Analysis of friction coupling and the Painlevé paradox in multibody systems


Multibody models are useful to describe the macroscopic motion of the elements of physical systems. Modeling contact in such systems can be challenging, especially if friction at the contact interface is taken into account. Furthermore, the dynamics equations of multibody systems with contacts and Coulomb friction may become ill-posed due to friction coupling, as in the Painlevé paradox, where a solution for system dynamics may not be found. Here, the dynamics problem is considered following a general approach so that friction phenomena, such as dynamic jamming, can be analyzed. The effect of the contact forces on the velocity field of the system is the cornerstone of the proposed formulation, which is used to analyze friction coupling in multibody systems with a single contact. In addition, we introduce a new representation of the so-called generalized friction cone, a quadratic form defined in the contact velocity space. The geometry of this cone can be used to determine the critical cases where the solvability of the system dynamic equations can be compromised. Moreover, it allows for assessing friction coupling at the contact interface, and obtaining the values of the friction coefficient that can make the dynamics formulation inconsistent. Finally, the classical Painlevé example of a single rod and the multibody model of an articulated arm are used to illustrate how the proposed cone can detect the cases where the dynamic equations have no solution, or multiple solutions.

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  1. 1.

    Interestingly, the original example used by Painlevé in [2] consists of a cylinder with one of its bases in contact with a slope, which is sliding down the slope with a velocity parallel to the surface. Although the mass distribution of the cylinder is different from the one of a rod, the same conclusions can be drown assuming one single contact point at the edge of the cylinder base.

  2. 2.

    The original text in French by Painlevé [3] can be translated as [28]: “Two rigid bodies, which under given conditions would not produce any pressure on one another if they were ideally smooth, would likewise not act on one another if they were rough.”


  1. 1.

    Berger, E.: Friction modeling for dynamic system simulation. Appl. Mech. Rev. 55(6), 535 (2002)

    Article  Google Scholar 

  2. 2.

    Painlevé, P.: Sur les lois du frottement de glissement. C. R. Hebd. Séances Acad. Sci. 121, 112 (1895)

    MATH  Google Scholar 

  3. 3.

    Painlevé, P.: Sur les lois du frottement de glissement. C. R. Hebd. Séances Acad. Sci. 141, 401 (1905)

    MATH  Google Scholar 

  4. 4.

    Moreau, J.: Quadratic programming in mechanics: dynamics of one sided constraints. SIAM J. Control 4(1), 153 (1966)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Lötstedt, P.: Mechanical systems of rigid bodies subject to unilateral constraints. SIAM J. Appl. Math. 42(2), 281 (1982)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Lemke, C.: On complementary pivot theory. In: Mathematics of the Decision Sciences. Lectures in Applied Mathematics, vol. 2, p. 95 (1968)

    Google Scholar 

  7. 7.

    Júdice, J., Pires, F.: Basic-set algorithm for a generalized linear complementarity problem. J. Optim. Theory Appl. 74(3), 391 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Murty, K.: Linear Complementarity, Linear and Nonlinear Programming. Heldermann, Berlin (1988)

    Google Scholar 

  9. 9.

    Brogliato, B.: Nonsmooth Mechanics. Springer, Berlin (1999)

    Google Scholar 

  10. 10.

    Panagiotopoulos, P.D.: In: Hemivariational Inequalities, pp. 99–134. Springer, Berlin (1993)

    Google Scholar 

  11. 11.

    Glocker, C.: Set-Valued Force Laws. Springer, Troy, New York, USA (2001)

    Google Scholar 

  12. 12.

    Stewart, D.E., Trinkle, J.C.: An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and Coulomb friction. Int. J. Numer. Methods Eng. 39, 2673 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Anitescu, M., Potra, F.A.: Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity problems. Nonlinear Dyn. 14, 231 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Lötstedt, P.: Coulomb friction in two-dimensional rigid body systems. Z. Angew. Math. Mech. 61(12), 605 (1981)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Pfeiffer, F., Glocker, C.: Multibody Dynamics with Unilateral Contacts. Wiley, New York (1996)

    Google Scholar 

  16. 16.

    Stewart, D.E.: Rigid-body dynamics with friction and impact. SIAM Rev. 42, 3 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Acary, V., Cadoux, F., Lemaréchal, C., Malick, J.: A formulation of the linear discrete coulomb friction problem via convex optimization. Z. Angew. Math. Mech. 91(2), 155 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Kaufman, D.M., Sueda, S., James, D.L., Pai, D.K.: Staggered projections for frictional contact in multibody systems. ACM Trans. Graph. 27(5), 164 (2008)

    Article  Google Scholar 

  19. 19.

