Research Papers

Direct Sensitivity Analysis of Multibody Systems: A Vehicle Dynamics Benchmark

[+] Author and Article Information
Alfonso Callejo

Department of Aerospace Engineering,
University of Maryland,
College Park, MD 20742
e-mail: callejo@umd.edu

Daniel Dopico

Laboratorio de Ingeniería Mecánica,
Universidade da Coruña,
Mendizábal s/n,
Ferrol 15403, Spain
e-mail: ddopico@udc.es

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 1, 2018; final manuscript received November 3, 2018; published online January 7, 2019. Assoc. Editor: Kyung Choi.

J. Comput. Nonlinear Dynam 14(2), 021004 (Jan 07, 2019) (9 pages) Paper No: CND-18-1250; doi: 10.1115/1.4041960 History: Received June 01, 2018; Revised November 03, 2018

Algorithms for the sensitivity analysis of multibody systems are quickly maturing as computational and software resources grow. Indeed, the area has made substantial progress since the first academic methods and examples were developed. Today, sensitivity analysis tools aimed at gradient-based design optimization are required to be as computationally efficient and scalable as possible. This paper presents extensive verification of one of the most popular sensitivity analysis techniques, namely the direct differentiation method (DDM). Usage of such method is recommended when the number of design parameters relative to the number of outputs is small and when the time integration algorithm is sensitive to accumulation errors. Verification is hereby accomplished through two radically different computational techniques, namely manual differentiation and automatic differentiation, which are used to compute the necessary partial derivatives. Experiments are conducted on an 18-degree-of-freedom, 366-dependent-coordinate bus model with realistic geometry and tire contact forces, which constitutes an unusually large system within general-purpose sensitivity analysis of multibody systems. The results are in good agreement; the manual technique provides shorter runtimes, whereas the automatic differentiation technique is easier to implement. The presented results highlight the potential of manual and automatic differentiation approaches within general-purpose simulation packages, and the importance of formulation benchmarking.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.


