Multibody System Dynamics

, Volume 46, Issue 1, pp 41–62 | Cite as

Modeling and analysis of multiple impacts in multibody systems under unilateral and bilateral constrains based on linear projection operators

  • Farhad AghiliEmail author


This paper presents a unifying dynamics formulation for nonsmooth multibody systems (MBSs) subject to changing topology and multiple impacts based on a linear projection operator. An oblique projection matrix ubiquitously yields all characteristic variables of such systems as follows: (i) the constrained acceleration before jump discontinuity from the projection of unconstrained acceleration, (ii) post-impact velocity from the projection of pre-impact velocity, (iii) impulse during impact from the projection of pre-impact momentum, (iv) generalized constraint force from the projection of generalized input force, and (v) post-impact kinetic energy from pre-impact kinetic energy based on projected inertia matrix. All solutions are presented in closed-form with elegant geometrical interpretations. The formulation is general enough to be applicable to MBSs subject to simultaneous multiple impacts with nonidentical restitution coefficients, changing topology, i.e., unilateral constraints becoming inactive or vice versa, or even when the overall constraint Jacobian becomes singular. Not only do the solutions always exist regardless of the constraint condition, but also the condition number for a generalized constraint inertia matrix is minimized in order to reduce numerical sensitivity in computation of the projection matrix to roundoff errors. The model is proven to be energetically consistent if a global restitution coefficient is assumed. In the case of nonidentical restitution coefficients, the set of energetically consistent restitution matrices is characterized by using a linear matrix inequality (LMI).


Impact dynamics Constrained multibody systems Unilateral constraints Impact Lagrangian systems Non-smooth mechanics 



