints: theory and a DAE framework. Multibody Syst. Dyn. 11, 343–364 (2004) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Book, W.J.: Recursive Lagrangian dynamics of flexible manipulator arms. Int. J. Robot. Res. 3(3), 87–101 (1984) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bottasso, C.L., Croce, A.: Optimal control of multibody systems using an energy preserving direct transcription method. Multibody Syst. Dyn. 12, 17–45 (2004) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bottasso, C.L., Croce, A., Ghezzi, L., Faure, P.: On the solution of inverse dynamics and trajectory optimization problems for multibody systems. Multibody Syst. Dyn. 11, 1–22 (2004) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Brüls, O., Arnold, M., Cardona, A.: Two Lie group formulations for dynamic multibody systems with large rotations. In: Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Washington, DC, USA (2011) Google Scholar
  12. 12.
    Brüls, O., Bastos, G.J., Seifried, R.: A stable inversion method for feedforward control of constrained flexible multibody systems. J. Comput. Nonlinear Dyn. 9, 011014 (2014). CrossRefGoogle Scholar
  13. 13.
    Brüls, O., Cardona, A., Arnold, M.: Lie group generalized-\(\alpha \) time integration of constrained flexible multibody systems. Mech. Mach. Theory 48, 121–137 (2012) CrossRefGoogle Scholar
  14. 14.
    Cannon, R., Schmitz, E.: Initial experiments on the end-point control of a flexible one-link robot. Int. J. Robot. Res. 3(3), 62–75 (1984) CrossRefGoogle Scholar
  15. 15.
    De Luca, A.: Feedforward/feedback laws for the control of flexible robots. In: Proceedings of the IEEE International Conference on Robotics & Automation (2000) Google Scholar
  16. 16.
    Devasia, S., Chen, D., Paden, B.: Nonlinear inversion-based output tracking. IEEE Trans. Autom. Control 41(7), 930–942 (1996) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Franke, R., Malzahn, J., Nierobisch, T., Hoffmann, F., Bertram, T.: Vibration control of a multi-link flexible robot arm with fiber-Bragg-grating sensors. In: Proceedings of IEEE International Conference on Robotics and Automation (2009) Google Scholar
  18. 18.
    Geradin, M., Cardona, A.: Flexible Multibody Dynamics: a Finite Element Approach. Wiley, New York (2001) Google Scholar
  19. 19.
    Kwon, D.S., Book, W.J.: A time-domain inverse dynamic tracking control of a single link flexible manipulator. J. Dyn. Syst. Meas. Control 116, 193–200 (1994) CrossRefGoogle Scholar
  20. 20.
    Lismonde, A., Sonneville, V., Brüls, O.: Trajectory planning of soft link robots with improved intrinsic safety. In: Proceedings of the 20th World Congress of the International Federation of Automatic Control (2017) Google Scholar
  21. 21.
    Lynch, K.M., Park, F.C.: Modern Robotics: Mechanics, Planning, and Control. Cambridge University Press, Cambridge (2017) Google Scholar
  22. 22.
    Malzahn, J., Ruderman, M., Phung, A.S., Hoffmann, F., Bertram, T.: Input shaping and strain gauge feedback vibration control of an elastic robotic arm. In: Proceedings of IEEE Conference on Control and Fault Tolerant Systems (2010) Google Scholar
  23. 23.
    Manara, S., Gabiccini, M., Artoni, A., Diehl, M.: On the integration of singularity-free representations of SO(3) for direct optimal control. Nonlinear Dyn. 90(2), 1223–1241 (2017) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Martins, J., Botto, M.A., Costa, J.S.D.: Modeling of flexible beams for robotic manipulators. Multibody Syst. Dyn. 7, 79–100 (2002) CrossRefGoogle Scholar
  25. 25.
    Moberg, S.: Modeling and control of flexible manipulators. Ph.D. thesis, Linköping University (2010) Google Scholar
  26. 26.
    Moberg, S., Hanssen, S.: Inverse dynamics of flexible manipulators. In: Proceedings of the Multibody Dynamics, ECCOMAS Thematic Conference (2009) Google Scholar
  27. 27.
    Murray, R.M., Li, Z., Sastry, S.S.: A Mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton (1994) zbMATHGoogle Scholar
  28. 28.
    Seifried, R.: Dynamics of Underactuated Multibody Systems: Modeling, Control and Optimal Design. Solid Mechanics and Its Applications. Springer, Berlin (2014) CrossRefGoogle Scholar
  29. 29.
    Seifried, R., Eberhard, P.: Design of feed-forward control for underactuated multibody systems with kinematic redundancy. In: Motion and Vibration Control: Selected Papers from MOVIC 2008 (2009) Google Scholar
  30. 30.
    Singer, N., Seering, W.P.: Preshaping command inputs to reduce system vibration. J. Dyn. Syst. Meas. Control 112, 76–82 (1990). CrossRefGoogle Scholar
  31. 31.
    Solis, J.F.P., Navarro, G.S., Linares, R.C.: Modeling and tip position control of a flexible link robot: experimental results. Comput. Sist. 12(4), 421–435 (2009) Google Scholar
  32. 32.
    Sonneville, V.: A geometric local frame approach for flexible multibody systems. Ph.D. thesis, University of Liège (2015) Google Scholar
  33. 33.
    Sonneville, V., Brüls, O.: A formulation on the special Euclidean group for dynamic analysis of multibody systems. J. Comput. Nonlinear Dyn. 9, 041002 (2014). CrossRefzbMATHGoogle Scholar
  34. 34.
    Sonneville, V., Cardona, A., Brüls, O.: Geometrically exact beam finite element formulated on the special Euclidean group SE(3). Comput. Methods Appl. Mech. Eng. 268, 451–474 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Staufer, P., Gattringer, H.: Passivity-based tracking control of a flexible link robot. In: Multibody System Dynamics, Robotic and Control, pp. 95–112. Springer, Vienna (2013) CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Aerospace and Mechanical EngineeringUniversity of LiègeLiègeBelgium
  2. 2.Aerospace EngineeringUniversity of MarylandCollege ParkUSA

Personalised recommendations