Nonlinear Dynamics

, Volume 93, Issue 4, pp 2039–2056 | Cite as

Direct sensitivity analysis of multibody systems with holonomic and nonholonomic constraints via an index-3 augmented Lagrangian formulation with projections

  • Daniel Dopico
  • Francisco GonzálezEmail author
  • Alberto Luaces
  • Mariano Saura
  • Daniel García-Vallejo
Original Paper


Optimizing the dynamic response of mechanical systems is often a necessary step during the early stages of product development cycle. This is a complex problem that requires to carry out the sensitivity analysis of the system dynamics equations if gradient-based optimization tools are used. These dynamics equations are often expressed as a highly nonlinear system of ordinary differential equations or differential-algebraic equations, if a dependent set of generalized coordinates with its corresponding kinematic constraints is used to describe the motion. Two main techniques are currently available to perform the sensitivity analysis of a multibody system, namely the direct differentiation and the adjoint variable methods. In this paper, we derive the equations that correspond to the direct sensitivity analysis of the index-3 augmented Lagrangian formulation with velocity and acceleration projections. Mechanical systems with both holonomic and nonholonomic constraints are considered. The evaluation of the system sensitivities requires the solution of a tangent linear model that corresponds to the Newton–Raphson iterative solution of the dynamics at configuration level, plus two additional nonlinear systems of equations for the velocity and acceleration projections. The method was validated in the sensitivity analysis of a set of examples, including a five-bar linkage with spring elements, which had been used in the literature as benchmark problem for similar multibody dynamics formulations, a point-mass system subjected to nonholonomic constraints, and a full-scale vehicle model.


Sensitivity analysis Multibody system dynamics Index-3 augmented Lagrangian method Projections 



The support of the Spanish Ministry of Economy and Competitiveness (MINECO) under Project DPI2016-81005-P, the Galician Government through Grant ED431B2016/031, and the postdoctoral research contract Juan de la Cierva No. JCI-2012-12376 is greatly acknowledged.


