Multibody System Dynamics

, Volume 39, Issue 3, pp 267–290 | Cite as

A new representation of systems with frictional unilateral constraints and its Baumgarte-like relaxation



This paper proposes a new representation of multibody mechanical systems involving three-dimensional frictional unilateral constraints. The new representation is of the form of a differential algebraic inclusion (DAI) employing a normal cone with a non-Euclidean, singular norm metric. It can be seen as a generalization of a differential algebraic equation (DAE) using Lagrange multipliers, which has been used to represent mechanical systems with equality constraints. The paper also presents an approach to approximate the aforementioned DAI by another form of DAI, which can be equivalently converted into an ordinary differential equation (ODE). The approach can be seen as a generalization of the Baumgarte stabilization, which was originally developed for DAEs. The new DAI representation and its ODE approximation are illustrated with some simple examples.


Baumgarte stabilization Frictional unilateral constraints Differential algebraic inclusions 


  1. 1.
    Acary, V.: Higher order event capturing time-stepping schemes for nonsmooth multibody systems with unilateral constraints and impacts. Appl. Numer. Math. 62(10), 1259–1275 (2012) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Acary, V.: Projected event-capturing time-stepping schemes for nonsmooth mechanical systems with unilateral contact and Coulomb’s friction. Comput. Methods Appl. Mech. Eng. 256(1), 224–250 (2013) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Acary, V., Brogliato, B.: Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics. Lecture Notes in Applied and Computational Mechanics, vol. 35. Springer, Berlin (2008) MATHGoogle Scholar
  4. 4.
    Acary, V., Brogliato, B., Goeleven, D.: Higher order Moreau’s sweeping process: mathematical formulation and numerical simulation. Math. Program. 113(1), 133–217 (2008) MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    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), 155–175 (2011) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Alart, P., Curnier, A.: A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput. Methods Appl. Mech. Eng. 92(3), 353–375 (1991) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Anitescu, M.: Optimization based simulation of nonsmooth rigid multibody. Math. Program. 105(1), 113–143 (2006) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Anitescu, M., Potra, F.A.: Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity problems. Nonlinear Dyn. 14(3), 231–247 (1997) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Anitescu, M., Tasora, A.: An iterative approach for cone complementarity problems for nonsmooth dynamics. Comput. Optim. Appl. 47(2), 207–235 (2010) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Arnold, M., Brüls, O.: Convergence of the generalized-\(\alpha\) scheme for constrained mechanical systems. Multibody Syst. Dyn. 18(2), 185–202 (2007) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Ascher, U.M., Chin, H., Petzold, L.R., Reich, S.: Stabilization of constrained mechanical systems with DAEs and invariant manifolds. Mech. Struct. Mach. 23(2), 135–157 (1995) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bauchau, O.A., Laulusa, A.: Review of contemporary approaches for constraint enforcement in multibody systems. J. Comput. Nonlinear Dyn. 3(1), 011005 (2007) CrossRefGoogle Scholar
  13. 13.
    Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. Eng. 1(1), 1–16 (1972) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Bertails-Descoubes, F., Cadoux, F., Daviet, G., Acary, V.: A nonsmooth Newton solver for capturing exact Coulomb friction in fiber assemblies. ACM Trans. Graph. 30(1), 6:1–6:14 (2011) CrossRefGoogle Scholar
  15. 15.
    Bonnefon, O., Daviet, G.: Quartic formulation of Coulomb 3D frictional contact. Tech. rep., INRIA Technical Report, RT-0400 (2011) Google Scholar
  16. 16.
    Brogliato, B.: Nonsmooth Mechanics: Models, Dynamics and Control, 2nd edn. Springer, Berlin (1999) CrossRefMATHGoogle Scholar
  17. 17.
