Eliminating constraint drift in the numerical simulation of constrained dynamical systems

Computer Methods in Applied Mechanics and Engineering

Volumn 198, Issue 37-40, 2009, Pages 3151-3160

  Braun, David J ^a   Goldfarb, Michael ^a  

^a VANDERBILT UNIVERSITY   (United States)

Author keywords

Constrained dynamics; Numerical integration; Pseudoinverse

Indexed keywords

BASIC IDEA; COMPUTATIONAL ENVIRONMENTS; CONSTRAINED DYNAMICAL SYSTEM; CONSTRAINED DYNAMICS; CONSTRAINED MOTION; EQUATION OF MOTION; ERROR ACCUMULATION; EXPLICIT EQUATIONS; FINITE WORDS; INITIAL CONDITIONS; ITERATIVE CONSTRAINT; KINEMATIC SINGULARITY; MATRIX; MULTIBODY SIMULATIONS; NUMERICAL INTEGRATION; NUMERICAL SIMULATION; ODE SOLVER; PSEUDOINVERSE; REDUNDANT CONSTRAINTS; VIRTUAL FORCES;

DYNAMICAL SYSTEMS; DYNAMICS;

EQUATIONS OF MOTION;

EID: 67949108051     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2009.05.013     Document Type: Article

References (56)

