Nonlinear Dynamics

, Volume 49, Issue 1–2, pp 217–232 | Cite as

The bouncing motion appearing in a robotic system with unilateral constraint

Original Article


The hopping or bouncing motion can be observed when robotic manipulators are sliding on a rough surface. Making clear the reason of generating such phenomenon is important for the control and dynamical analysis for mechanical systems. In particular, such phenomenon may be related to the problem of Painlevé paradox. By using LCP theory, a general criterion for identifying the bouncing motion appearing in a planar multibody system subject to single unilateral constraint is established, and found its application to a two-link robotic manipulator that comes in contact with a rough constantly moving belt. The admissible set in state space that can assure the manipulator keeping contact with the rough surface is investigated, and found which is influenced by the value of the friction coefficient and the configuration of the system. Painlevé paradox can cause either multiple solutions or non-existence of solutions in calculating contact force. Developing some methods to fill in the flaw is also important for perfecting the theory of rigid-body dynamics. The properties of the tangential impact relating to the inconsistent case of Painlevé paradox have been discovered in this paper, and a jump rule for determining the post-states after the tangential impact finishes is developed. Finally, the comprehensively numerical simulation for the two-link robotic manipulator is carried out, and its dynamical behaviors such as stick-slip, the bouncing motion due to the tangential impact at contact point or the external forces, are exhibited.


