Contact Simulation

  • Evan DrumwrightEmail author
  • Jeffrey C. Trinkle
Reference work entry


This chapter discusses numerous topics related to simulating multi-rigid bodies undergoing contact, including rigid and pseudo-rigid models of contact, complementarity problems, the Coulomb friction model, rigid body impacts, coordinate selection for rigid bodies and multibodies, integrating the equations of motion, constructing Jacobian matrices for unilateral and bilateral constraints and time-stepping and event-driven simulation methods, and determining contact data from geometric representations of rigid bodies. The material is approached starting from foundational models and moves toward practical implementation. The chapter concludes with further reading, which includes both current research directions and open problems.


  1. 1.
    V. Acary, Toward higher order event-capturing schemes and adaptive time-step strategie for nonsmooth multibody systems. Technical report RR-7151, INRIA (2009)Google Scholar
  2. 2.
    M. Anitescu, Optimization-based simulation of nonsmooth dynamics. Math. Program. Ser. A 105, 113–143 (2006)Google Scholar
  3. 3.
    M. Anitescu, G.D. Hart, A constraint-stabilized time-stepping approach for rigid multibody dynamics with joints, contacts, and friction. Int. J. Numer. Methods Eng. 60(14), 2335–2371 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    M. Anitescu, F.A. Potra, Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity problems. Nonlinear Dyn. 14, 231–247 (1997)Google Scholar
  5. 5.
    U.M. Ascher, L. Petzold, Computer Methods for Ordinary Differential Equations and Differential Algebraic Equations (SIAM, Philadelphia, 1998)Google Scholar
  6. 6.
    U.M. Ascher, H. Chin, L.R. Petzold, S. Reich, Stabilization of constrained mechanical systems with DAEs and invariant manifolds. J. Mech. Struct. Mach. 23, 135–158 (1995)MathSciNetCrossRefGoogle Scholar
  7. 7.
    D. Baraff, Fast contact force computation for nonpenetrating rigid bodies, in Proceedings of SIGGRAPH, Orlando, Jul 1994Google Scholar
  8. 8.
    D. Baraff, Linear-time dynamics using lagrange multipliers, in Proceedings of Computer Graphics, New Orleans, Aug 1996Google Scholar
  9. 9.
    J. Baumgarte, Stabilization of constraints and integrals of motion in dynamical systems. Comput. Math. Appl. Mech. Engr. 1, 1–16 (1972)MathSciNetCrossRefGoogle Scholar
  10. 10.
    S. Berard, Using Simulation for Planning and Design of Robotic Systems with Intermittent Contact. PhD thesis, Rennselaer Polytechnic Institute (2009)Google Scholar
  11. 11.
    B. Brogliato, A.A. Ten Dam, L. Paoli, F. Génot, S. Abadie, Numerical simulation of finite dimensional multibody nonsmooth mechanical systems. ASME Appl. Mech. Rev. 55(2), 107–150 (2002)CrossRefGoogle Scholar
  12. 12.
    S. Cameron, Enhancing GJK: computing minimum and penetration distances between convex polyhedra, in Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), Albuquerque, Apr 1997Google Scholar
  13. 13.
    C. Canudas de Wit, H. Olsson, K.J. Ȧström, A new model for control of systems with friction. IEEE Trans. Autom. Control 40(3), 419–425 (1995)Google Scholar
  14. 14.
    E. Catto, Soft constraints reinventing the spring, in Game Developer’s Conference (2011)Google Scholar
  15. 15.
    A. Chatterjee, On the realism of complementarity conditions in rigid-body collisions. Nonlinear Dyn. 20, 159–168 (1999)Google Scholar
  16. 16.
    A. Chatterjee, A. Ruina, A new algebraic rigid body collision law based on impulse space considerations. ASME J. Appl. Mech. 65(4), 939–951 (1998)CrossRefGoogle Scholar
  17. 17.
    B. Chazelle, Convex partitions of polyhedra: a lower bound and worst-case optimal algorithm. SIAM J. Comput. 13, 488–507 (1984)MathSciNetCrossRefGoogle Scholar
  18. 18.
    B. Cheng Yi, E.M. Drumwright, Determining contact data for time stepping rigid body simulations with convex polyhedral geometries, in Proceedings of International Conference on Simulation, Modeling, and Programming for Autonomous Robots (SIMPAR), San Francisco (2016)Google Scholar
  19. 19.
    M.B. Cline, D.K. Pai, Post-stabilization for rigid body simulation with contact and constraints, in Proceedings of IEEE International Conference on Robotics and Automation (ICRA) (2003), pp. 3744–3751Google Scholar
  20. 20.
    M.J. Coleman, A. Ruina, An uncontrolled toy that can walk but cannot stand still. Phys. Rev. Lett. 80(16), 3658–3661 (1998)CrossRefGoogle Scholar
  21. 21.
