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Eliminating constraint drift in the numerical simulation of constrained dynamical systems. (English) Zbl 1229.70003

Summary: By means of the Udwadia-Kalaba approach we propose an explicit equation of constrained motion developed to simulate constrained dynamical systems without error accumulation due to constraint drift. The basic idea is to embed a small virtual force and a small virtual impulse to the equation of motion, in order to avoid the drift typically experienced in constrained multibody simulations. The embedded correction terms are selected to minimally alter the dynamics in an acceleration and kinetic energy norm sense. The formulation allows one to use a standard ODE solver, avoiding the need for iterative constraint stabilization. The equation is based on the pseudoinverse of a constraint matrix such that it can be used under redundant constraints and kinematic singularities. The proposed method takes into account the finite word-length of the computational environment, and also accommodates possibly inconsistent initial conditions.

MSC:

70-08 Computational methods for problems pertaining to mechanics of particles and systems
70E55 Dynamics of multibody systems

Software:

DASSL; RODAS
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J.L. Lagrange, Mecanique Analytique, Mme Ve Courcier, Paris, 1787.; 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, 232-235 (1829) · ERAM 004.0157cj
[3] G.A. Maggi, Principii della Teoria Mathematica del Movimento dei Corpi: Corso di Meccanica Razionale, Ulrico Hoepli, Milano, 1896.; G.A. Maggi, Principii della Teoria Mathematica del Movimento dei Corpi: Corso di Meccanica Razionale, Ulrico Hoepli, Milano, 1896. · JFM 26.0777.01
[4] Gibbs, J. W., On the fundamental formulae of dynamics, Am. J. Math., 2, 1, 49-64 (1879) · JFM 11.0643.01
[5] Appell, P., Sur une forme generale des equations de la dynamique, C.R. Acad. Sci. Paris, 129, 459-460 (1899) · JFM 30.0641.03
[6] Kane, T. R.; Levinson, D. A., Dynamics: Theory and Applications (1985), McGraw-Hill: McGraw-Hill New York · Zbl 0546.70015
[7] Udwadia, F. E.; Kalaba, R. E., A new perspective on constrained motion, Proc. R. Soc. Lond. A, 439, 407-410 (1992) · Zbl 0766.70011
[8] Udwadia, F. E.; Kalaba, R. E., Analytical Dynamics: A New Approach (1996), Cambridge University Press: Cambridge University Press Cambridge, England · Zbl 0875.70100
[9] Udwadia, F. E.; Kalaba, R. E., On the foundations of analytical dynamics, Int. J. Nonlinear Mech., 37, 1079-1090 (2002) · Zbl 1346.70011
[10] Pars, L. A., A Treatise on Analytical Dynamics (1965), John Wiley and Sons: John Wiley and Sons New York · Zbl 0125.12004
[11] Ju.I. Neimark, N.A. Fufaev, Dynamics of nonholonomic systems, AMS Providence, 1972.; Ju.I. Neimark, N.A. Fufaev, Dynamics of nonholonomic systems, AMS Providence, 1972. · Zbl 0245.70011
[12] Gantmacher, F., Lectures in Analytical Mechanics (1975), Mir Publisher: Mir Publisher Moscow, (English translation)
[13] Goldstein, H., Classical Mechanics (1980), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0491.70001
[14] Chetaev, N. G., Theoretical Mechanics (1989), Mir Publisher: Mir Publisher Moscow, (English translation) · Zbl 0715.70002
[15] Lurie, A. I., Analytical Mechanics (2002), Springer-Verlag: Springer-Verlag New York · Zbl 1015.70001
[16] Baumgarte, J., Stabilization of constraints and integrals of motion in dynamical systems, Comput. Method Appl. Mech. Engrg., 1, 1-16 (1972) · Zbl 0262.70017
[17] Gear, C. W.; Leimkuhler, B.; Gupta, G. K., Automatic integration of Euler-Lagrange equations with constraints, J. Comput. Appl. Math., 12-13, 77-90 (1985) · Zbl 0576.65072
[18] Lötstedt, P.; Petzold, L., Numerical solution of nonlinear differential equations with algebraic constraints. I: Convergence results for backward differentiation formulas, Math. Comput., 46, 174, 491-516 (1986) · Zbl 0601.65060
[19] Führer, C.; Leimkuhler, B., Numerical solution of differential-algebraic equations for constrained mechanical motion, Numer. Math., 59, 55-69 (1991) · Zbl 0701.70003
[20] Petzold, L. R., Numerical solution of differential-algebraic equations in mechanical systems simulation, Physica D, 60, 1-4, 269-279 (1992) · Zbl 0779.70002
[21] ten Dam, A. A., Stable numerical integration of dynamical systems subject to equality state-space constraints, J. Engrg. Math., 26, 315-337 (1992) · Zbl 0765.70003
[22] Eich, E., Convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints, SIAM J. Numer. Anal., 30, 5, 1467-1482 (1993) · Zbl 0785.65079
[23] Bayo, E.; Ledesma, R., Augmented lagrangian and mass-orthogonal projection methods for constrained multibody dynamics, Nonlinear Dynam., 9, 1-2, 113-130 (1996)
[24] Blajer, W., Elimination of constraint violation and accuracy aspects in numerical simulation of multibody systems, Multibody Syst. Dynam., 7, 265-284 (2002) · Zbl 1007.70006