    Brogliato, B., Goeleven, D.: Singular mass matrix and redundant constraints in unilaterally constrained Lagrangian and Hamiltonian systems. Multibody Syst. Dyn. 35(1), 39–61 (2015) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Chatterjee, A.: Rigid body collisions: some general considerations, new collision laws, and some experimental data. Ph.D. thesis, Cornell University (1997) Google Scholar
  19. 19.
    Daviet, G., Bertails-Descoubes, F., Boissieux, L.: A hybrid iterative solver for robustly capturing Coulomb friction in hair dynamics. ACM Trans. Graph. 30(6), 139:1–139:11 (2011) CrossRefGoogle Scholar
  20. 20.
    De Saxcé, G., Feng, Z.Q.: New inequality and functional for contact with friction: the implicit standard material approach. Mech. Struct. Mach. 19(3), 301–325 (1991) MathSciNetCrossRefGoogle Scholar
  21. 21.
    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), 225–245 (1998) MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Flores, P., Machado, M., Seabra, E., da Silva, M.T.: A parametric study on the Baumgarte stabilization method for forward dynamics of constrained multibody systems. J. Comput. Nonlinear Dyn. 6(1), 011019 (2011) CrossRefGoogle Scholar
  23. 23.
    García de Jalón, J., Gutiérrez-López, M.D.: Multibody dynamics with redundant constraints and singular mass matrix: existence, uniqueness, and determination of solutions for accelerations and constraint forces. Multibody Syst. Dyn. 30(3), 311–341 (2013) MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Génot, F., Brogliato, B.: New results on Painlevé paradoxes. Eur. J. Mech. A, Solids 18(4), 653–677 (1999) MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Glocker, C., Pfeiffer, F.: Complementarity problems in multibody systems with planar friction. Arch. Appl. Mech. 63(7), 452–463 (1993) MATHGoogle Scholar
  26. 26.
    Haddouni, M., Acary, V., Beley, J.D.: Comparison of index-3, index-2 and index-1 DAE solvers for nonsmooth multibody systems with unilateral and bilateral constraints. In: Proceedings of ECCOMAS Thematic Conference on Multibody Dynamics 2013, pp. 133–142 (2013) Google Scholar
  27. 27.
    Jean, M.: The non-smooth contact dynamics method. Comput. Methods Appl. Mech. Eng. 177(3–4), 235–257 (1999) MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Joli, P., Séguy, N., Feng, Z.Q.: A modular modeling approach to simulate interactively multibody systems with a Baumgarte/Uzawa formulation. J. Comput. Nonlinear Dyn. 3(1), 011011 (2007) CrossRefGoogle Scholar
  29. 29.
    Kanno, Y., Martins, J.A.C., Pinto da Costa, A.: Three-dimensional quasi-static frictional contact by using second-order cone linear complementarity problem. Int. J. Numer. Methods Eng. 65(1), 62–83 (2006) MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Kikuuwe, R., Fujimoto, H.: Incorporating geometric algorithms in impedance- and admittance-type haptic rendering. In: Proceedings of the Second Joint Eurohaptics Conference and Symposium on Haptic Interfaces for Virtual Environment and Teleoperator Systems (World Haptics Conference 2007), pp. 249–254 (2007) CrossRefGoogle Scholar
  31. 31.
    Kikuuwe, R., Takesue, N., Sano, A., Mochiyama, H., Fujimoto, H.: Admittance and impedance representations of friction based on implicit Euler integration. IEEE Trans. Robot. 22(6), 1176–1188 (2006) CrossRefGoogle Scholar
  32. 32.
    Kim, Y., Kim, S.H., Kwak, Y.K.: Dynamic analysis of a nonholonomic two-wheeled inverted pendulum robot. J. Intell. Robot. Syst. 44(1), 25–46 (2005) CrossRefGoogle Scholar
  33. 33.
    Klarbring, A.: A mathematical programming approach to three-dimensional contact problems with friction. Comput. Methods Appl. Mech. Eng. 58(2), 175–200 (1986) MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Klarbring, A., Björkman, G.: A mathematical programming approach to contact problems with friction and varying contact surface. Comput. Struct. 30(5), 1185–1198 (1988) CrossRefMATHGoogle Scholar
  35. 35.
    Kobilarov, M., Crane, K., Desbrun, M.: Lie group integrators for animation and control of vehicles. ACM Trans. Graph. 28(1), 16:1–16:14 (2009) Google Scholar
  36. 36.