    M.J. Coleman, M. Garcia, K. Mombaur, A. Ruina, Prediction of stable walking for a toy that cannot stand. Phys. Rev. E 64(2) (2001)Google Scholar
  22. 22.
    R.W. Cottle, J.-S. Pang, R. Stone, The Linear Complementarity Problem (Academic, Boston, 1992)Google Scholar
  23. 23.
    D. Dobkin, J. Hershberger, D. Kirkpatrick, S. Suri, Computing the intersection-depth of polyhedra. Algorithmica 9, 518–533 (1993)MathSciNetCrossRefGoogle Scholar
  24. 24.
    E. Drumwright, D.A. Shell, Modeling contact friction and joint friction in dynamic robotic simulation using the principle of maximum dissipation, in Proceedings of Workshop on the Algorithmic Foundations of Robotics (WAFR) (2010)zbMATHGoogle Scholar
  25. 25.
    E. Drumwright, D. Shell, Extensive analysis of linear complementarity problem (LCP) solver performance on randomly generated rigid body contact problems, in Proceedings of IEEE/RSJ International Conference of Intelligent Robots and Systems (IROS), Vilamoura, Oct 2012Google Scholar
  26. 26.
    C. Ericson, Real-Time Collision Detection (Morgan Kaufmann, San Francisco, 2005)Google Scholar
  27. 27.
  28. 28.
    R. Featherstone, Robot Dynamics Algorithms (Kluwer, Boston, 1987)CrossRefGoogle Scholar
  29. 29.
    R. Featherstone, Rigid Body Dynamics Algorithms (Springer, New York, 2008)CrossRefGoogle Scholar
  30. 30.
    S.F. Frisken, R.N. Perry, A.P. Rockwood, T.R. Jones, Adaptively sampled distance fields: a general representation of shape for computer graphics, in Computer Graphics (Proceedings of ACM SIGGRAPH) (2000)Google Scholar
  31. 31.
    E.G. Gilbert, D.W. Johnson, S.S. Keerthi, A fast procedure for computing the distance between complex objects in three-dimensional space. IEEE J. Robot. Autom. 4(2), 193–203 (1988)CrossRefGoogle Scholar
  32. 32.
    S. Gottschalk, M.C. Lin, D. Manocha, OBB-tree: a hierarchical structure for rapid interference detection, in Proceedings of ACM SIGGRAPH (1996)Google Scholar
  33. 33.
    S. Goyal, Planar sliding of a rigid body with dry friction: limit surfaces and dynamics of motion. PhD thesis, Cornell University (1988)Google Scholar
  34. 34.
    G.D. Hart, M. Anitescu, An O(m + n) measure of penetration depth between convex polyhedral bodies for rigid multibody dynamics. Technical report, ANL/MCS-P1753-0510 (2010)Google Scholar
  35. 35.
    J.M. Hsu, S.C. Peters, Extending open dynamics engine for the DARPA virtual robotics challenge, in Proceedings of Simulation, Modeling, and Programming for Autonomous Robots (SIMPAR) (2014)Google Scholar
  36. 36.
    A.P. Ivanov, On multiple impact. J. Appl. Math. Mech. 59(6), 887–902 (1995)MathSciNetCrossRefGoogle Scholar
  37. 37.
    K.L. Johnson, Contact Mechanics (Cambridge University Press, Cambridge, 1987)Google Scholar
  38. 38.
    T.R. Kane, D.A. Levinson, Dynamics: Theory and Applications (McGraw-Hill, New York, 1985)Google Scholar
  39. 39.
    Y.J. Kim, M.A. Otaduy, M.C. Lin, D. Manocha, Fast penetration depth computation for physically-based animation, in Proceedings of Symposium on Computer Animation (SCA) (2002)Google Scholar
  40. 40.
    C. Lacoursière, Splitting methods for dry frictional contact problems in rigid multibody systems: preliminary performance results, in Proceedings of SIGRAD, ed. by M. Ollila, Nov 2003, pp. 11–16Google Scholar
  41. 41.
    C. Lacoursière, Ghosts and machines: regularized variational methods for interactive simulations of multibodies with dry frictional contacts. PhD thesis, Umeå University (2007)Google Scholar
  42. 42.
    P. Löstedt, Mechanical systems of rigid bodies subject to unilateral constraints. SIAM J. Appl. Math. 42(2), 281–296 (1982)MathSciNetCrossRefGoogle Scholar
  43. 43.
    P. Löstedt, Numerical simulation of time-dependent contact friction problems in rigid body mechanics. SIAM J. Sci. Stat. Comput. 5(2), 370–393 (1984)Google Scholar
  44. 44.
