Lyapunov stable penalty methods for imposing holonomic constraints in multibody system dynamics

This paper presents stability and convergence results on a novel approach for imposing holonomic constraints for a class of multibody system dynamics. As opposed to some recent techniques that employ a penalty functional to approximate the Lagrange multipliers, the method herein defines a penalized dynamical system using penalty-augmented kinetic and potential energies, as well as a penalty dependent constraint violation dissipation function. In as much as the governing equations are not typically cocreive, the usual convergence criteria for linear variational boundary value problems are not directly applicable. Still numerical simulations by various researchers suggest that the method is convergent and stable. Despite the fact that the governing equations are nonlinear, the theoretical convergence of the formulation is guaranteed if the multibody system is natural and conservative. Likewise, stability and asymptotic stability results for the penalty formulation are derived from well-known stability results available from classical mechanics. Unfortunately, the convergence theorem is not directly applicable to dissipative multibody systems, such as those encountered in control applications. However, it is shown that the approximate solutions of a typical dissipative system converge to a nearby collection of trajectories that can be characterized precisely using a Lyapunov/Invariance Principle analysis. In short, the approach has many advantages as an alternative to other computational techniques:

(1)	Explicit constraint violation bounds can be derived for a large class of nonlinear multibody dynamics problems
(2)	Sufficient conditions for the Lyapunov stability, and asymptotic stability, of the penalty formulation are derived for a large class of multibody systems
(3)	The method can be shown to be relatively insensitive to singular configurations by selecting the penalty parameters to dissipate constraint violation energy
(4)	The Invariance Principle can be employed in the method, in certain cases, to derive the asymptotic behavior of the constraint violation for dissipative multibody systems by identifying constraint violation limit cycles

Just as importantly, these results for nonlinear systems can be sharpened considerably for linear systems:

(5)	Explicit spectral error estimates can be obtained for substructure synthesis
(6)	The penalty equations can be shown to be optimal in the sense that the terms represent feedback that minimizes a measure of the constraint violation