The present paper illustrates the potential of the adjoint method for a wide range of optimization problems in multibody dynamics such as inverse dynamics and parameter identification. Although the equations and matrices included show a complicated structure, the additional effort when combining the standard forward solver to the adjoint backward solver is kept in limits. Therefore, the adjoint method shows an efficient way to incorporate inverse dynamics to engineering multibody applications, e.g., trajectory tracking or parameter identification in the field of robotics. The present paper studies examples for both, parameter identification and optimal control, and shows the potential of the adjoint method in solving classical optimization problems in multibody dynamics.
Issue Section:
Research Papers
References
1.
Haug
, E.
, and Ehle
, P.
, 1982
, “Second-Order Design Sensitivity Analysis of Mechanical System Dynamics
,” Int. J. Numer. Methods Eng.
, 18
(11
), pp. 1699
–1717
.10.1002/nme.16201811092.
Haug
, E.
, Wehage
, R.
, and Mani
, N.
, 1984
, “Design Sensitivity Analysis of Large-Scaled Constrained Dynamic Mechanical Systems
,” Trans. ASME
, 106
(2
), pp. 156
–162
.10.1115/1.32608563.
Bestle
, D.
, and Eberhard
, P.
, 1992
, “Analyzing and Optimizing Multibody Systems
,” Mech. Struct. Mach.
, 20
(1
), pp. 67
–92
.10.1080/089054592089051614.
Eberhard
, P.
, 1996
, “Adjoint Variable Method for Sensitivity Analysis of Multibody Systems Interpreted as a Continuous, Hybrid Form of Automatic Differentiation
,” Computational Differentiation Techniques, Applications, and Tools, M. Berz, C. Bischof, G. Corliss, and A. Griewank, A., eds., SIAM, Philadelphia
, PA, pp. 319
–328
.5.
Lions
, J.
, 1971
, Optimal Control of Systems Governed by Partial Differential Equations
, Springer-Verlag
, New York
.6.
Giles
, M.
, and Pierce
, N.
, 2000
, “An Introduction to the Adjoint Approach to Design
,” Flow, Turbul. Combust.
, 65
(3–4
), pp. 393
–415
.10.1023/A:10114304100757.
Jameson
, A.
, 2003
, “Aerodynamic Shape Optimization Using the Adjoint Method” (VKI Lecture Series on Aerodynamic Drag Prediction and Reduction)
, von Karman Institute of Fluid Dynamics
, Rhode St. Genese, Belgium
, pp. 3
–7
.8.
Bryson
, A. E.
, and Ho
, Y. C.
, 1975
, Applied Optimal Control
, Hemisphere
, Washington, DC
.9.
Anderson
, W.
, and Venkatakrishnan
, V.
, 1999
, “Aerodynamic Design Optimization on Unstructured Grid With a Continuous Adjoint Formulation
,” Comput. Fluid
, 28
(4–5), pp. 443
–480
. 10.1016/S0045-7930(98)00041-310.
Nadarajah
, S.
, and Jameson
, A.
, 2000
, “A Comparison of the Continuous and Discrete Adjoint Approach to Automatic Aerodynamic Optimisation
,” AIAA
Paper No. 2000-0067. 11.
Oberai
, A.
, Gokhale
, N.
, and Feijóo
, G.
, 2003
, “Solution of Inverse Problems in Elasticity Imaging Using the Adjoint Method
,” Inverse Probl.
, 19
(2
), pp. 297
–313
.10.1088/0266-5611/19/2/30412.
McNamara
, A.
, Treuille
, A.
, Popović
, Z.
, and Stam
, J.
, 2004
, “Fluid Control using the Adjoint Method
,” ACM Trans. Graphics (TOG) - Proc ACM SIGGRAPH
, 23
(3
), pp. 449
–456
.10.1145/1015706.101574413.
Taylor
, T.
, Palacios
, F.
, Duraisamy
, K.
, and Alonso
, J.
, 2013
, “A Hybrid Adjoint Approach Applied to Turbulent Flow Simulations
,” 21st AIAA Computational Fluid Dynamics Conference
, San Diego
, June 24–27.14.
Bottasso
, C.
, Croce
, A.
, Ghezzi
, L.
, and Faure
, P.
, 2004
, “On the Solution of Inverse Dynamics and Trajectory Optimization Problems for Multibody Systems
,” Multibody Syst. Dyn.
, 11
(1
), pp. 1
–22
.10.1023/B:MUBO.0000014875.66058.7415.
Bertolazzi
, E.
, Biral
, F.
, and Lio
, M. D.
, 2005
, “Symbolic-Numeric Indirect Method for Solving Optimal Control Problems for Large Multibody Systems
,” Multibody Syst. Dyn.
, 13
(2
), pp. 233
–252
.10.1007/s11044-005-3987-416.
Schaffer
, A.
, 2005
, “On the Adjoint Formulation of Design Sensitivity Analysis of Multibody Dynamics
,” Ph.D. dissertation, University of Iowa, Iowa City, IA, http://ir.uiowa.edu/etd/9317.
Petzold
, L.
, Li
, S.
, Cao
, Y.
, and Serban
, R.
, 2006
, “Sensitivity Analysis for Differential-Algebraic Equations and Partial Differential Equations
,” Comput. Chem. Eng.
