Advertisement

Modeling three-dimensional surface-to-surface rigid contact and impact

  • Abhishek Chatterjee
  • Alan BowlingEmail author
Article
  • 66 Downloads

Abstract

This work presents a rigid body framework for analyzing three-dimensional surface contacts and impacts as a simultaneous multi-point impact problem with friction. A method is developed to address the indeterminacy issue typically associated with multi-point contact and impact analysis. This is accomplished using the constraints on impulses and contact forces defined by the Coulomb friction law and rigid body constraints. The proposed approach relies on a global interpretation of Stronge’s energetic coefficient of restitution (ECOR) to maintain energetic consistency. A key aspect of this work involves addressing the three-dimensionality of this problem, which requires a numerical integration in the impulse domain to address the slip/no-slip behavior in the tangential plane of the impact. This work also models the transition to contact after a series of impacts, and proposes a method for enforcing frictional contact constraints. Several examples of simulation results using the proposed method are presented here.

Keywords

Rigid body Impact Contact 3D model Coulomb friction Stronge hypothesis 

Notes

References

  1. 1.
    Wang, Y.-T., Kumar, V., Abel, J.: Dynamics of rigid bodies undergoing multiple frictional contacts. In: Proceedings 1992 IEEE International Conference on Robotics and Automation, pp. 2764–2769. IEEE, New York (1992) Google Scholar
  2. 2.
    Kraus, P.R., Kumar, V.: Compliant contact models for rigid body collisions. In: Proceedings 1997 IEEE International Conference on Robotics and Automation, vol. 2, pp. 1382–1387. IEEE, New York (1997) Google Scholar
  3. 3.
    Jia, Y.-B.: Energy-based modeling of tangential compliance in 3-dimensional impact. In: Algorithmic Foundations of Robotics IX, pp. 267–284. Springer, Berlin (2011) Google Scholar
  4. 4.
    Gonthier, Y., McPhee, J., Lange, C., Piedboeuf, J.-C.: A regularized contact model with asymmetric damping and dwell-time dependent friction. Multibody Syst. Dyn. 11(3), 209–233 (2004) zbMATHGoogle Scholar
  5. 5.
    Sharf, I., Zhang, Y.: A contact force solution for non-colliding contact dynamics simulation. Multibody Syst. Dyn. 16(3), 263–290 (2006) MathSciNetzbMATHGoogle Scholar
  6. 6.
    Lankarani, H.: Contact force model with hysteresis damping for impact analysis of multibody systems. J. Mech. Des. 112(3), 369–376 (1990) Google Scholar
  7. 7.
    Gilardi, G., Sharf, I.: Literature survey of contact dynamics modeling. Mech. Mach. Theory 37(10), 1213–1239 (2002) MathSciNetzbMATHGoogle Scholar
  8. 8.
    Darboux, G.: Etude géométrique sur les percussions et le choc des corps. Bull. Sci. Math. Astron. 4(1), 126–160 (1880) MathSciNetzbMATHGoogle Scholar
  9. 9.
    Whittaker, E.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 2nd edn. Cambridge University Press, Cambridge (1917) Google Scholar
  10. 10.
    Keller, J.: Impact with friction. J. Appl. Mech. 53(1), 1–4 (1986) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Djerassi, S.: Three-dimensional, one-point collision with friction. Multibody Syst. Dyn. 27(2), 173–195 (2012) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Stronge, W.: Impact Mechanics. Cambridge University Press, Cambridge (2000) zbMATHGoogle Scholar
  13. 13.
    Han, I., Gilmore, B.: Multi-body impact motion with friction-analysis, simulation, and experimental validation. J. Mech. Des. 115(3), 412–422 (1993) Google Scholar
  14. 14.
    Pfeiffer, F., Glocker, C.: Multibody Dynamics with Unilateral Contacts, vol. 9. John Wiley & Sons, New York (1996) zbMATHGoogle Scholar
  15. 15.
    Pfeiffer, F.: Mechanical System Dynamics, vol. 40. Springer, Berlin (2008) zbMATHGoogle Scholar
  16. 16.
    Brogliato, B.: Nonsmooth Mechanics: Models, Dynamics and Control, 2nd edn. Springer, London (1999) zbMATHGoogle Scholar