    Y. Lu, J.C. Trinkle, On the convergence of fixed-point iteration in solving complementarity problems arising in robot locomotion and manipulation, in Proceedings of IEEE/RSJ International Conference on Intelligent Robots & Systems (IROS) (2014)Google Scholar
  45. 45.
    M. Machado, P. Moreira, P. Flores, H. Lankarani, Compliant contact force models in multibody dynamics: evolution of the Hertz contact theory. Mech. Mach. Theory 53, 99–121 (2012)CrossRefGoogle Scholar
  46. 46.
    D. Meltz, Y. Or, E. Rimon, Experimental verification and graphical characterization of dynamic jamming in frictional rigid-body mechanics, in Proceedings of IEEE International Conference on Robotics and Automation, Rome (2007)Google Scholar
  47. 47.
    B. Mirtich, Impulse-based dynamic simulation of rigid body systems. PhD thesis, University of California, Berkeley (1996)Google Scholar
  48. 48.
    B. Mirtich, V-Clip: fast and robust polyhedral collision detection. ACM Trans. Graph. 17(3), 177–208 (1998)CrossRefGoogle Scholar
  49. 49.
    M.D.P. Monteiro-Marques, Differential inclusions in nonsmooth mechanical problems: shocks and dry friction, in Progress in Nonlinear Differential Equations and Their Applications, vol. 9 (Birkhäuser Verlag, Basel, 1993)Google Scholar
  50. 50.
    J.J. Moreau, Standard inelastic shocks and the dynamics of unilateral constraints, in C.I.S.M. Courses and Lectures, ed. by G. del Piero, F. Maceri, vol. 288 (Springer, Vienna, 1985), pp. 173–221CrossRefGoogle Scholar
  51. 51.
    K.G. Murty, Linear Complementarity, Linear and Nonlinear Programming (Heldermann, Berlin, 1988)Google Scholar
  52. 52.
    P.E. Nikravesh, Computer-Aided Analysis of Mechanical Systems (Prentice Hall, Englewood Cliffs, 1988)Google Scholar
  53. 53.
    G. Nützi, A. Schweizer, M. Möller, C. Glocker, Projective jacobi and gauss-seidel on the GPU for non-smooth multi-body systems, in Proceedings of International Conference Multibody Systems, Nonlinear Dynamics, and Control, Buffalo (2014)Google Scholar
  54. 54.
    P. Painlevé, Sur le lois du frottement de glissemment. C. R. Académie des Sciences Paris 121, 112–115 (1895)Google Scholar
  55. 55.
    M. Posa, R. Tedrake, Direct trajectory optimization of rigid body dynamical systems through contact, in Proceedings of Workshop on Algorithmic Foundations of Robotics (WAFR), Boston (2012)Google Scholar
  56. 56.
    F.A. Potra, M. Anitescu, B. Gavrea, J. Trinkle, A linearly implicit trapezoidal method for stiff multibody dynamics with contact, joints, and friction. Int. J. Numer. Methods Eng. 66(7), 1079–1124 (2006)Google Scholar
  57. 57.
    W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes in C, 2nd edn. (Cambridge University Press, Cambridge, 1992)Google Scholar
  58. 58.
    S. Roy, Recent advances in numerical methods for fluid dynamics and heat transfer. J. Fluid Eng. 127(4):629–630 (2005)CrossRefGoogle Scholar
  59. 59.
    L. Saab, O.E. Ramos, F. Keith, N. Mansard, P. Souères, J.-Y. Fourquest, Dynamic whole-body motion generation under rigid contacts and other unilateral constraints. IEEE Trans. Robot. 29(2), 346–362 (2013)CrossRefGoogle Scholar
  60. 60.
    P.J. Schneider, D.H. Eberly, Geometric Tools for Computer Graphics (Morgan Kaufman, San Francisco, 2003)CrossRefGoogle Scholar
  61. 61.
    L. Sciavicco, B. Siciliano, Modeling and Control of Robot Manipulators, 2nd edn. (Springer, London, 2000)CrossRefGoogle Scholar
  62. 62.
    J.A. Sethian, A fast marching level set method for monotonically advancing fronts. Proc. Natl. Acad. Sci. 93, 1591–1595 (1996)MathSciNetCrossRefGoogle Scholar
  63. 63.
    A.A. Shabana, Computational Dynamics, 2nd edn. (Wiley, New York, 2001)Google Scholar
  64. 64.
    J. Shepherd, S. Zapolsky, E.M. Drumwright, Fast multi-body simulations of robots controlled with error feedback, in Proceedings of International Conference on Simulation, Modeling, and Programming for Autonomous Robots (SIMPAR), 2016Google Scholar
  65. 65.
    C.E. Smith, P.-P. Liu, Coefficients of restitution. ASME J. Appl. Mech. 59, 963–969 (1992)CrossRefGoogle Scholar
  66. 66.