    Anitescu, M., Tasora, A.: An iterative approach for cone complementarity problems for nonsmooth dynamics. Comput. Optim. Appl. 47(2), 207 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    De Saxcé, G., Feng, Z.Q.: The bipotential method: a constructive approach to design the complete contact law with friction and improved numerical algorithms. Math. Comput. Model. 28(4–8), 225 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Li, J., Daviet, G., Narain, R., Bertails-Descoubes, F., Overby, M., Brown, G.E., Boissieux, L.: An implicit frictional contact solver for adaptive cloth simulation. ACM Trans. Graph. 37(4), 52 (2018)

    Google Scholar 

  22. 22.

    Charles, A., Ballard, P.: The formulation of dynamical contact problems with friction in the case of systems of rigid bodies and general discrete mechanical systems—Painlevé and kane paradoxes revisited. Z. Angew. Math. Phys. 67(4), 99 (2016)

    Article  MATH  Google Scholar 

  23. 23.

    Moreau, J.J.: In: Nonsmooth Mechanics and Applications. CISM Courses & Lectures, pp. 1–82. Springer, Berlin (1988)

    Google Scholar 

  24. 24.

    Batlle, J.A.: On Newton’s and Poisson’s rules of percussive dynamics. J. Appl. Mech. 60(2), 376 (1993)

    Article  MATH  Google Scholar 

  25. 25.

    Zhao, Z., Liu, C., Ma, W., Chen, B.: Experimental investigation of the Painlevé paradox in a robotic system. J. Appl. Mech. 75(4), 041006 (2008)

    Article  Google Scholar 

  26. 26.

    Batlle, J.A., Cardona, S.: The jamb (self-locking) process in three-dimensional collisions. J. Appl. Mech. 65(2), 417 (1998)

    Article  Google Scholar 

  27. 27.

    Stronge, W.J.: In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 431, pp. 169–181. The Royal Society, London (1990)

    Google Scholar 

  28. 28.

    Génot, F., Brogliato, B.: New results on Painlevé paradoxes. Tech. Rep. RR-3366, INRIA (1998).

  29. 29.

    Stewart, D.E.: Convergence of a time-stepping scheme for rigid-body dynamics and resolution of Painlevé’s problem. Arch. Ration. Mech. Anal. 145(3), 215 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Shen, Y., Stronge, W.: Painlevé paradox during oblique impact with friction. Eur. J. Mech. A, Solids 30(4), 457 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Leine, R.I., Brogliato, B., Nijmeijer, H.: Periodic motion and bifurcations induced by the Painlevé paradox. Eur. J. Mech. A, Solids 21(5), 869 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Or, Y., Rimon, E.: Investigation of Painlevé’s paradox and dynamic jamming during mechanism sliding motion. Nonlinear Dyn. 67(2), 1647 (2012)

    Article  MATH  Google Scholar 

  33. 33.

    Erdmann, M.: On a representation of friction in configuration space. Int. J. Robot. Res. 13(3), 240 (1994)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Flores, P.: Concepts and Formulations for Spatial Multibody Dynamics. Springer, Berlin (2015)

    Google Scholar 

  35. 35.

    Wit, C.C.D., Olsson, H., Astrom, K.J., Lischinsky, P.: A new model for control of systems with friction. IEEE Trans. Autom. Control 40(3), 419 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Kövecses, J.: Dynamics of mechanical systems and the generalized free-body diagram—Part I: general formulation. J. Appl. Mech. 75, 061012 (2008)

    Article  Google Scholar 

  37. 37.

    Kövecses, J., Font-Llagunes, J.M.: An eigenvalue problem for the analysis of variable topology mechanical systems. J. Comput. Nonlinear Dyn. 4(3), 031006 (2009)

    Article  Google Scholar 

  38. 38.

    Font-Llagunes, J.M., Barjau, A., Pàmies-Vilà, R., Kövecses, J.: Dynamic analysis of impact in swing-through crutch gait using impulsive and continuous contact models. Multibody Syst. Dyn. 28(3), 257 (2012)

    MathSciNet  Article  Google Scholar 

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Correspondence to Albert Peiret.

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Peiret, A., Kövecses, J. & Font-Llagunes, J.M. Analysis of friction coupling and the Painlevé paradox in multibody systems. Multibody Syst Dyn 45, 361–378 (2019).

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  • Coulomb friction
  • Friction coupling
  • Painlevé paradox
  • Generalized friction cone