  17. 17.
    Brogliato, B., Ten Dam, A., et al.: Numerical simulation of finite dimensional multibody nonsmooth mechanical systems. Appl. Mech. Rev. 55(2), 107–149 (2002) Google Scholar
  18. 18.
    Flickinger, D., Bowling, A.: Simultaneous oblique impacts and contacts in multibody systems with friction. Multibody Syst. Dyn. 23(3), 249–261 (2010) MathSciNetzbMATHGoogle Scholar
  19. 19.
    Huněk, I.: On a penalty formulation for contact-impact problems. Comput. Struct. 48(2), 193–203 (1993) zbMATHGoogle Scholar
  20. 20.
    Simo, J.C., Laursen, T.: An augmented Lagrangian treatment of contact problems involving friction. Comput. Struct. 42(1), 97–116 (1992) MathSciNetzbMATHGoogle Scholar
  21. 21.
    Papadopoulos, P., Solberg, J.: A Lagrange multiplier method for the finite element solution of frictionless contact problems. Math. Comput. Model. 28(4), 373–384 (1998) zbMATHGoogle Scholar
  22. 22.
    Brogliato, B.: Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction. Multibody Syst. Dyn. 32(2), 175–216 (2014) MathSciNetzbMATHGoogle Scholar
  23. 23.
    Liu, C., Zhao, Z., Brogliato, B.: Frictionless multiple impacts in multibody systems. I. Theoretical framework. Proc. R. Soc. A, Math. Phys. Eng. Sci., 464, 3193–3211. (2008) MathSciNetzbMATHGoogle Scholar
  24. 24.
    Stewart, D.: Rigid-body dynamics with friction and impact. SIAM Rev. 42(1), 3–39 (2000) MathSciNetzbMATHGoogle Scholar
  25. 25.
    Chakraborty, N., Berard, S., Akella, S., Trinkle, J.C.: An implicit time-stepping method for multibody systems with intermittent contact. In: Robotics: Science and Systems (2007) Google Scholar
  26. 26.
    Anitescu, M., Potra, F.A., Stewart, D.E.: Time-stepping for three-dimensional rigid body dynamics. Comput. Methods Appl. Mech. Eng. 177(3), 183–197 (1999) MathSciNetzbMATHGoogle Scholar
  27. 27.
    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) MathSciNetzbMATHGoogle Scholar
  28. 28.
    Liu, T., Wang, M.Y.: Computation of three-dimensional rigid-body dynamics with multiple unilateral contacts using time-stepping and Gauss–Seidel methods. IEEE Trans. Autom. Sci. Eng. 2(1), 19–31 (2005) Google Scholar
  29. 29.
    Moreau, J.: Numerical aspects of the sweeping process. Comput. Methods Appl. Mech. Eng. 177(3), 329–349 (1999) MathSciNetzbMATHGoogle Scholar
  30. 30.
    Zhao, Z., Liu, C., Brogliato, B.: Planar dynamics of a rigid body system with frictional impacts. II. Qualitative analysis and numerical simulations. Proc. R. Soc. A, Math. Phys. Eng. Sci. 465(2107), 2267–2292 (2009) MathSciNetzbMATHGoogle Scholar
  31. 31.
    Rodriguez, A., Bowling, A.: Solution to indeterminate multi-point impact with frictional contact using constraints. Multibody Syst. Dyn. 28(4), 313–330 (2012) MathSciNetGoogle Scholar
  32. 32.
    Rodriguez, A., Bowling, A.: Study of Newton’s cradle using a new discrete approach. Multibody Syst. Dyn. 33(1), 61–92 (2015) MathSciNetGoogle Scholar
  33. 33.
    Chatterjee, A., Rodriguez, A., Bowling, A.: Analytic solution for planar indeterminate impact problems using an energy constraint. Multibody Syst. Dyn. 42(3), 347–379 (2018) MathSciNetzbMATHGoogle Scholar
  34. 34.
    Rodriguez, A.: Dynamic simulation of multibody systems in simultaneous, indeterminate contact and impact with friction. PhD dissertation, UTA (2014) Google Scholar
  35. 35.
    Rodriguez, A., Chatterjee, A., Bowling, A.: Solution to three-dimensional indeterminate contact and impact with friction using rigid body constraints. In: ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, V006T10A037 (2015). American Society of Mechanical Engineers Google Scholar
  36. 36.
    Chatterjee, A., Bowling, A.: Resolving the unique invariant slip-direction in rigid three-dimensional multi-point impacts at stick–slip transitions. In: ASME 2018 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, V006T09A008 (2018). American Society of Mechanical Engineers Google Scholar
  37. 37.
