The contact problem in Lagrangian systems with redundant frictional bilateral and unilateral constraints and singular mass matrix. The all-sticking contacts problem

  • Bernard BrogliatoEmail author
  • Jozsef Kovecses
  • Vincent Acary


In this article we analyze the following problem: given a mechanical system subject to (possibly redundant) bilateral and unilateral constraints with set-valued Coulomb’s friction, provide conditions such that the state, which consists of all contacts sticking in both tangential and normal directions, is solvable. The analysis uses complementarity problems, variational inequalities, and linear algebra, hence it provides criteria which are, in principle, numerically tractable. An algorithm and several illustrating examples are proposed.


Lagrangian systems Set-valued friction Complementarity conditions Contact problem Redundant constraints Singular mass matrix Variational inequality Tangent cone Normal cone Force closure Form closure 



  1. 1.
    Abadie, M.: Dynamic simulation of rigid bodies: modelling of frictional contacts. In: Brogliato, B. (ed.) Impacts in Mechanical Systems. Analysis and Modelling. Lecture Notes in Physics, vol. 551, pp. 61–144. Springer, Berlin (2000) zbMATHCrossRefGoogle Scholar
  2. 2.
    Acary, V., Brogliato, B.: Numerical Methods for Nonsmooth Dynamical Systems. Lecture Notes in Applied and Computational Mechanics, vol. 35. Springer, Berlin (2008) zbMATHCrossRefGoogle Scholar
  3. 3.
    Addi, K., Brogliato, B., Goeleven, D.: A qualitative mathematical analysis of a class of linear variational inequalities via semi-complementarity problems. Applications in electronics. Math. Program., Ser. A 126(1), 31–67 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Adly, S., Nacry, F., Thibault, L.: Preservation of prox-regularity of sets and application to constrained optimization. SIAM J. Optim. 26(1), 448–473 (2016) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Akhadkar, N., Acary, V., Brogliato, B.: Multibody systems with 3D revolute joints with clearances: an industrial case study with an experimental validation. Multibody Syst. Dyn. 42(3), 249–282 (2018) zbMATHCrossRefGoogle Scholar
  6. 6.
    Anitescu, M., Cremer, J.F., Potra, F.A.: Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity problems. Nonlinear Dyn. 24, 405–437 (1997) MathSciNetzbMATHGoogle Scholar
  7. 7.
    Anitescu, M., Cremer, J.F., Potra, F.A.: On the existence of solutions to complementarity formulations of contact problems with friction. In: Ferris, M.C., Pang, J.S. (eds.) Complementarity and Variational Problems. State of the Art, pp. 12–21. SIAM, Philadelphia (1997) zbMATHGoogle Scholar
  8. 8.
    Aubin, J.P.: Applied Functional Analysis. Wiley, New York (1979) zbMATHGoogle Scholar
  9. 9.
    Audren, H., Kheddar, A.: 3-D robust stability polyhedron in multicontact. IEEE Trans. Robot. 34(2), 388–403 (2018) CrossRefGoogle Scholar
  10. 10.
    Balkcom, D., Trinkle, J.: Computing wrench cones for planar rigid body contact tasks. Int. J. Robot. Res. 21(2), 1053–1066 (2002) CrossRefGoogle Scholar
  11. 11.
    Baraff, D.: Issues in computing contact forces for non-penetrating rigid bodies. Algorithmica 10(2–4), 292–352 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bernstein, D.S.: Matrix Mathematics. Theory, Facts, and Formulas with Application to Linear Systems Theory. Princeton University Press, Princeton (2005) zbMATHGoogle Scholar
  13. 13.
    Bertsekas, D.P.: Convex Optimization Theory. Athena Scientific, Belmont (2009) zbMATHGoogle Scholar
  14. 14.
    Bicchi, A.: Hands for dexterous manipulation and robust grasping: a difficulty road toward simplicity. IEEE Trans. Robot. Autom. 16(6), 652–662 (2000) CrossRefGoogle Scholar
  15. 15.
    Blumentals, A., Brogliato, B., Bertails-Descoubes, F.: The contact problem in Lagrangian systems subject to bilateral and unilateral constraints, with or without sliding Coulomb’s friction: a tutorial. Multibody Syst. Dyn. 38, 43–76 (2016) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Bretl, T., Lall, S.: Testing static equilibrium for legged robots. IEEE Trans. Robot. 24(4), 794–807 (2008) CrossRefGoogle Scholar
  17. 17.