  1. 1.
    Anderson, K.S., Hsu, Y.: Analytical fully-recursive sensitivity analysis for multibody dynamic chain systems. Multibody Syst. Dyn. 8, 1–27 (2002). MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Banerjee, J.M., McPhee, J.: Multibody Dynamics. Computational Methods and Applications. In: Samin, J.C., Fisette, P. (eds.) Computational Methods in Applied Sciences, chap. Symbolic Sensitivity Analysis of Multibody Systems, vol. 28, pp. 123–146. Springer, Berlin (2013).
  3. 3.
    Bauchau, O.A., Laulusa, A.: Review of contemporary approaches for constraint enforcement in multibody systems. J. Comput. Nonlinear Dyn. 3(1), 011,005–011,005 (2007). CrossRefGoogle Scholar
  4. 4.
    Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. Eng. 1(1), 1–16 (1972). MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bayo, E., García de Jalon, J., Serna, M.: A modified Lagrangian formulation for the dynamic analysis of constrained mechanical systems. Comput. Methods Appl. Mech. Eng. 71(2), 183–195 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bayo, E., Ledesma, R.: Augmented Lagrangian and mass-orthogonal projection methods for constrained multibody dynamics. Nonlinear Dyn. 9(1–2), 113–130 (1996). MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bestle, D., Seybold, J.: Sensitivity analysis of constrained multibody systems. Arch. Appl. Mech. 62, 181–190 (1992). zbMATHGoogle Scholar
  8. 8.
    Brenan, K., Campbell, S., Petzold, L.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. North-Holland, New York (1989)zbMATHGoogle Scholar
  9. 9.
    Callejo, A.: Dynamic response optimization of vehicles through efficient multibody formulations and automatic differentiation techniques. Ph.D. thesis, Escuela Técnica Superior de Ingenieros Industriales. Universidad Politécnica de Madrid (2013)Google Scholar
  10. 10.
    Callejo, A., García de Jalón, J., Luque, P., Mántaras, D.A.: Sensitivity-based, multi-objective design of vehicle suspension systems. J. Comput. Nonlinear Dyn. 10(3), 031,008 (2015). CrossRefGoogle Scholar
  11. 11.
    Cao, Y., Li, S., Petzold, L.: Adjoint sensitivity analysis for differential-algebraic equations: algorithms and software. J. Comput. Appl. Math. 149(1), 171–191 (2002). MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Chang, C.O., Nikravesh, P.E.: Optimal design of mechanical systems with constraint violation stabilization method. J. Mech. Trans. Autom. Des. 107(4), 493–498 (1985). CrossRefGoogle Scholar
  13. 13.
    Cuadrado, J., Cardenal, J., Morer, P., Bayo, E.: Intelligent simulation of multibody dynamics: space-state and descriptor methods in sequential and parallel computing environments. Multibody Syst. Dyn. 4(1), 55–73 (2000). zbMATHCrossRefGoogle Scholar
  14. 14.
    Cuadrado, J., Gutiérrez, R., Naya, M., Morer, P.: A comparison in terms of accuracy and efficiency between a MBS dynamic formulation with stress analysis and a non-linear FEA code. Int. J. Numer. Methods Eng. 51(9), 1033–1052 (2001). zbMATHCrossRefGoogle Scholar
  15. 15.
    Dias, J., Pereira, M.: Sensitivity analysis of rigid-flexible multibody systems. Multibody Syst. Dyn. 1, 303–322 (1997). zbMATHCrossRefGoogle Scholar
  16. 16.
    Dopico, D., González, F., Cuadrado, J., Kövecses, J.: Determination of holonomic and nonholonomic constraint reactions in an index-3 augmented Lagrangian formulation with velocity and acceleration projections. J. Comput. Nonlinear Dyn. 9(4), 041,006 (2014). CrossRefGoogle Scholar
  17. 17.
    Dopico, D., Luaces, A., González, M., Cuadrado, J.: Dealing with multiple contacts in a human-in-the-loop application. Multibody Syst. Dyn. 25(2), 167–183 (2011). MathSciNetCrossRefGoogle Scholar
  18. 18.
    Dopico, D., Luaces, A., Lugrís, U., Saura, M., González, F., Sanjurjo, E., Pastorino, R.: Mbslim: Multibody Systems in Laboratorio de Ingeniería Mecánica (2009–2016).
  19. 19.
    Dopico, D., Sandu, A., Sandu, C., Zhu, Y.: Multibody Dynamics. Computational Methods and Applications. In: Terze, Z. (ed.) Computational Methods in Applied Sciences. Sensitivity Analysis of Multibody Dynamic Systems Modeled by ODEs and DAEs, vol. 35, pp. 1–32. Springer, Berlin (2014). Google Scholar
  20. 20.
    Dopico, D., Zhu, Y., Sandu, A., Sandu, C.: Direct and adjoint sensitivity analysis of ordinary differential equation multibody formulations. J. Comput. Nonlinear Dyn. 10(1), 1–8 (2014). Google Scholar
  21. 21.
    Eich-Soellner, E., Führer, C.: Numerical Methods in Multibody Dynamics. B.G.Teubner Stuttgart (1998)Google Scholar
  22. 22.
    Flores, P., Machado, M., Seabra, E., Tavares da Silva, M.: A parametric study on the Baumgarte stabilization method for forward dynamics of constrained multibody systems. J. Comput. Nonlinear Dyn. 6(1), 011,019 (2011). CrossRefGoogle Scholar
  23. 23.
    García de Jalón, J., Bayo, E.: Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge. Springer, New York (1994)CrossRefGoogle Scholar
  24. 24.
    García Orden, J.: Energy considerations for the stabilization of constrained mechanical systems with velocity projection. Nonlinear Dyn. 60, 49–62 (2010). MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    García Orden, J., Conde, S.: Controllable velocity projection for constraint stabilization in multibody dynamics. Nonlinear Dyn. 68, 245–257 (2012). MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    García Orden, J., Dopico, D.: Multibody Dynamics: Computational Methods and Applications. In: Computational Methods in Applied Sciences. On the Stabilizing Properties of Energy-Momentum Integrators and Coordinate Projections for Constrained Mechanical Systems, vol. 4, pp. 49–68. Springer, Berlin (2007).