    Djerassi, S.: Collision with friction; Part A: Newton’s hypothesis. Multibody Syst. Dyn. 21(1), 37–54 (2009) MathSciNetzbMATHGoogle Scholar
  38. 38.
    Djerassi, S.: Collision with friction; Part B: Poisson’s and Stronge’s hypotheses. Multibody Syst. Dyn. 21(1), 55–70 (2009) MathSciNetzbMATHGoogle Scholar
  39. 39.
    Djerassi, S.: Stronge’s hypothesis-based solution to the planar collision-with-friction problem. Multibody Syst. Dyn. 24(4), 493–515 (2010) MathSciNetzbMATHGoogle Scholar
  40. 40.
    Kane, T., Levinson, D.: Dynamics: Theory and Applications. McGraw-Hill, New York (1985) Google Scholar
  41. 41.
    Marghitu, D., Stoenescu, E.: Rigid body impact with moment of rolling friction. Nonlinear Dyn. 50(3), 597–608 (2007) MathSciNetzbMATHGoogle Scholar
  42. 42.
    Bergés, P., Bowling, A.: Rebound, slip, and compliance in the modeling and analysis of discrete impacts in legged locomotion. J. Vib. Control 17(12), 1407–1430 (2006) MathSciNetzbMATHGoogle Scholar
  43. 43.
    Najafabadi, S., Kovecses, J., Angeles, J.: Generalization of the energetic coefficient of restitution for contacts in multibody systems. J. Comput. Nonlinear Dyn. 3(4), 70–84 (2008) Google Scholar
  44. 44.
    Yilmaz, C., Gharib, M., Hurmuzlu, Y.: Solving frictionless rocking block problem with multiple impacts. Proc. R. Soc. A, Math. Phys. Eng. Sci. 465(2111), 3323–3339 (2009) MathSciNetzbMATHGoogle Scholar
  45. 45.
    Brake, M.: An analytical elastic-perfectly plastic contact model. Int. J. Solids Struct. 49(22), 3129–3141 (2012) Google Scholar
  46. 46.
    Jackson, R.L., Green, I., Marghitu, D.B.: Predicting the coefficient of restitution of impacting elastic-perfectly plastic spheres. Nonlinear Dyn. 60(3), 217–229 (2010) zbMATHGoogle Scholar
  47. 47.
    Zait, Y., Zolotarevsky, V., Kligerman, Y., Etsion, I.: Multiple normal loading-unloading cycles of a spherical contact under stick contact condition. J. Tribol. 132(4), 1–7 (2010) zbMATHGoogle Scholar
  48. 48.
    Zhang, F., Yeddanapudi, M., Mosterman, P.J.: Zero-crossing location and detection algorithms for hybrid system simulation. IFAC Proc. Vol. 41(2), 7967–7972 (2008) Google Scholar
  49. 49.
    Mosterman, P.J.: An overview of hybrid simulation phenomena and their support by simulation packages. In: International Workshop on Hybrid Systems: Computation and Control, pp. 165–177. Springer, Berlin (1999) Google Scholar
  50. 50.
    Utkin, V.: Chattering problem. IFAC Proc. Vol. 44(1), 13374–13379 (2011) Google Scholar
  51. 51.
    Aljarbouh, A., Caillaud, B.: Chattering-free simulation of hybrid dynamical systems with the functional mock-up interface 2.0. In: The First Japanese Modelica Conferences, vol. 124, pp. 95–105 (2016) Google Scholar
  52. 52.
    Pennestrı, V.P., Valentini, P.: Coordinate reduction strategies in multibody dynamics: a review. In: Proceedings of the Conference on Multibody System Dynamics (2007) Google Scholar
  53. 53.
    Bauchau, O.A., Laulusa, A.: Review of contemporary approaches for constraint enforcement in multibody systems. J. Comput. Nonlinear Dyn. 3(1), 011005 (2008) Google Scholar
  54. 54.
    Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. Eng. 1(1), 1–16 (1972) MathSciNetzbMATHGoogle Scholar
  55. 55.
    Ostermeyer, G.-P.: On Baumgarte stabilization for differential algebraic equations. In: Real-Time Integration Methods for Mechanical System Simulation, pp. 193–207. Springer, Berlin (1990) Google Scholar
  56. 56.
    Nikravesh, C., Nikravesh, P.: An adaptive constraint violation stabilization method for dynamic analysis of mechanical systems. J. Mech. Transm. Autom. Des. 107, 488–492 (1985) Google Scholar
  57. 57.
    Park, K., Chiou, J.: Stabilization of computational procedures for constrained dynamical systems. J. Guid. Control Dyn. 11(4), 365–370 (1988) MathSciNetzbMATHGoogle Scholar
  58. 58.