    Brogliato, B.: Inertial couplings between unilateral and bilateral holonomic constraints in frictionless Lagrangian systems. Multibody Syst. Dyn. 29, 289–325 (2013) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Brogliato, B.: Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction. Multibody Syst. Dyn. 32, 175–216 (2014) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Brogliato, B.: Nonsmooth Mechanics. Models, Dynamics and Control, 3rd edn. Communications and Control Engineering. Springer International Publishing Switzerland, Cham (2016). Erratum/addendum at zbMATHGoogle Scholar
  20. 20.
    Brogliato, B., Goeleven, D.: Singular mass matrix and redundant constraints in unilaterally constrained Lagrangian and Hamiltonian systems. Multibody Syst. Dyn. 35, 39–61 (2015) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Brogliato, B., Thibault, L.: Existence and uniqueness of solutions for non-autonomous complementarity dynamical systems. J. Convex Anal. 17(3–4), 961–990 (2010) MathSciNetzbMATHGoogle Scholar
  22. 22.
    Caron, S., Pham, Q.C., Nakamura, Y.: ZMP support areas for multicontact mobility under frictional constraints. IEEE Trans. Robot. 33(1), 67–80 (2017) CrossRefGoogle Scholar
  23. 23.
    Chen, X., Xiang, S.: Perturbation bounds of P-matrix linear complementarity problems. SIAM J. Optim. 18(4), 1250–1265 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Choudhury, D., Horn, R., Pierce, S.: Quasi-positive definite operators and matrices. Linear Algebra Appl. 99, 161–176 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Cottle, R., Pang, J., Stone, R.: The Linear Complementarity Problem. Computer Science and Scientific Computing. Academic Press, San Diego (1992) zbMATHGoogle Scholar
  26. 26.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I. Operations Research. Springer, New York (2003) zbMATHGoogle Scholar
  27. 27.
    Fraczek, J., Wojtyra, M.: On the unique solvability of a direct dynamics problem for mechanisms with redundant constraints and Coulomb friction. Mech. Mach. Theory 46, 312–334 (2011) zbMATHCrossRefGoogle Scholar
  28. 28.
    Gholami, F., Nasri, M., Kovecses, J., Teichmann, M.: A linear complementarity formulation for contact problems with regularized friction. Mech. Mach. Theory 105, 568–582 (2016) CrossRefGoogle Scholar
  29. 29.
    Glocker, C.: Set-Valued Force Laws: Dynamics of Non-Smooth Systems. Springer, Berlin (2001) zbMATHCrossRefGoogle Scholar
  30. 30.
    Glocker, C., Pfeiffer, F.: Dynamical systems with unilateral contacts. Nonlinear Dyn. 3(4), 245–259 (1992) CrossRefGoogle Scholar
  31. 31.
    Glocker, C., Pfeiffer, F.: Complementarity problems in multibody systems with planar friction. Arch. Appl. Mech. 63(7), 452–463 (1993) zbMATHGoogle Scholar
  32. 32.
    Goeleven, D.: Complementarity and Variational Inequalities in Electronics. Mathematical Analysis and Its Applications. Academic Press, San Diego (2017) zbMATHGoogle Scholar
  33. 33.
    Han, L., Trinkle, J., Li, Z.X.: Grasp analysis as linear matrix inequality problems. IEEE Trans. Robot. Autom. 16(6), 663–674 (2000) CrossRefGoogle Scholar
  34. 34.
    Higashimori, M., Kimura, M., Ishii, I., Kaneko, M.: Dynamic capturing strategy for a 2-D stick-shaped object based on friction independent collision. IEEE Trans. Robot. 23(3), 541–552 (2007) CrossRefGoogle Scholar
  35. 35.
    Hiriart-Urruty, J.B., Lemaréchal, C.: Fundamentals of Convex Analysis. Grundlehren Text Editions. Springer, Berlin (2001) zbMATHCrossRefGoogle Scholar
  36. 36.
    de Jalón, J.G., Gutteriez-Lopez, 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) MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    de Jalón, J.G., Unda, J., Avello, A.: Natural coordinates for the computer analysis of multibody systems. Comput. Methods Appl. Mech. Eng. 56(3), 309–327 (1986) zbMATHCrossRefGoogle Scholar
  38. 38.