    B. Smith, D.M. Kaufman, E. Vouga, R. Tamstorf, E. Grinspun, Reflections on simultaneous impact. ACM Trans. Graph. (Proc. SIGGRAPH) 31(4), 106:1–106:12 (2012)CrossRefGoogle Scholar
  67. 67.
    P. Song, M. Yashima, V. Kumar, Dynamic simulation for grasping and whole arm manipulation, in Proceedings of IEEE International Conference on Robotics and Automation, San Francisco, Apr 2000Google Scholar
  68. 68.
    D.E. Stewart, Convergence of a time-stepping scheme for rigid-body dynamics and resolution of Painlevé’s problem. Arch. Ration. Mech. Anal. 145, 215–260 (1998)MathSciNetCrossRefGoogle Scholar
  69. 69.
    D.E. Stewart, J.C. Trinkle, 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)MathSciNetCrossRefGoogle Scholar
  70. 70.
    D. Stewart, J.C. Trinkle, An implicit time-stepping scheme for rigid body dynamics with Coulomb friction, in Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), San Francisco, Apr 2000Google Scholar
  71. 71.
    D. Stoianovici, Y. Hurmuzlu, A critical study of the applicability of rigid-body collision theory. ASME J. Appl. Mech. 63, 307–316 (1996)CrossRefGoogle Scholar
  72. 72.
    W.J. Stronge, Rigid body collisions with friction. Proc. R. Soc. Lond. A 431(169–181) (1990)MathSciNetCrossRefGoogle Scholar
  73. 73.
    C. Studer, C. Glocker, Simulation of non-smooth mechanical systems with many unilateral constraints, in Proceedings of EUROMECH Nonlinear Oscillation Conference (ENOC) (2005)Google Scholar
  74. 74.
    C. Studer, R.I. Leine, C. Glocker, Step size adjustment and extrapolation for time-stepping schemes in non-smooth dynamics. Int. J. Numer. Methods Eng. 76, 1747–1781 (2008)MathSciNetCrossRefGoogle Scholar
  75. 75.
    J.R. Taylor, E.M. Drumwright, State estimation of a wild robot toward validation of rigid body simulation, in Proceedings of International Conference on Simulation, Modeling, and Programming for Autonomous Robots (SIMPAR), San Francisco (2016)Google Scholar
  76. 76.
    J.R. Taylor, E.M. Drumwright, J. Hsu, Analysis of grasping failures in multi-rigid body simulations, in Proceedings of International Conference on Simulation, Modeling, and Programming for Autonomous Robots (SIMPAR), San Francisco (2016)Google Scholar
  77. 77.
    E. Todorov, A convex, smooth and invertible contact model for trajectory optimization, in Proceedings of IEEE International Conference on Robotics and Automation (ICRA), Shanghai (2011)Google Scholar
  78. 78.
    E. Todorov, Analytically-invertible dynamics with contacts and constraints: theory and implementation in MuJoCo, in Proceedings of IEEE International Conference on Robotics and Automation (2014)Google Scholar
  79. 79.
    J. Trinkle, J.-S. Pang, S. Sudarsky, G. Lo, On dynamic multi-rigid-body contact problems with Coulomb friction. Zeithscrift fur Angewandte Mathematik und Mechanik 77(4), 267–279 (1997)MathSciNetCrossRefGoogle Scholar
  80. 80.
    G. van den Bergen, Proximity queries and penetration depth computation on 3D game objects, in Proceedings of Game Developer’s Conference (2001)Google Scholar
  81. 81.
    M.W. Walker, D.E. Orin, Efficient dynamic computer simulation of robotic mechanisms. ASME J. Dyn. Syst. Meas. Control. 104, 205–211 (1982)CrossRefGoogle Scholar
  82. 82.
    Y.-T. Wang, V. Kumar, Simulation of mechanical systems with multiple frictional contacts. ASME J. Mech. Des. 116, 571–580 (1994)CrossRefGoogle Scholar
  83. 83.
    J. Williams, Y. Lu, J. C. Trinkle, A complementarity based contact model for geometrically accurate treatment of polytopes in simulation, in Proceedings of ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (2014)Google Scholar
  84. 84.
    S. Zapolsky, E.M. Drumwright, Adaptive integration for controlling speed vs. accuracy in multi-rigid body simulation, in Proceedings of IEEE/RSJ International Conference on Intelligent Robots & Systems (IROS) (2015)Google Scholar
  85. 85.
    S. Zapolsky, E. Drumwright, I. Havoutis, J. Buchli, C. Semini, Inverse dynamics for a quadruped robot locomoting on slippery surfaces, in Proceedings of International Conference Climbing Walking Robots (CLAWAR), Sydney (2013)Google Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Toyota Research InstituteLos AltosUSA
  2. 2.National Robotics InitiativeThe National Science FoundationArlingtonUSA

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