1. J.L. Lagrange, Mecanique Analytique, Mme Ve Courcier, Paris, 1787.
2. Gauss C.F. Über ein neues algemeines grundgesetz der mechanik. Zeitschrift für die Reine und Angewandte Mathematik 4 (1829) 232-235
3. G.A. Maggi, Principii della Teoria Mathematica del Movimento dei Corpi: Corso di Meccanica Razionale, Ulrico Hoepli, Milano, 1896.
4. Gibbs J.W. On the fundamental formulae of dynamics. Am. J. Math. 2 1 (1879) 49-64
5. Appell P. Sur une forme generale des equations de la dynamique. C.R. Acad. Sci. Paris 129 (1899) 459-460
6. Kane T.R., and Levinson D.A. Dynamics: Theory and Applications (1985), McGraw-Hill, New York
7. Udwadia F.E., and Kalaba R.E. A new perspective on constrained motion. Proc. R. Soc. Lond. A 439 (1992) 407-410
8. Udwadia F.E., and Kalaba R.E. Analytical Dynamics: A New Approach (1996), Cambridge University Press, Cambridge, England
9. Udwadia F.E., and Kalaba R.E. On the foundations of analytical dynamics. Int. J. Nonlinear Mech. 37 (2002) 1079-1090
10. Pars L.A. A Treatise on Analytical Dynamics (1965), John Wiley and Sons, New York
11. Ju.I. Neimark, N.A. Fufaev, Dynamics of nonholonomic systems, AMS Providence, 1972.
12. Gantmacher F. Lectures in Analytical Mechanics (1975), Mir Publisher, Moscow (English translation)
13. Goldstein H. Classical Mechanics (1980), Addison-Wesley, Reading, MA
14. Chetaev N.G. Theoretical Mechanics (1989), Mir Publisher, Moscow (English translation)
15. Lurie A.I. Analytical Mechanics (2002), Springer-Verlag, New York
16. Baumgarte J. Stabilization of constraints and integrals of motion in dynamical systems. Comput. Method Appl. Mech. Engrg. 1 (1972) 1-16
17. Gear C.W., Leimkuhler B., and Gupta G.K. Automatic integration of Euler-Lagrange equations with constraints. J. Comput. Appl. Math. 12-13 (1985) 77-90
18. Lötstedt P., and Petzold L. Numerical solution of nonlinear differential equations with algebraic constraints. I: Convergence results for backward differentiation formulas. Math. Comput. 46 174 (1986) 491-516
19. Führer C., and Leimkuhler B. Numerical solution of differential-algebraic equations for constrained mechanical motion. Numer. Math. 59 (1991) 55-69
20. Petzold L.R. Numerical solution of differential-algebraic equations in mechanical systems simulation. Physica D 60 1-4 (1992) 269-279
21. ten Dam A.A. Stable numerical integration of dynamical systems subject to equality state-space constraints. J. Engrg. Math. 26 (1992) 315-337
22. Eich E. Convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints. SIAM J. Numer. Anal. 30 5 (1993) 1467-1482
23. Bayo E., and Ledesma R. Augmented lagrangian and mass-orthogonal projection methods for constrained multibody dynamics. Nonlinear Dynam. 9 1-2 (1996) 113-130
24. Blajer W. Elimination of constraint violation and accuracy aspects in numerical simulation of multibody systems. Multibody Syst. Dynam. 7 (2002) 265-284
25. 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 (2005) 834-849
26. Schiehlen W. Multibody system dynamics: roots and perspectives. Multibody Syst. Dynam. 1 (1997) 149-188
27. Brogliato B., ten Dam A.A., Paoli L., Génot F., and Abadie M. Numerical simulation of finite dimensional multibody nonsmooth mechanical systems. ASME Appl. Mech. Rev. 55 2 (2002) 107-150
28. J. D'Alembert, Traite de Dynamique, Paris, 1743.
29. Moreau J.J. Quadratic programming in mechanics: dynamics of one-sided constraints. J. SIAM Control 4 1 (1966) 153-158
30. Lötstedt P. Mechanical systems of rigid bodies subject to unilateral constraints. SIAM J. Appl. Math. 42 2 (1982) 281-296
31. Ben-Israel A., and Greville T.N.E. Generalized Inverse: Theory and Applications (2003), Springer
32. Gear C.W. Differential-algebraic equation index transformations. SIAM J. Sci. Stat. Comput. 9 (1988) 39-47
33. L. Petzold, DASSL: A Differential/Algebraic System Solver. .
34. Brenan K.E., Campbell S.L., and Petzold L.R. Numerical Solutions of Initial-Value Problems in Differential-Algebraic Equations (1989), Elsevier Science, NY
35. de Jalón J.G., and Bayo E. Kinematic and Dynamic Simulation of Multibody Systems The Real-Time Challenge (1994), Springer-Verlag, New York
36. Hairer E., and Wanner G. Solving Ordinary Differential Equations. II: Stiff and Differential-Algebraic Problems. Series in Computational Mathematics. second ed. (1996), Springer-Verlag
37. Golub G., and Loan C.V. Matrix Computations. third ed. (1996), The John Hopkins University Press
38. Leimkuhler B., Petzold L.R., and Gear C.W. Approximation methods for the consistent initialization of differential-algebraic equations. SIAM J. Numer. Anal. 28 1 (1991) 205-226
39. Nikravesh P.E. Initial condition correction in multibody dynamics. Multibody Syst. Dynam. 18 (2008) 107-115
40. de Jalón J.G., Unda J., and Avello A. Natural coordinates for the computer analysis of multibody systems. Comput. Method Appl. Mech. Engrg. 59 (1986) 309-327
41. Kraus C., Winckler M., and Bock H.G. Modeling mechanical DAE using natural coordinates. Math. Comput. Model. Dynam. Syst. 7 2 (2001) 145-158
42. Haug E.J. Computer aided kinematics and dynamics of mechanical systems. Basic Methods vol. I (1989), Allyn and Bacon, Boston
43. Serban R., Negrut D., Haug E.J., and Potra F.A. A topology based approach for exploiting sparsity in multibody dynamics in cartesian formulation. Mech. Struct. Mach. 25 3 (1997) 379-396
44. Arnold M., Fuchs A., and Führer C. Efficient corrector iteration for DAE time integration in multibody dynamics. Comput. Method Appl. Mech. Engrg. 195 50-51 (2006) 6958-6973
45. Laulusa A., and Bauchau O.A. Review of classical approaches for constraint enforcement in multibody systems. J. Comput. Nonlinear Dynam. 3 (2008) 011004
46. Bauchau O.A., and Laulusa A. Review of contemporary approaches for constraint enforcement in multibody systems. J. Comput. Nonlinear Dynam. 3 (2008) 011005
47. Baumgarte J. A new method of stabilization for holonomic constraints. J. Appl. Mech. 50 (1983) 869-870
48. Chang C.O., and Nikravesh P.E. An adaptive constraint violation stabilization method for dynamic analysis of mechanical systems. J. Mech. Trans. Automat. Des. 107 (1985) 488-492
49. Ascher U.M., Chin H., and Reich S. Stabilization of DAEs and invariant manifolds. Numer. Math. 67 (1994) 131-149
50. Lin S.T., and Hong M.H. Stabilization method for numerical integration of multibody mechanical systems. J. Mech. Des. 120 (1998) 565-572
51. Blajer W. A geometrical interpretation and uniform matrix formulation of multibody system dynamics. Z. Angew. Math. Mech. 81 4 (2001) 247-259
52. Wehage R.A., and Haug E.J. Generalized coordinate partitioning for dimension reduction in analysis of constrained dynamic systems. J. Mech. Des. 104 (1982) 247-255
53. Bayo E., and Avello A. Singularity free augmented lagrangian algorithms for constraint multibody dynamics. Nonlinear Dynam. 5 (1994) 209-231
54. Udwadia F.E. A new perspective on the tracking control of nonlinear structural and mechanical systems. Proc. R. Soc. Lond. A 459 (2003) 1783-1800
55. Winter D.A. Biomechanics and Motor Control of Human Movement. second ed. (1990), Wiley-Interscience, New York
56. Blajer W., and Kolodziejczyk K. A geometric approach to solving problems of control constraints: theory and a DAE framework. Multibody Syst. Dynam. 11 (2004) 343-364