    Bayo, E., De Jalon, J.G., Serna, M.A.: A modified Lagrangian formulation for the dynamic analysis of constrained mechanical systems. Comput. Methods Appl. Mech. Eng. 71(2), 183–195 (1988) MathSciNetzbMATHGoogle Scholar
  59. 59.
    Wehage, R., Haug, E.: Generalized coordinate partitioning for dimension reduction in analysis of constrained dynamic systems. J. Mech. Des. 104(1), 247–255 (1982) Google Scholar
  60. 60.
    García de Jalón, J., Unda, J., Avello, A., Jiménez, J.: Dynamic analysis of three-dimensional mechanisms in “natural” coordinates. J. Mech. Transm. Autom. Des. 109(4), 460–465 (1987) Google Scholar
  61. 61.
    Liang, C.G., Lance, G.M.: A differentiable null space method for constrained dynamic analysis I. J. Mech. Transm. Autom. Des. 109(3), 405–411 (1987) Google Scholar
  62. 62.
    Kim, S., Vanderploeg, M.: QR decomposition for state space representation of constrained mechanical dynamic systems. J. Mech. Trans. 108(2), 183–188 (1986) Google Scholar
  63. 63.
    Amirouche, F., Ider, S.: Coordinate reduction in the dynamics of constrained multibody system a new approach. J. Appl. Mech. 55, 899 (1988) MathSciNetzbMATHGoogle Scholar
  64. 64.
    Righetti, L., Buchli, J., Mistry, M., Schaal, S.: Inverse dynamics control of floating-base robots with external constraints: a unified view. In: 2011 IEEE International Conference on Robotics and Automation (ICRA), pp. 1085–1090. IEEE, New York (2011) Google Scholar
  65. 65.
    Mistry, M., Buchli, J., Schaal, S.: Inverse dynamics control of floating base systems using orthogonal decomposition. In: 2010 IEEE International Conference on Robotics and Automation (ICRA), pp. 3406–3412. IEEE, New York (2010) Google Scholar
  66. 66.
    Glocker, C., Studer, C.: Formulation and preparation for numerical evaluation of linear complementarity systems in dynamics. Multibody Syst. Dyn. 13(4), 447–463 (2005) MathSciNetzbMATHGoogle Scholar
  67. 67.
    Bowling, A.: Dynamic performance, mobility, and agility of multi-legged robots. J. Dyn. Syst. Meas. Control 128(4), 765–777 (2006) Google Scholar
  68. 68.
    Pfeiffer, F., Glocker, C.: Multi-Body Dynamics with Unilateral Constraints. Wiley, New York (1996) zbMATHGoogle Scholar
  69. 69.
    Stronge, W.: Smooth dynamics of oblique impact with friction. Int. J. Impact Eng. 51, 36–49 (2013) Google Scholar
  70. 70.
    Christoph, G.: Energy consistency conditions for standard impacts. Multibody Syst. Dyn. 29(1), 77–117 (2013) MathSciNetzbMATHGoogle Scholar
  71. 71.
    Christoph, G.: Energy consistency conditions for standard impacts. Multibody Syst. Dyn. 32(4), 445–509 (2014) MathSciNetzbMATHGoogle Scholar
  72. 72.
    Boulanger, G.: Sur le choc avec frottement des corps non parfaitement élastiques. Rev. Sci. 77, 325–327 (1939) zbMATHGoogle Scholar
  73. 73.
    Routh, E.J., et al.: Dynamics of a System of Rigid Bodies. Dover, New York (1960) zbMATHGoogle Scholar
  74. 74.
    Shampine, L.F., Reichelt, M.W.: The Matlab ode suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997) MathSciNetzbMATHGoogle Scholar
  75. 75.
    Dormand, J., Prince, P.: A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6(1), 19–26 (1980) MathSciNetzbMATHGoogle Scholar
  76. 76.
    Liu, T.: Non-jamming conditions in multi-contact rigid-body dynamics. Multibody Syst. Dyn. 22(3), 269–295 (2009) MathSciNetzbMATHGoogle Scholar
  77. 77.
    Zhao, Z., Liu, C., Brogliato, B.: Energy dissipation and dispersion effects in granular media. Phys. Rev. E 78(3), 031307 (2008) MathSciNetGoogle Scholar
  78. 78.
    Liu, C., Zhao, Z., Brogliato, B.: Variable structure dynamics in a bouncing dimer. PhD dissertation, INRIA (2008) Google Scholar
  79. 79.
    Liu, C., Zhao, Z., Brogliato, B.: Frictionless multiple impacts in multibody systems. II. Numerical algorithm and simulation results. Proc. R. Soc. A, Math. Phys. Eng. Sci., 465, 1–23 (2009) MathSciNetzbMATHGoogle Scholar
  80. 80.
    Peña, F., Prieto, F., Lourenço, P.B., Campos Costa, A., Lemos, J.V.: On the dynamics of rocking motion of single rigid-block structures. Earthq. Eng. Struct. Dyn. 36(15), 2383–2399 (2007) Google Scholar