    Klepp, H.J.: Existence and uniqueness of solutions for accelerations for multibody systems with friction. Z. Angew. Math. Mech. 75(1), 679–689 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Laulusa, A., Bauchau, O.A.: Review of classical approaches for constraint enforcement in multibody systems. J. Comput. Nonlinear Dyn. 3(1), 011004 (2008) CrossRefGoogle Scholar
  40. 40.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Leine, R.I., van de Wouw, N.: Stability and Convergence of Mechanical Systems with Unilateral Constraints. Lecture Notes in Applied and Computational Mechanics, vol. 36. Springer, Berlin (2008) zbMATHCrossRefGoogle Scholar
  42. 42.
    Lötstedt, P.: Coulomb friction in two-dimensional rigid body systems. Z. Angew. Math. Mech. 61, 605–615 (1981) MathSciNetCrossRefGoogle Scholar
  43. 43.
    Lötstedt, P.: Mechanical systems of rigid bodies subject to unilateral constraints. SIAM J. Appl. Math. 42(2), 281–296 (1982) MathSciNetCrossRefGoogle Scholar
  44. 44.
    Ma, S., Wang, T.: Planar multiple-contact problems subject to unilateral and bilateral kinetic constraints with static Coulomb friction. Nonlinear Dyn. 94, 99–121 (2018) zbMATHCrossRefGoogle Scholar
  45. 45.
    Moreau, J.J.: Application of convex analysis to some problems of dry friction. In: Zorski, H. (ed.) Trends in Applications of Pure Mathematics to Mechanics, vol. 2, pp. 263–280. Pitman, London (1979) Google Scholar
  46. 46.
    Negrut, D., Serban, R., Tasora, A.: Posing multibody dynamics with friction and contact as a differential complementarity problem. J. Comput. Nonlinear Dyn. 13, 014503 (2018) CrossRefGoogle Scholar
  47. 47.
    Nikolic, M., Borovac, B., Rakovic, M.: Dynamic balance preservation and prevention of sliding for humanoid robots in the presence of multiple spatial contacts. Multibody Syst. Dyn. 42, 197–218 (2018) MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Or, Y., Rimon, E.: Analytic characterization of a class of three-contact frictional equilibrium postures in three-dimensional gravitational environments. Int. J. Robot. Res. 29(1), 3–22 (2010) CrossRefGoogle Scholar
  49. 49.
    Or, Y., Rimon, E.: Characterization of frictional multi-legged equilibrium postures on uneven terrains. Int. J. Robot. Res. 36(1), 105–128 (2017) CrossRefGoogle Scholar
  50. 50.
    Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Birkhäuser, Basel (1985) zbMATHCrossRefGoogle Scholar
  51. 51.
    Panagiotopoulos, P.D., Al-Fahed, A.M.: Robot hand grasping and related problems: optimal control and identification. Int. J. Robot. Res. 13(2), 127–136 (1994) CrossRefGoogle Scholar
  52. 52.
    Pang, J., Trinkle, J.: Stability characterization of rigid body contact problems with Coulomb friction. Z. Angew. Math. Mech. 80(10), 643–663 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Pang, J.S., Trinkle, J.C.: Complementarity formulation and existence of solutions of dynamic rigid-body contact problems with Coulomb friction. Math. Program. 73(2), 199–226 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Pang, J.S., Trinkle, J.C., Lo, G.: A complementarity approach to a quasistatic multi-rigid-body contact problem. J. Comput. Optim. Appl. 5(2), 139–154 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Pekal, M., Fraczek, J.: Comparison of selected formulations for multibody systems dynamics with redundant constraints. Arch. Mech. Eng. LXII(1), 93–112 (2016) CrossRefGoogle Scholar
  56. 56.
    Pfeiffer, F.: Non-smooth engineering dynamics. Meccanica 51(12), 3167–3184 (2016) MathSciNetCrossRefGoogle Scholar
  57. 57.
    Pfeiffer, F., Glocker, C.: Multibody Dynamics with Unilateral Contacts. Nonlinear Science. Wiley, New York (1996) zbMATHCrossRefGoogle Scholar
  58. 58.
