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Order-(n+m) direct differentiation determination of design sensitivity for constrained multibody dynamic systems

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Abstract

With the complexity and large dimensionality of many modern multibody dynamic applications, the efficiency of the sensitivity evaluation methods used can greatly impact the overall computation cost and as such can greatly limit the usefulness of the sensitivity information. Most current direct differentiation approaches suffer from prohibitive computational cost, which may be as great as O(n4+n2m2+nm3) (for systems with n generalized coordinates and m algebraic constraints). This paper presents a concise and computationally efficient sensitivity analysis scheme to facilitate such sensitivity calculations. A unique feature of this scheme is its use of recursive procedures to directly embed the algebraic constraint relations in forming and simultaneously solving a minimal set of equations. This results in far fewer operations than more traditional, or so-called O(n), counterparts. The algorithm determines the derivatives of generalized accelerations in O(n+m) operations overall. The resulting equations are “exact” to integration accuracy and enforce constraints exactly at both the velocity and acceleration levels.

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References

  1. Anderson, K.S. 1990: Recursive Derivation of Explicit Equations of Motion for Efficient Dynamic/Control Simulation of Large Multibody Systems, Ph.D. Dissertation, Stanford University, CA

  2. Anderson, K.S. 1992: An order-n formulation for motion simulation of general constrained multi-rigid-body systems. Computers and Structures 43(3), 565–572

    Google Scholar 

  3. Anderson, K.S. 2001: Improved order-n Performance Algorithm for the Simulation of Constrained Multi-Rigid-Body Dynamic Systems. Proceedings ASME Design Engineering Technical Conference (DETC 2001), Pittsburgh Sept. 9–13, 2001. Paper DETC2001/VIB-21334

  4. Anderson, K.S.; Critchley, J.H. 2003: Improved Order-n Performance Algorithm for the Simulation of Constrained Multi-Rigid-Body Systems. Multibody System Dynamics 9, 185–212

    Google Scholar 

  5. Barthelemy, J.-F.M.; Hall, L.E. 1995: Automatic differentiation as a tool in engineering design. Structural Optimization 9, 76–82

    Google Scholar 

  6. Bestle, D.; Eberhard, P. 1992: Analyzing and optimizing multibody systems. Mechanics of Structures and Machines 20(1), 67–92

  7. Bestle, D.; Seybold, J. 1992: Sensitivity analysis of constrained multibody systems. Archive of Applied Mechanics 62, 181–190

    Google Scholar 

  8. Bischof, C.H. 1996: On the automatic differentiation of computer programs and an application to multibody systems. IUTAM Symposium on Optimization of Mechanical Systems, Kluwer Academic, pp. 41–48

  9. Chang, C.O.; Nikravesh, P.E. 1985: Optimal design of mechanical systems with constraint violation stabilization method. Journal of Mechanisms, Transmissions, and Automation in Design 107, 493–498

    Google Scholar 

  10. Dias, J.M.P.; Pereira, M.S. 1997: Sensitivity analysis of rigid-flexible multibody systems. Multibody System Dynamics 1(3), 303–322

    Google Scholar 

  11. Eberhard, P. 1996: Adjoint variable method for sensitivity analysis of multibody systems interpreted as a continuous, hybrid form of automatic differentiation. Proceedings of the 2nd International Workshop on Computational Differentiation, Santa Fe, NM, USA, 12–14 Feb, pp. 319–328

  12. Featherstone, R. 1983: The calculation of robot dynamics using articulated-body inertias. International Journal of Robotics Research 2(1), 13–30

    Google Scholar 

  13. Hollerbach, J.M. 1980: A recursive lagrangian formulation of manipulator dynamics and a comparative study of dynamics formulation complexity. IEEE Transaction on Systems, Man, and Cybernetics SMC-10(11), 730–736

    Google Scholar 

  14. Haug, E.J.; Arora, J.S. 1979: Applied Optimal Design. New York: John Wiley & Sons, Inc.

  15. Haug, E.J.; Wehage, R.A.; Mani, N.K. 1984: Design sensitivity analysis of large-scaled constrained dynamic mechanical systems. Transactions of the ASME 106, 156–162

    Google Scholar 

  16. Hsu, Y. 2001: Order-n Analytic Sensitivity Analysis for Multibody Dynamic Systems Optimization. Ph.D. Dissertation Department of Mechanical Engineering, Aeronautical Engineering and Mechanics. Rensselaer Polytechnic Institute, Nov.

  17. Hsu, Y.; Anderson, K.S. 2001: Low operational order analytic sensitivity analysis for tree-type multibody dynamic systems. Journal of Guidance, Control, and Dynamics 24(6), 1133–1143

    Google Scholar 

  18. Hsu, Y.; Anderson, K.S. 2002: Recursive Sensitivity Analysis for Constrained Multi-rigid-body Dynamic Systems Design Optimization. Structural and Multidisciplinary Optimization 24(4), 312–324

  19. Jain, A.; Rodriguez, G. 1999: Sensitivity analysis for multibody systems using spatial operators. International Conference on Scientific Computation and Differential Equations (SciCADE’99), Aug., Fraser Island, Queensland, Australia.

  20. Kane, T.R.; Levinson, D.A. 1985: Dynamics: Theory and Applications. New York: McGraw-Hill, Inc.

  21. Pagalday, J.M.; Aranburu, I. 1996: Multibody dynamics optimization by direct differentiation methods using object oriented programming. IUTAM Symposium on Optimization of Mechanical Systems, Kluwer Academic, pp. 213–220

  22. Pradhan, S.; Modi, V.J.; Misra, A.K. 1997: Order n formulation for flexible multibody systems in tree topology: lagrangian approach. Journal of Guidance, Control, and Dynamics 20(4), 665–672

    Google Scholar 

  23. Rosenthal, D.E. 1990: An order n formulation for robotic systems. Journal of Astronautical Sciences 38(4), 511–529

    Google Scholar 

  24. Schaechter, D.B.; Levinson, D.A.; Kane, T.R. 1997: AUTOLEV User’s Manual Version 3.2, Online Dynamics, Inc., Sunnyvale, CA

  25. Serban, R.; Haug, E.J. 1998: Analytical derivatives for multibody system analyses. Mechanics of Structures and Machines 26(2), 145–173

    Google Scholar 

  26. Tak, T. 1990: A recursive approach to design sensitivity analysis of multibody systems using direct differentiation. Ph.D. Dissertation, The University of Iowa

  27. Stejskal, V.; Valášek, M. 1996: Kinematics and Dynamics of Machinery. New York: Marcel Dekker, Inc.

  28. Wehage, R.A.; Haug, E.J. 1982: Generalized coordinate partitioning for dimension reduction in analysis of constrained dynamic systems. Transaction of ASME, Journal of Machine Design 104, 247–255

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Authors and Affiliations

  1. Rensselaer Polytechnic Institute, Department of Mechanical, Aerospace, and Nuclear Engineering, 12180-3590, Troy, NY, USA

    K.S. Anderson & Y. Hsu

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  1. K.S. Anderson
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  2. Y. Hsu
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Correspondence to K.S. Anderson.

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Anderson, K., Hsu, Y. Order-(n+m) direct differentiation determination of design sensitivity for constrained multibody dynamic systems. Struct Multidisc Optim 26, 171–182 (2004). https://doi.org/10.1007/s00158-003-0336-1

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  • DOI: https://doi.org/10.1007/s00158-003-0336-1

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