González, M. , Dopico, D. , Lugrís, U. , and Cuadrado, J. , 2006, “ A Benchmarking System for MBS Simulation Software: Problem Standardization and Performance Measurement,” Multibody Syst. Dyn., 16(2), pp. 179–190. [CrossRef]
Saltelli, A. , Chan, K. , and Scott, E. M. , 2000, Sensitivity Analysis, Wiley, New York.
Haug, E. J. , and Arora, J. S. , 1979, Applied Optimal Design: Mechanical and Structural Systems, Wiley, New York.
Haftka, R. T. , and Adelman, H. M. , 1989, “ Recent Developments in Structural Sensitivity Analysis,” Struct. Optim., 1(3), pp. 137–151. [CrossRef]
Barthelemy, B. , and Haftka, R. T. , 1990, “ Accuracy Analysis of the Semi-Analytical Method for Shape Sensitivity Calculation,” Mech. Struct. Mach., 18(3), pp. 407–432. [CrossRef]
Olhoff, N. , and Rasmussen, J. , 1991, “ Study of Inaccuracy in Semi-Analytical Sensitivity Analysis—A Model Problem,” Struct. Optim., 3(4), pp. 203–213. [CrossRef]
Newman, J. C. , Anderson, W. K. , and Whitfield, D. L. , 1998, “ Multidisciplinary Sensitivity Derivatives Using Complex Variables,” Mississippi State University, Starkville, MS, Technical Report No. MSSU-EIRS-ERC-98-08.
Martins, J. R. R. A. , Sturdza, P. , and Alonso, J. J. , 2003, “ The Complex-Step Derivative Approximation,” Trans. Math. Software, 29(3), pp. 245–262. [CrossRef]
Griewank, A. , 1989, “ On Automatic Differentiation,” Mathematical Programming: Recent Developments and Applications, M. Iri , and K. Tanabe , eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 83–108.
Griewank, A. , and Walther, A. , 2008, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Society for Industrial and Applied Mathematics, Philadelphia, PA.
Bischof, C. H. , 1996, “ On the Automatic Differentiation of Computer Programs and an Application to Multibody Systems,” IUTAM Symposium on Optimization of Mechanical Systems (Solid Mechanics and its Applications, Vol. 43), D. Bestle , and W. Schiehlen , eds., Springer, Dordrecht, The Netherlands, pp. 41–48.
Dürrbaum, A. , Klier, W. , and Hahn, H. , 2002, “ Comparison of Automatic and Symbolic Differentiation in Mathematical Modeling and Computer Simulation of Rigid-Body Systems,” Multibody Syst. Dyn., 7(4), pp. 331–355. [CrossRef]
Neto, M. A. , Ambrósio, J. A. C. , and Leal, R. P. , 2009, “ Sensitivity Analysis of Flexible Multibody Systems Using Composite Materials Components,” Int. J. Numer. Methods Eng., 77(3), pp. 386–413. [CrossRef]
Callejo, A. , and García de Jalón, J. , 2014, “ A Hybrid Direct-Automatic Differentiation Method for the Computation of Independent Sensitivities in Multibody Systems,” Int. J. Numer. Methods Eng., 100(12), pp. 933–952. [CrossRef]
Callejo, A. , Narayanan, S. H. K. , García de Jalón, J. , and Norris, B. , 2014, “ Performance of Automatic Differentiation Tools in the Dynamic Simulation of Multibody Systems,” Adv. Eng. Software, 73, pp. 35–44. [CrossRef]
Krishnaswami, P. , Whage, R. A. , and Haug, E. J. , 1983, “ Design Sensitivity Analysis of Constrained Dynamic Systems by Direct Differentiation,” SIAM, Iowa City, IA, Technical Report No. 83-5.
Krishnaswami, P. , and Bhatti, M. A. , 1984, “ A General Approach for Design Sensitivity Analysis of Constrained Dynamic Systems,” ASME Paper No. 84-DET-132.
Chang, C. O. , and Nikravesh, P. E. , 1985, “ Optimal Design of Mechanical Systems With Constraint Violation Stabilization Method,” ASME J. Mech. Transm. Autom. Des., 107(4), pp. 493–498. [CrossRef]
Haug, E. J. , 1987, “ Design Sensitivity Analysis of Dynamic Systems,” Computer Aided Optimal Design: Structural and Mechanical Systems, Springer, Berlin, pp. 705–755.
Serban, R. , and Freeman, J. S. , 1996, “ Direct Differentiation Methods for the Design Sensitivity of Multibody Dynamic Systems,” ASME Paper No. 96-DETC/DAC-1087.
Dopico, D. , Sandu, A. , and Sandu, C. , 2015, “ Direct and Adjoint Sensitivity Analysis of ODE Multibody Formulations,” ASME J. Comput. Nonlinear Dyn., 10(1), p. 011012. [CrossRef]