[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, 834-849 (2005)
[26] Schiehlen, W., Multibody system dynamics: roots and perspectives, Multibody Syst. Dynam., 1, 149-188 (1997) · Zbl 0901.70009
[27] Brogliato, B.; ten Dam, A. A.; Paoli, L.; Génot, F.; Abadie, M., Numerical simulation of finite dimensional multibody nonsmooth mechanical systems, ASME Appl. Mech. Rev., 55, 2, 107-150 (2002)
[28] J. D’Alembert, Traite de Dynamique, Paris, 1743.; 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, 153-158 (1966)
[30] Lötstedt, P., Mechanical systems of rigid bodies subject to unilateral constraints, SIAM J. Appl. Math., 42, 2, 281-296 (1982) · Zbl 0489.70016
[31] Ben-Israel, A.; Greville, T. N.E., Generalized Inverse: Theory and Applications (2003), Springer · Zbl 1026.15004
[32] Gear, C. W., Differential-algebraic equation index transformations, SIAM J. Sci. Stat. Comput., 9, 39-47 (1988) · Zbl 0637.65072
[33] L. Petzold, DASSL: A Differential/Algebraic System Solver. <http://www.netlib.org/ode/ddassl.f>; L. Petzold, DASSL: A Differential/Algebraic System Solver. <http://www.netlib.org/ode/ddassl.f>
[34] Brenan, K. E.; Campbell, S. L.; Petzold, L. R., Numerical Solutions of Initial-Value Problems in Differential-Algebraic Equations (1989), Elsevier Science: Elsevier Science NY · Zbl 0699.65057
[35] de Jalón, J. G.; Bayo, E., Kinematic and Dynamic Simulation of Multibody Systems The Real-Time Challenge (1994), Springer-Verlag: Springer-Verlag New York
[36] Hairer, E.; Wanner, G., Solving Ordinary Differential Equations. II: Stiff and Differential-Algebraic Problems, Series in Computational Mathematics (1996), Springer-Verlag · Zbl 0859.65067
[37] Golub, G.; Loan, C. V., Matrix Computations (1996), The John Hopkins University Press
[38] Leimkuhler, B.; Petzold, L. R.; Gear, C. W., Approximation methods for the consistent initialization of differential-algebraic equations, SIAM J. Numer. Anal., 28, 1, 205-226 (1991) · Zbl 0725.65076
[39] Nikravesh, P. E., Initial condition correction in multibody dynamics, Multibody Syst. Dynam., 18, 107-115 (2008) · Zbl 1120.70008
[40] de Jalón, J. G.; Unda, J.; Avello, A., Natural coordinates for the computer analysis of multibody systems, Comput. Method Appl. Mech. Engrg., 59, 309-327 (1986) · Zbl 0577.70004
[41] Kraus, C.; Winckler, M.; Bock, H. G., Modeling mechanical DAE using natural coordinates, Math. Comput. Model. Dynam. Syst., 7, 2, 145-158 (2001) · Zbl 1060.70005
[42] Haug, E. J., Computer aided kinematics and dynamics of mechanical systems, Basic Methods, vol. I (1989), Allyn and Bacon: Allyn and Bacon Boston
[43] Serban, R.; Negrut, D.; Haug, E. J.; Potra, F. A., A topology based approach for exploiting sparsity in multibody dynamics in cartesian formulation, Mech. Struct. Mach., 25, 3, 379-396 (1997)
[44] Arnold, M.; Fuchs, A.; Führer, C., Efficient corrector iteration for DAE time integration in multibody dynamics, Comput. Method Appl. Mech. Engrg., 195, 50-51, 6958-6973 (2006) · Zbl 1120.70300
[45] Laulusa, A.; Bauchau, O. A., Review of classical approaches for constraint enforcement in multibody systems, J. Comput. Nonlinear Dynam., 3, 011004 (2008)
[46] Bauchau, O. A.; Laulusa, A., Review of contemporary approaches for constraint enforcement in multibody systems, J. Comput. Nonlinear Dynam., 3, 011005 (2008)
[47] Baumgarte, J., A new method of stabilization for holonomic constraints, J. Appl. Mech., 50, 869-870 (1983) · Zbl 0534.70011
[48] Chang, C. O.; Nikravesh, P. E., An adaptive constraint violation stabilization method for dynamic analysis of mechanical systems, J. Mech. Trans. Automat. Des., 107, 488-492 (1985)
[49] Ascher, U. M.; Chin, H.; Reich, S., Stabilization of DAEs and invariant manifolds, Numer. Math., 67, 131-149 (1994) · Zbl 0791.65051
[50] Lin, S. T.; Hong, M. H., Stabilization method for numerical integration of multibody mechanical systems, J. Mech. Des., 120, 565-572 (1998)
[51] Blajer, W., A geometrical interpretation and uniform matrix formulation of multibody system dynamics, Z. Angew. Math. Mech., 81, 4, 247-259 (2001) · Zbl 0986.70005
[52] Wehage, R. A.; Haug, E. J., Generalized coordinate partitioning for dimension reduction in analysis of constrained dynamic systems, J. Mech. Des., 104, 247-255 (1982)
[53] Bayo, E.; Avello, A., Singularity free augmented lagrangian algorithms for constraint multibody dynamics, Nonlinear Dynam., 5, 209-231 (1994)
[54] Udwadia, F. E., A new perspective on the tracking control of nonlinear structural and mechanical systems, Proc. R. Soc. Lond. A, 459, 1783-1800 (2003) · Zbl 1046.70016
[55] Winter, D. A., Biomechanics and Motor Control of Human Movement (1990), Wiley-Interscience: Wiley-Interscience New York
[56] Blajer, W.; Kolodziejczyk, K., A geometric approach to solving problems of control constraints: theory and a DAE framework, Multibody Syst. Dynam., 11, 343-364 (2004) · Zbl 1066.70018
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