, 30
(10–12
), pp. 1553
–1559
.10.1016/j.compchemeng.2006.05.01518.
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
.10.1016/S0377-0427(02)00528-919.
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
.10.1137/S106482750138063020.
Shabana
, A.
, 1997
, “Definition of the Slopes and the Finite Element Absolute Nodal Coordinate Formulation
,” Multibody Syst. Dyn.
, 1
(3
), pp. 339
–348
.10.1023/A:100974080046321.
Held
, A.
, and Seifried
, R.
, 2013
, “Gradient-Based Optimization of Flexible Multibody Systems Using the Absolute Nodal Coordinate Formulation
,” Proceedings of the ECCOMAS Thematic Conference Multibody Dynamics 2013
, Zagreb, Croatia
, July 1–4.22.
Pi
, T.
, Zhang
, Y.
, and Chen
, L.
, 2012
, “First Order Sensitivity Analysis of Flexible Multibody Systems Using the Absolute Nodal Coordinate Formulation
,” Multibody Syst. Dyn.
, 27
(2
), pp. 153
–171
.10.1007/s11044-011-9269-423.
Ding
, J.-Y.
, Pan
, Z.-K.
, and Chen
, L.-Q.
, 2007
, “Second Order Adjoint Sensitivity Analysis of Multibody Systems Described by Differential-Algebraic Equations
,” Multibody Syst. Dyn.
, 18
(4
), pp. 599
–617
.10.1007/s11044-007-9080-424.
Vyasarayani
, C.
, Uchida
, T.
, and McPhee
, J.
, 2012
, “Nonlinear Parameter Identification in Multibody Systems Using Homotopy Continuation
,” ASME J. Comput. Nonlinear Dyn.
, 7
(1
), p. 011012
.10.1115/1.400488525.
Ding
, J.-Y.
, Pan
, Z.-K.
, and Chen
, L.-Q.
, 2012
, “Parameter Identification of Multibody Systems Based on Second Order Sensitivity Analysis
,” Int. J. Nonlinear Mech.
, 47
(10
), pp. 1105
–1110
.10.1016/j.ijnonlinmec.2011.09.00926.
Özyurt
, D. B.
, and Barton
, P. I.
, 2005
, “Cheap Second Order Directional Derivatives of Stiff ODE Embedded Functionals
,” SIAM J. Sci. Comput.
, 26
(2
), pp. 1725
–1743
.10.1137/03060158227.
Steiner
, W.
, and Reichl
, S.
, 2011
, “A contribution to Inverse Dynamic Problems in Multibody Systems
,” Multibody Dynamics 2011, ECCOMAS Thematic Conference, Brussels, Belgium, July 4–7
.28.
Steiner
, W.
, and Reichl
, S.
, 2012
, “The Optimal Control Approach to Dynamical Inverse Problems
,” ASME J. Dyn. Syst., Meas., Control
, 134
(2
), p. 021010
.10.1115/1.400536529.
Gear
, C.
, Gupta
, G.
, and Leimkuhler
, B.
, 1985
, “Automatic Integration of the Euler-Lagrange Equations With Constraints
,” J. Comp. Appl. Math.
, 12/13
, pp. 77
–90
.10.1016/0377-0427(85)90008-130.
Hilbert
, H.
, Hughes
, T.
, and Taylor
, R.
, 1977
, “Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics
,” Earthquake Eng. Struct. Dyn.
, 5
(3
), pp. 283
–292
.10.1002/eqe.429005030631.
Negrut
, D.
, Rampalli
, R.
, Ottarsson
, G.
, and Sajdak
, A.
, 2005
, “On the Use of the HHT Method in the Context of Index 3 Differential Algebraic Equations of Multibody Dynamics
,” ASME
Paper No. DETC2005-85096. 10.1115/DETC2005-8509632.
Süli
, E.
, and Mayers
, D.
, 2003
, An Introduction to Numerical Analysis
, Cambridge University Press, Cambridge, UK.33.
Kirk
, D.
, 2004
, Optimal Control Theory
, Dover Publications
, Mineola, NY
.34.
Rao
, S.
, 2009
, Engineering Optimization: Theory and Practice
, 4th ed., Wiley
, Hoboken, NJ
.35.
Blajer
, W.
, and Kołodziejczyk
, K.
, 2004
, “A Geometric Approach to Solving Problems of Control Constraints: Theory and a DAE Framework
,” Multibody Syst. Dyn.
, 11
(4
), pp. 343
–364
.10.1023/B:MUBO.0000040800.40045.5136.
Betsch
, P.
, Uhlar
, S.
, and Quasem
, M.
, 2009
, “Numerical Integration of Mechanical Systems With Mixed Holonomic and Control Constraints
,” Proceedings of the ECCOMAS Thematic Conference on Multibody Dynamics
, K.
Arczewski
, J.
Fraczek
, and M.
Wojtyra
, eds., Warsaw University of Technology, Poland, June 29–July 2.37.
Glück
, T.
, Eder
, A.
, and Kugi
, A.
, 2013
, “Swing-Up Control of a Triple Pendulum on a Cart With Experimental Validation
,” Automatica
, 49
(3
), pp. 801
–808
.10.1016/j.automatica.2012.12.006Copyright © 2015 by ASME
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