Dopico, D. , González, F. , Luaces, A. , Saura, M. , and García-Vallejo, D. , 2018, “ Direct Sensitivity Analysis of Multibody Systems With Holonomic and Nonholonomic Constraints Via an Index-3 Augmented Lagrangian Formulation With Projections,” Nonlinear Dyn., 93(4), pp. 2039–2056.
Haug, E. J. , and Arora, J. S. , 1978, “ Design Sensitivity Analysis of Elastic Mechanical Systems,” Comput. Methods Appl. Mech. Eng., 15(1), pp. 35–62. [CrossRef]
Bestle, D. , and Eberhard, P. , 1992, “ Analyzing and Optimizing Multibody Systems,” J. Struct. Mech., 20(1), pp. 67–92.
Bestle, D. , and Seybold, J. , 1992, “ Sensitivity Analysis of Constrained Multibody Systems,” Arch. Appl. Mech., 62(3), pp. 181–190.
Cao, Y. , Li, S. , and Petzold, L. , 2002, “ Adjoint Sensitivity Analysis for Differential-Algebraic Equations: Algorithms and Software,” J. Comput. Appl. Math., 149(1), pp. 171–191. [CrossRef]
Cao, Y. , Li, S. , Petzold, L. , and Serban, R. , 2003, “ Adjoint Sensitivity Analysis for Differential-Algebraic Equations: The Adjoint DAE System and Its Numerical Solution,” SIAM J. Sci. Comput., 24(3), pp. 1076–1089. [CrossRef]
Maly, T. , and Petzold, L. R. , 1996, “ Numerical Methods and Software for Sensitivity Analysis of Differential-Algebraic Systems,” J. Appl. Numer. Math., 20(1–2), pp. 57–79. [CrossRef]
García de Jalón, J. , and Bayo, E. , 1994, Kinematic and Dynamic Simulation of Multibody Systems. The Real-Time Challenge, Springer-Verlag, New York.
García de Jalón, J. , Callejo, A. , and Hidalgo, A. F. , 2012, “ Efficient Solution of Maggi's Equations,” ASME J. Comput. Nonlinear Dyn., 7(2), p. 021003. [CrossRef]
Griewank, A. , Juedes, D. , and Utke, J. , 1996, “ ADOL-C, A Package for the Automatic Differentiation of Algorithms Written in C/C++,” ACM Trans. Math. Software, 22(2), pp. 131–167. [CrossRef]
Kowarz, A. , and Walther, A. , 2006, “ Optimal Checkpointing for Time-Stepping Procedures in ADOL-C,” Computational Science—ICCS 2006, V. N. Alexandrov , G. D. van Albada , P. M. A. Sloot , and J. Dongarra , eds., Springer, Berlin, pp. 541–549.
Callejo, A. , 2013, “ Dynamic Response Optimization of Vehicles Through Efficient Multibody Formulations and Automatic Differentiation Techniques,” Ph.D. thesis, Universidad Politécnica de Madrid, Madrid, Spain. http://oa.upm.es/22555/2/ALFONSO_CALLEJO_GOENA_VERSION_REVISADA.pdf
Callejo, A. , García de Jalón, J. , Luque, P. , and Mántaras, D. A. , 2015, “ Sensitivity-Based, Multi-Objective Design of Vehicle Suspension Systems,” ASME J. Comput. Nonlinear Dyn., 10(3), p. 031008. [CrossRef]
Callejo, A. , and García de Jalón, J. , 2015, “ Vehicle Suspension Identification Via Algorithmic Computation of State and Design Sensitivities,” ASME J. Mech. Des., 137(2), p. 021403. [CrossRef]
Zhu, Y. , Dopico, D. , Sandu, C. , and Sandu, A. , 2015, “ Dynamic Response Optimization of Complex Multibody Systems in a Penalty Formulation Using Adjoint Sensitivity,” ASME J. Comput. Nonlinear Dyn., 10(3), pp. 1–9.
Dopico, D. , Luaces, A. , Lugrís, U. , Saura, M. , González, F. , Sanjurjo, E. , and Pastorino, R. , 2009–2016, “ MBSLIM: Multibody Systems en Laboratorio de Ingeniería Mecánica,” (epub).


Grahic Jump Location
Fig. 2

Static equilibrium dynamic response: wheel z-coordinates

Grahic Jump Location
Fig. 3

Static equilibrium sensitivities: z-coordinate of chassis COG with respect to front and rear stiffness (ks), damping coefficient (cd), and natural length (l0)

Grahic Jump Location
Fig. 4

Steering profile and chassis y-position

Grahic Jump Location
Fig. 5

Lane change dynamic response: angular velocity of wheels

Grahic Jump Location
Fig. 6

Static equilibrium sensitivities: z-coordinate of chassis COG with respect to front and rear stiffness (ks), damping coefficient (cd), and natural length (l0)



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In