A globally stable convergent algorithm for the integration of constrained mechanical systems

In this paper the problem of simulation of constrained mechanical systems is addressed. In modeling multi-body mechanical systems, the Lagrange formulation produces a redundant set of differential-algebraic equations, the integration of which can lead to several difficulties, for example the drift of the “constraint violation”. One of the most popular approaches to alleviate this issue is the so-called Baumgarte's method that relies on a linear feedback mechanism. This method can however lead to numerical instabilities when applied to nonlinear (mechanical) systems. The objective of this study is to propose a new method that ensures existence of solutions and makes the constraint manifold asymptotically attractive. The proposed technique is illustrated by means of a simple example.

[1]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[2]  C. W. Gear,et al.  Automatic integration of Euler-Lagrange equations with constraints , 1985 .

[3]  Hooshang Hemami,et al.  Modeling of Nonholonomic Dynamic Systems With Applications , 1981 .

[4]  I. Neĭmark,et al.  Dynamics of Nonholonomic Systems , 1972 .

[5]  Per Lötstedt.  Mechanical Systems of Rigid Bodies Subject to Unilateral Constraints , 1982 .

[6]  C. W. Gear,et al.  ODE METHODS FOR THE SOLUTION OF DIFFERENTIAL/ALGEBRAIC SYSTEMS , 1984 .

[7]  Georg-P. Ostermeyer.  On Baumgarte Stabilization for Differential Algebraic Equations , 1990 .

[8]  R. M. Howe,et al.  Geometric Elimination of Constraint Violations in Numerical Simulation of Lagrangian Equations , 1994 .

[9]  C. W. Gear,et al.  Differential-Algebraic Equations , 1984 .

[10]  G. Webb,et al.  Lie Algebraic Control for the Stabilization of Nonlinear Multibody System Dynamics Simulation , 1999 .

[11]  Alessandro Astolfi,et al.  On the stability of constrained mechanical systems , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[12]  O. Bauchau,et al.  Review of Contemporary Approaches for Constraint Enforcement in Multibody Systems , 2008 .

[13]  R. Kalaba,et al.  A new perspective on constrained motion , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[14]  Danielle C. Tarraf,et al.  On the nature and stability of differential-algebraic systems , 2002, Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301).

[15]  Gian Antonio Maggi,et al.  Principii della teoria matematica del movimento dei corpi : corso di meccanica razionale , 1896 .

[16]  Firdaus E. Udwadia,et al.  On constrained motion , 1992, Appl. Math. Comput..

[17]  C. W. Gear,et al.  Differential-algebraic equations index transformations , 1988 .

[18]  B. Leimkuhler,et al.  Numerical solution of differential-algebraic equations for constrained mechanical motion , 1991 .

[19]  W. Blajer.  Elimination of Constraint Violation and Accuracy Aspects in Numerical Simulation of Multibody Systems , 2002 .

[20]  A. Kurdila,et al.  Role of Maggi's Equations in Computational Methods for Constrained Multibody Systems , 1990 .

[21]  J. Baumgarte.  Stabilization of constraints and integrals of motion in dynamical systems , 1972 .

[22]  O. Bauchau,et al.  Review of Classical Approaches for Constraint Enforcement in Multibody Systems , 2008 .

[23]  Parviz E. Nikravesh,et al.  An adaptive constraint violation stabilization method for dynamic analysis of mechanical systems , 1985 .