  27. 27.
    Gear, C., Leimkuhler, B., Gupta, G.: Automatic integration of euler-lagrange equations with constraints. J. Comput. Appl. Math. 12(13), 77–90 (1985). MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Géradin, M., Cardona, A.: Flexible Multibody Dynamics. A Finite Element Approach. Wiley, Chinchester (2001)Google Scholar
  29. 29.
    González, F., Dopico, D., Pastorino, R., Cuadrado, J.: Behaviour of augmented Lagrangian and Hamiltonian methods for multibody dynamics in the proximity of singular configurations. Nonlinear Dyn. 85(3), 1491–1508 (2016). MathSciNetCrossRefGoogle Scholar
  30. 30.
    González, F., Kövecses, J.: Use of penalty formulations in dynamic simulation and analysis of redundantly constrained multibody systems. Multibody Syst. Dyn. 29(1), 57–76 (2013). MathSciNetCrossRefGoogle Scholar
  31. 31.
    Gutiérrez-López, M.D., Callejo, A., García de Jalón, J.: Computation of independent sensitivities using Maggi’s formulation. In: The 2nd joint international conference on multibody system dynamics. Stuttgart, Germany (2012)Google Scholar
  32. 32.
    Haug, E.: Computer aided optimal design: structural and mechanical systems, chap. Design sensitivity analysis of dynamic systems, in NATO ASI series. Series F, Computer and systems sciences. vol. 27, pp. 705–755. Springer, Berlin (1987).
  33. 33.
    Haug, E.J.: Computer Aided Kinematics and Dynamics of Mechanical Systems: Basic Methods. Allyn and Bacon. Prentice Hall College Div, Boston (1989)Google Scholar
  34. 34.
    Kane, T.R., Levinson, D.A.: Dynamics—Theory and Applications. McGraw-Hill, New York (1985)Google Scholar
  35. 35.
    Naya, M., Cuadrado, J., Dopico, D., Lugris, U.: An efficient unified method for the combined simulation of multibody and hydraulic dynamics: comparison with simplified and co-integration approaches. Arch. Mech. Eng. 58(2), 223–243 (2011). CrossRefGoogle Scholar
  36. 36.
    Newmark, N.M.: A method of computation for structural dynamics. J. Eng. Mech. Div. ASCE 85(EM3), 67–94 (1959)Google Scholar
  37. 37.
    Pagalday, J.M., Avello, A.: Optimization of multibody dynamics using object oriented programming and a mixed numerical-symbolic penalty formulation. Mech. Mach. Theory 32(2), 161–174 (1997). CrossRefGoogle Scholar
  38. 38.
    Pagalday, J., Aranburu, I., Avello, A., García de Jalón, J.: Multibody dynamics optimization by direct differentiation methods using object oriented programming. In: IUTAM symposium on optimization of mechanical systems. pp. 213–220. Springer, Berlin (1996).
  39. 39.
    Pastorino, R., Sanjurjo, E., Luaces, A., Naya, M.A., Desmet, W., Cuadrado, J.: Validation of a real-time multibody model for an X-by-wire vehicle prototype through field testing. J. Comput. Nonlinear Dyn. 10(3), 031,006 (2015). CrossRefGoogle Scholar
  40. 40.
    Paul, B., Krajcinovic, D.: Computer analysis of machines with planar motion. ASME J. Appl. Mech. 37(3), 697–712 (1970). CrossRefGoogle Scholar
  41. 41.
    Schaffer, A.: Stability of the adjoint differential-algebraic equation of the index-3 multibody system equation of motion. SIAM J. Sci. Comput. 26(4), 1432–1448 (2005). MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Schaffer, A.: Stabilized index-1 differential-algebraic formulations for sensitivity analysis of multi-body dynamics. Proc. Inst. Mech. Eng. Part K- J Multi-Body Dyn. 220(3), 141–156 (2006). Google Scholar
  43. 43.
    Schiehlen, W.: Multibody system dynamics: roots and perspectives. Multibody Syst. Dyn. 1(2), 149–188 (1997). MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Serna, M.A., Avilés, R., García de Jalón, J.: Dynamic analysis of plane mechanisms with lower pairs in basic coordinates. Mech. Mach. Theory 17(6), 397–403 (1982). CrossRefGoogle Scholar
  45. 45.
    Sheth, P., Uicker, J.: IMP (integrated mechanism program): a computer-aided design analysis system for mechanisms and linkages. J. Eng. Ind. 94, 454–464 (1972). CrossRefGoogle Scholar
  46. 46.
    Smith, D., Chace, M., Rubens, A.: The automatic generation of a mathematical model for machinery systems. J. Eng. Ind. 95, 629–635 (1973). CrossRefGoogle Scholar
  47. 47.
    Wehage, R., Haug, E.: Generalized coordinate partitioning for dimension reduction in analysis of constrained mechanical systems. J. Mech. Des. 104(1), 247–255 (1982). CrossRefGoogle Scholar
  48. 48.
    Zhu, Y., Dopico, D., Sandu, C., Sandu, A.: Dynamic response optimization of complex multibody systems in a penalty formulation using adjoint sensitivity. J. Comput. Nonlinear Dyn. 10(3), 1–9 (2015). Google Scholar

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Authors and Affiliations

  1. 1.Laboratorio de Ingeniería MecánicaUniversidade da CoruñaFerrolSpain
  2. 2.Departamento de Ingeniería MecánicaUniversidad Politécnica de CartagenaCartagenaSpain
  3. 3.Departamento de Ingeniería Mecánica y FabricaciónUniversidad de SevillaSevilleSpain

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