Dynamical simulation Friction Painlevé paradox Stick-slip Tangential impact 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Painlevé, P.: Sur les lois du frottement de glissement. Comptes Rendu Séances ĺAcad. Sci. 121, 112–115 (1895)Google Scholar
  2. 2.
    Klein, F.: Zu Painleves kritik der coulombschen reibungsgesetze. Z. Math. Phys. 58, 186–191 (1909)Google Scholar
  3. 3.
    Delassus, E.: Considérations sur le frottement de glissement. Nouv. Ann. Math. (4èmé série) 20, 485–496 (1920)Google Scholar
  4. 4.
    Delassus, E.: Sur les lois du frottement de glissement. Bull. Soc. Math. France 51, 22–33 (1923)Google Scholar
  5. 5.
    Lötstedt, P.: Coulomb friction in two-dimensional rigid-body systems. Z. Angew. Math. Mech. 61, 605–615 (1981)Google Scholar
  6. 6.
    Lötstedt, P.: Mechanical systems of rigid bodies subject to unilateral constraints. ASME J. Appl. Math. 42, 281–296 (1982)Google Scholar
  7. 7.
    Erdmann, M.: On a representation of friction in conguration space. Int. J. Robotics Res. 13(3), 240–271 (1994)Google Scholar
  8. 8.
    Moreau, J.J.: Unilateral Contact and Dry Friction in Finite Freedom Dynamics. Nonsmooth Mechanics and Applications, pp. 1–82. Springer, Berlin Heidelberg New York (1988)Google Scholar
  9. 9.
    Wang, Y., Mason, M.T.: Two-dimensional rigid-body collisions with friction. J. Appl. Mech. 59, 635–642 (1992)Google Scholar
  10. 10.
    Baraff, D.: Coping with friction for non-penetrating rigid body simulation. Comput. Graph. 25(4), 31–40 (1991)Google Scholar
  11. 11.
    Glocker, C., Pheiffer, F.: Multibody Dynamics with Unilateral Contacts. Wiley, New York (1996)Google Scholar
  12. 12.
    Payr, M., Glocker, C.: Oblique frictional impact of a bar: analysis and comparison of different impact laws. Nonlinear Dyn. 41, 361–383 (2005)Google Scholar
  13. 13.
    Brogliato, B.: Nonsmooth Mechanics, 2nd edn. Springer, Berlin Heidelberg New York (1999)Google Scholar
  14. 14.
    Leine, R.I., Brogliato, B., Nijmeijer, H.: Periodic motion and bifurcations induced by the Painlevé paradox. Eur. J. Mech. A Solids 21, 869–896 (2002)Google Scholar
  15. 15.
    Génot, F., Brogliato, B.: New results on Painlevé paradoxes. Eur. J. Mech. A Solids 18, 653–677 (1999)Google Scholar
  16. 16.
    Ivanov, A.P.: The problem of constrainted Impact. J. Appl. Math. Mech. 61(3), 341–253 (1997)Google Scholar
  17. 17.
    Ivanov, A.P.: Singularities in the dynamics of systems with non-ideal constraints. J. Appl. Math. Mech. 67(2), 185–192 (2003)Google Scholar
  18. 18.
    Brach, R.M.: Impact coefficients and tangential impacts. ASME J. Appl. Mech. 64, 1014–1016 (1997)Google Scholar
  19. 19.
    Zhen, Z., Bin, C., Liu, C., Hai, J.: Impact model resolution on Painlev's paradox. ACTA Mech. Sin. 20(6), 659–660 (2004)Google Scholar
  20. 20.
    Zhen, Z., Liu, C., Bin, C.: The numerical method for three-dimensional impact with friction of multi-rigid-body system. Sci. China Ser. G Phys. Astr. 49(1), 102–118 (2006)Google Scholar
  21. 21.
    Peng, S., Kraus, P., Kumar, V., Dupont, P.: Analysis of rigid-body dynamic models for simulation of systems with frictional contacts. J. Appl. Mech. 68, 118–128 (2001)Google Scholar
  22. 22.
    Grigoryan, S.S.: The solution to the Painlevé paradox for dry friction. Doklady Phys. 46(7), 499–503 (2001)Google Scholar
  23. 23.
    Stewart, D.E.: Rigid-body dynamics with friction and impact. SIAM Rev. 42(1), 3–39 (2000)Google Scholar
  24. 24.
    Stewart, D.E.: Convergence of a time-stepping scheme for rigid-body dynamics and resolution of Painlevé’s problem. Arch. Rational Mech. Anal. 145, 215–260 (1998)Google Scholar
  25. 25.
    Wilms, E.V., Cohen, H.: The occurence of Painlevé’s paradox in the motion of a rotating shaft. J. Appl. Mech. 64 (1997)Google Scholar
  26. 26.
    Ibrahim, R.A.: Friction-induced vibration, chatter, squeal and chaos. Part II: Dynamics and modeling. ASME Appl. Mech. Rev. 47(7), 227–253 (1994)Google Scholar
  27. 27.
    Brogliato, B.: Some perspectives on the analysis and control of complementarity systems. IEEE Trans. Autom. control 48(6), 918–935 (2003)Google Scholar
  28. 28.
    Schiehlen, W., Seiferied, R.: Three approaches for elastodynamic contact in multibody systems. Multibody Syst. Dyn. 12, 1–16 (2004)Google Scholar
  29. 29.
    Hu, B., Eberhard, P., Schiehlen, W.: Comparison of analytical and experimental results for longitudinal impacts on elastic rods. J. Vib. Control 9, 157–174 (2003)Google Scholar
  30. 30.
    Keller, J.B.: Impact with friction. Trans. ASME J. Appl. Mech. 53, 1–4 (1986)Google Scholar
  31. 31.
    Stronge, W.J.: Impact Mechanics. Cambridge University Press, Cambridge, UK (2000)Google Scholar
  32. 32.
    Bhatt, V., Koechling, J.: Partitioning the parameter space according to different behaviors during three-dimensional. ASME J. Appl. Mech. 62, 740–746 (1995)Google Scholar
  33. 33.
    Batlle, J.A.: Rough balanced collisions. ASME J. Appl. Mech. 63, 168–172 (1996)Google Scholar

Copyright information

© Springer Science + Business Media B.v. 2006

Authors and Affiliations

  1. 1.Department of Mechanics and Engineering SciencePeking UniversityBeijingChina

Personalised recommendations