    Leine, R.I., Glocker, C.: A set-valued force law for spatial Coulomb–Contensou friction. Eur. J. Mech. A, Solids 22(2), 193–216 (2003) MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Leine, R.I., Van Campen, D.H., Glocker, C.: Nonlinear dynamics and modeling of various wooden toys with impact and friction. J. Vib. Control 9(1–2), 25–78 (2003) MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Lunk, C., Simeon, B.: Solving constrained mechanical systems by the family of Newmark and \(\alpha\)-methods. Z. Angew. Math. Mech. 86(10), 772–784 (2006) MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Marquis-Favre, W., Bideaux, E., Scavarda, S.: A planar mechanical library in the AMESim simulation software. Part I: Formulation of dynamics equations. Simul. Model. Pract. Theory 14(1), 25–46 (2006) CrossRefGoogle Scholar
  40. 40.
    Mazhar, H., Heyn, T., Pazouki, A., Melanz, D., Seidl, A., Bartholomew, A., Tasora, A., Negrut, D.: Chrono: a parallel multi-physics library for rigid-body, flexible-body, and fluid dynamics. Mech. Sci. 4(1), 49–64 (2013) CrossRefGoogle Scholar
  41. 41.
    Moreau, J.J.: Unilateral contact and dry friction in finite freedom dynamics. In: Moreau, J., Panagiotopoulos, P.D. (eds.) Nonsmooth Mechanics and Applications, pp. 1–82. Springer, Berlin (1988) CrossRefGoogle Scholar
  42. 42.
    Nakaoka, S., Hattori, S., Kanehiro, F., Kajita, S., Hirukawa, H.: Constraint-based dynamics simulator for humanoid robots with shock absorbing mechanisms. In: Proceedings of the 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 3641–3647 (2007) CrossRefGoogle Scholar
  43. 43.
    Or, Y., Rimon, E.: Investigation of Painlevé’s paradox and dynamic jamming during mechanism sliding motion. Nonlinear Dyn. 67(2), 1647–1668 (2012) CrossRefMATHGoogle Scholar
  44. 44.
    Payr, M., Glocker, C.: Oblique frictional impact of a bar: analysis and comparison of different impact laws. Nonlinear Dyn. 41(4), 361–383 (2005) MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Pfeiffer, F.: Unilateral problems of dynamics. Arch. Appl. Mech. 69(8), 503–527 (1999) CrossRefMATHGoogle Scholar
  46. 46.
    Schindler, T., Acary, V.: Timestepping schemes for nonsmooth dynamics based on discontinuous Galerkin methods: definition and outlook. Math. Comput. Simul. 95, 180–199 (2013) MathSciNetCrossRefGoogle Scholar
  47. 47.
    Schwager, T., Pöschel, T.: Coefficient of restitution and linear-dashpot model revisited. Granul. Matter 9(6), 465–469 (2007) CrossRefGoogle Scholar
  48. 48.
    Silcowitz, M., Niebe, S., Erleben, K.: Interactive rigid body dynamics using a projected Gauss–Seidel subspace minimization method. In: Richard, P., Braz, J. (eds.) Computer Vision, Imaging and Computer Graphics. Theory and Applications. Communications in Computer and Information Science, vol. 229, pp. 218–229. Springer, Berlin (2011) CrossRefGoogle Scholar
  49. 49.
    Song, P., Kraus, P., Kumar, V., Dupont, P.: Analysis of rigid-body dynamic models for simulation of systems with frictional contacts. J. Appl. Mech. 68(1), 118–128 (2001) CrossRefMATHGoogle Scholar
  50. 50.
    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(15), 2673–2691 (1996) MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Stronge, W.J.: Smooth dynamics of oblique impact with friction. Int. J. Impact Eng. 51, 36–49 (2013) CrossRefGoogle Scholar
  52. 52.
    Studer, C., Glocker, C.: Representation of normal cone inclusion problems in dynamics via non-linear equations. Arch. Appl. Mech. 76(5–6), 327–348 (2006) CrossRefMATHGoogle Scholar
  53. 53.
    van der Schaft, A.J., Schumacher, J.M.: Complementarity modeling of hybrid systems. IEEE Trans. Autom. Control 43(4), 483–490 (1998) MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Xiong, X., Kikuuwe, R., Yamamoto, M.: A differential-algebraic method to approximate nonsmooth mechanical systems by ordinary differential equations. Journal of Applied Mathematics 2013 (2013). Article ID 320276 Google Scholar

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© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Kyushu UniversityFukuokaJapan
  2. 2.INRIA Grenoble Rhônes-AlpesSaint-IsmierFrance

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