  81. 81.
    Peña, F., Lourenço, P.B., Campos-Costa, A.: Experimental dynamic behavior of free-standing multi-block structures under seismic loadings. J. Earthq. Eng. 12(6), 953–979 (2008) Google Scholar
  82. 82.
    Giouvanidis, A., Dimitrakopoulos, I.: Modelling contact in rocking structures with a nonsmooth dynamics approach. In: ECCOMAS Congress 2016-Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering (2016) Google Scholar
  83. 83.
    Zhang, H., Brogliato, B., Liu, C.: Dynamics of planar rocking-blocks with coulomb friction and unilateral constraints: comparisons between experimental and numerical data. Multibody Syst. Dyn. 32(1), 1–25 (2014) MathSciNetGoogle Scholar
  84. 84.
    Zhang, H., Brogliato, B.: The planar rocking-block: analysis of kinematic restitution laws, and a new rigid-body impact model with friction. PhD dissertation, INRIA (2011) Google Scholar
  85. 85.
    Johnson, K.L., Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1987) zbMATHGoogle Scholar
  86. 86.
    Wriggers, P., Zavarise, G.: Computational contact mechanics. In: Encyclopedia of Computational Mechanics (2004) Google Scholar
  87. 87.
    Jaeger, J.: New solutions in contact mechanics. Wit Pr/Computational Mechanics (2005) Google Scholar
  88. 88.
    Craig, J.: Introduction to Robotics: Mechanics and Control. Addison-Wesley Publishing Company, Inc., Boston (1989) zbMATHGoogle Scholar
  89. 89.
    Li, T.-Y.: Numerical solution of multivariate polynomial systems by homotopy continuation methods. Acta Numer. 6, 399–436 (1997) MathSciNetzbMATHGoogle Scholar
  90. 90.
    Morgan, A., Sommese, A.: Computing all solutions to polynomial systems using homotopy continuation. Appl. Math. Comput. 24(2), 115–138 (1987) MathSciNetzbMATHGoogle Scholar
  91. 91.
    Lee, E., Mavroidis, C.: Solving the geometric design problem of spatial 3R robot manipulators using polynomial homotopy continuation. J. Mech. Des. 124(4), 652–661 (2002) Google Scholar
  92. 92.
    Morgan, A., Sommese, A.: A homotopy for solving general polynomial systems that respects \(m\)-homogeneous structures. Appl. Math. Comput. 24(2), 101–113 (1987) MathSciNetzbMATHGoogle Scholar
  93. 93.
    Wampler, C.W., Morgan, A., Sommese, A.: Numerical continuation methods for solving polynomial systems arising in kinematics. J. Mech. Des. 112(1), 59–68 (1990) Google Scholar
  94. 94.
    Cox, D., Little, J., O’shea, D.: Ideals, Varieties, and Algorithms, vol. 3. Springer, Berlin (1992) zbMATHGoogle Scholar
  95. 95.
    Cox, D.A., Little, J., O’shea, D.: Using Algebraic Geometry, vol. 185. Springer, Berlin (2006) zbMATHGoogle Scholar
  96. 96.
    Sturmfels, B.: Solving Systems of Polynomial Equations, vol. 97. American Mathematical Soc., Providence (2002) zbMATHGoogle Scholar
  97. 97.
    Kapur, D., Lakshman, Y.N.: Elimination methods: an introduction. In: Symbolic and Numerical Computation for Artificial Intelligence (1992) Google Scholar
  98. 98.
    Kapur, D.: Using Gröbner bases to reason about geometry problems. J. Symb. Comput. 2(4), 399–408 (1986) zbMATHGoogle Scholar
  99. 99.
    Manocha, D.: Solving systems of polynomial equations. IEEE Comput. Graph. Appl. 14(2), 46–55 (1994) Google Scholar
  100. 100.
    Kukelova, Z., Bujnak, M., Pajdla, T.: Polynomial eigenvalue solutions to minimal problems in computer vision. IEEE Trans. Pattern Anal. Mach. Intell. 34(7), 1381–1393 (2012) Google Scholar
  101. 101.
    Jónsson, G., Vavasis, S.: Accurate solution of polynomial equations using Macaulay resultant matrices. Math. Comput. 74(249), 221–262 (2005) MathSciNetzbMATHGoogle Scholar
  102. 102.
    Stiller, P.: An Introduction to the Theory of Resultants, Mathematics and Computer Science, T&M University, Texas, College Station, TX (1996) Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringThe University of Texas at ArlingtonArlingtonUSA

Personalised recommendations