    Rockafellar, R.: Convex Analysis. Princeton University Press, Princeton (1970) zbMATHCrossRefGoogle Scholar
  59. 59.
    Rohde, C.A.: Generalized inverses of partitioned matrices. J. Soc. Ind. Appl. Math. 13(4), 1033–1035 (1965) MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Seifried, R.: Dynamics of Underactuated Multibody Systems. Solid Mechanics and Its Applications. Springer International Publishing Switzerland, Cham (2014) zbMATHCrossRefGoogle Scholar
  61. 61.
    Seon, J.A., Dahmouche, R., Gauthier, M.: Enhance in-hand dexterous micromanipulation by exploiting adhesion forces. IEEE Trans. Robot. 34(1), 113–125 (2018) CrossRefGoogle Scholar
  62. 62.
    Shi, J., Woodruff, J., Umbanhowar, P., Lynch, K.: Dynamic in-hand sliding manipulation. IEEE Trans. Robot. 33(4), 778–795 (2017) CrossRefGoogle Scholar
  63. 63.
    Simeon, B.: Computational Flexible Multibody Dynamics. A Differential-Algebraic Approach. Differential-Algebraic Equations Forum. Springer, Berlin (2013) zbMATHCrossRefGoogle Scholar
  64. 64.
    Studer, C., Glocker, C.: Representation of normal cone inclusion problems in dynamics via non-linear equations. Arch. Appl. Mech. 76(5), 327–348 (2006) zbMATHCrossRefGoogle Scholar
  65. 65.
    Tasora, A., Anitescu, M.: A matrix-free cone complementarity approach for solving large-scale, nonsmooth, rigid body dynamics. Comput. Methods Appl. Mech. Eng. 200(5–8), 439–453 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Tasora, A., Anitescu, M.: A complementarity-based rolling friction model for rigid contacts. Meccanica 48(7), 1643–1659 (2013) MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Tian, Y., Takane, Y.: Schur complements and Banachiewicz--Schur forms. Electron. J. Linear Algebra 13, 405–418 (2005) MathSciNetzbMATHGoogle Scholar
  68. 68.
    Trinkle, J.C.: On the stability and instantaneous velocity of grasped frictionless objects. IEEE Trans. Robot. Autom. 8(5), 560–572 (1992) CrossRefGoogle Scholar
  69. 69.
    Trinkle, J.C., Pang, J.S., Sudarsky, S., Lo, G.: On dynamic multi-rigid-body contact problems with Coulomb friction. J. Appl. Math. Mech./Z. Angew. Math. Mech. 77(4), 267–279 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    Trinkle, J.C., Tzitzouris, J.A., Pang, J.S.: Dynamic multi-rigid-body systems with concurrent distributed contacts: theory and examples. Philos. Trans., Math. Phys. Eng. Sci. 359(1789), 2575–2593 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    Vanderbei, R., Carpenter, T.: Symmetric indefinite systems for interior point methods. Math. Program. 58(1–3), 1–32 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Varkonyi, P.L., Or, Y.: Lyapunov stability of a rigid body with two frictional contacts. Nonlinear Dyn. 88, 363–393 (2017) MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Wojtyra, M.: Joint reactions in rigid body mechanisms with dependent constraints. Mech. Mach. Theory 44, 2265–2278 (2009) zbMATHCrossRefGoogle Scholar
  74. 74.
    Wojtyra, M.: Modeling of static friction in closed-loop kinetic chains–uniqueness and parametric sensitivity. Multibody Syst. Dyn. 39, 337–361 (2017) MathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    Wojtyra, M.: The Moore–Penrose inverse approach to modeling of multibody systems with redundant constraints. In: Uhl, T. (ed.) Advances in Mechanisms and Machine Science, Mechanisms and Machine Science, vol. 73, pp. 3087–3096. Springer Nature Switzerland AG, Cham (2019) CrossRefGoogle Scholar
  76. 76.
    Wojtyra, M., Fraczek, J.: Comparison of selected methods of handling redundant constraints in multibody systems simulations. J. Comput. Nonlinear Dyn. 8, 021,007 (2013) CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Inria, CNRS, Grenoble INP, LJKUniv. Grenoble AlpesGrenobleFrance
  2. 2.Mechanical Engineering Department, Macdonald Engineering BuildingMcGill UniversityMontrealCanada

Personalised recommendations