  1. 1.
    McClamroch, N.H., Wang, D.: Feedback stabilization and tracking in constrained robots. IEEE Trans. Autom. Control 33, 419–426 (1988) CrossRefzbMATHGoogle Scholar
  2. 2.
    Garcia de Jalón, J., Bayo, E.: Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge. Springer, New York (1994) CrossRefGoogle Scholar
  3. 3.
    Blajer, W., Schiehlen, W., Schirm, W.: A projective criterion to the coordinate partitioning method for multibody dynamics. Appl. Mech. 64, 86–98 (1994) zbMATHGoogle Scholar
  4. 4.
    Aghili, F.: Control of redundant mechanical systems under equality and inequality constraints on both input and constraint forces. J. Comput. Nonlinear Dyn. 6(3), 031013 (2011) CrossRefGoogle Scholar
  5. 5.
    Aghili, F., Su, C.: Impact dynamics in robotic and mechatronic systems. In: 2017 International Conference on Advanced Mechatronic Systems (ICAMechS), pp. 163–167 (2017) CrossRefGoogle Scholar
  6. 6.
    Gottschlich, S.N., Kak, A.C.: A dynamic approach to high-precision parts mating. IEEE Trans. Syst. Man Cybern. 19(4), 797–810 (1989) CrossRefGoogle Scholar
  7. 7.
    Dupree, K., Liang, C.H., Hu, G., Dixon, W.E.: Adaptive Lyapunov-based control of a robot and mass–spring system undergoing an impact collision. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 38(4), 1050–1061 (2008) CrossRefGoogle Scholar
  8. 8.
    Marhefka, D.W., Orin, D.E.: A compliant contact model with nonlinear damping for simulation of robotic systems. IEEE Trans. Syst. Man Cybern., Part A, Syst. Hum. 29(6), 566–572 (1999) CrossRefGoogle Scholar
  9. 9.
    Khulief, Y.A.: Modeling of impact in multibody systems: an overview. J. Comput. Nonlinear Dyn. 8(2), 021012 (2013) CrossRefGoogle Scholar
  10. 10.
    Brogliato, B.: Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction. Multibody Syst. Dyn. 32(2), 175–216 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hunt, K.H., Crossley, F.R.E.: Coefficient of restitution interpreted as damping in vibroimpact. J. Appl. Mech. 42, 440–445 (1975) CrossRefGoogle Scholar
  12. 12.
    Goldsmith, W.: Impact: The Theory and Physical Behavior of Colliding Solids. Edward Arnold, London (1960) zbMATHGoogle Scholar
  13. 13.
    Lankarani, H.M., Nikravesh, P.E.: A contact force model with hysteresis damping for impact analysis of multibody systems. J. Mech. Des. 112, 369–376 (1990) CrossRefGoogle Scholar
  14. 14.
    Hurmuzlu, Y., Chang, T.H.: Rigid body collisions of a special class of planar kinematic chains. IEEE Trans. Syst. Man Cybern. 22(5), 964–971 (1992) CrossRefzbMATHGoogle Scholar
  15. 15.
    Mu, X., Wu, Q.: On impact dynamics and contact events for biped robots via impact effects. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 36(6), 1364–1372 (2006) CrossRefGoogle Scholar
  16. 16.
    Liu, C., Zhao, Z., Brogliato, B.: Frictionless multiple impacts in multibody systems. I. Theoretical framework. Proc. R. Soc. A 464, 3193–3211 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    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 465, 2267–2292 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Yoshida, Y., Takeuchi, K., Miyamoto, Y., Sato, D., Nenchev, D.: Postural balance strategies in response to disturbances in the frontal plane and their implementation with a humanoid robot. IEEE Trans. Syst. Man Cybern. Syst. 44(6), 692–704 (2014) CrossRefGoogle Scholar
  19. 19.
    Schiehlen, W., Seifried, R., Eberhard, P.: Elastoplastic phenomena in multibody impact dynamics. Comput. Methods Appl. Mech. Eng. 195, 6874–6890 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zhang, X., Vu-Quoc, L.: Modeling the dependence of the coefficient of restitution on the impact velocity in elasto-plastic collisions. Int. J. Impact Eng. 27(3), 317–341 (2002) CrossRefGoogle Scholar
  21. 21.
    Najafabadi, S.M., Kovecses, J., Angeles, J.: Impacts in multibody systems: modeling and experiments. Multibody Syst. Dyn. 120, 163–176 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pena, F., Lourenco, 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) CrossRefGoogle Scholar
  23. 23.
    Aghili, F., Buehler, M., Hollerbach, J.M.: Dynamics and control of direct-drive robots with positive joint torque feedback. In: IEEE Int. Conf. Robotics and Automation, vol. 2, pp. 1156–1161 (1997) CrossRefGoogle Scholar
  24. 24.
    Arponen, T.: Regularization of constraint singularities in multibody systems. Multibody Syst. Dyn. 6, 355–375 (2001) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Blajer, W.: Augmented lagrangian formulation: geometrical interpretation and application to systems with singularities and redundancy. Multibody Syst. Dyn. 8, 141–159 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Aghili, F., Piedbœuf, J.-C.: Simulation of motion of constrained multibody systems based on projection operator. Multibody Syst. Dyn. 10, 3–16 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Muller, A.: A conservative elimination procedure for permanently redundant closure constraints in MBS models with relative coordinates. Multibody Syst. Dyn. 16, 309–330 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Aghili, F.: A unified approach for inverse and direct dynamics of constrained multibody systems based on linear projection operator: applications to control and simulation. IEEE Trans. Robot. 21(5), 834–849 (2005) CrossRefGoogle Scholar
  29. 29.
    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 (2010) CrossRefGoogle Scholar
  30. 30.
    Mistry, M., Righetti, L.: Operational space control of constrained and underactuated systems. In: Proceedings of Robotics: Science and Systems, Los Angeles, CA, USA (2011) Google Scholar
  31. 31.
    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 (2011) CrossRefGoogle Scholar
  32. 32.
    Ahmad, M., Ismail, K.A., Mat, F.: Impact models and coefficient of restitution: a review. J. Eng. Appl. Sci. 11(10), 6549–6555 (2016) Google Scholar
  33. 33.
    Ismail, K.A., Stronge, W.: Impact of viscoplastic bodies: dissipation and restitution. J. Appl. Mech. 75(6), 061011 (2008) CrossRefGoogle Scholar
  34. 34.
    Kangur, K., Kleis, I.: Experimental and theoretical determination of the coefficient of velocity restitution upon impact. Mech. Solids 23(5), 2–5 (1988) Google Scholar
  35. 35.
    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) CrossRefzbMATHGoogle Scholar
  36. 36.
    Stronge, W.: Unraveling paradoxical theories for rigid body collisions. J. Appl. Mech. 58, 1049–1055 (1991) CrossRefzbMATHGoogle Scholar
  37. 37.
    Stoianovici, D., Hurmuzlu, Y.: A critical study of the applicability of rigid-body collision theory. J. Appl. Mech. 63(2), 307–316 (1996) CrossRefGoogle Scholar
  38. 38.
    Lu, C.J., Kuo, M.C.: Coefficients of restitution based on a fractal surface model. J. Appl. Mech. 70(3), 339–345 (2003) CrossRefzbMATHGoogle Scholar
  39. 39.
    amd, J.G.V., Braatz, R.D.: A tutorial on linear and bilinear matrix inequalities. J. Process Control 10, 363–385 (2000) CrossRefGoogle Scholar
  40. 40.
    Gahinet, P., Nemirovski, A., Laub, A.J., Chilali, M.: The LMI control toolbox. In: Proceedings of the 3rd European Control Conference, Rome, Italy, 1995, pp. 3206–3211 (1995) Google Scholar
  41. 41.
    Delebecque, R.N.F., Ghaoui, L.E.: LMITOOL: A Package for LMI Optimization in Scilab—User’s Guide (1995) Google Scholar
  42. 42.
    Golub, G.H., Loan, C.F.V.: Matrix Computations. Johns Hopkins University Press, Baltimore (1996) zbMATHGoogle Scholar
  43. 43.
    Aghili, F.: Non-minimal order model of mechanical systems with redundant constraints for simulations and controls. IEEE Trans. Autom. Control 61(5), 1350–1355 (2016) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Crown 2018

Authors and Affiliations

  1. 1.Concordia UniversityMontrealCanada
  2. 2.Canadian Space Agency (CSA)MontrealCanada

Personalised recommendations