Constraint stabilization in nonlinear dynamics via control theoretic methods

Methods for the stabilization of constraint violation that arises in the simulation of nonlinear multibody dynamic systems have been studied over the past few decades. This paper derives constraint stabilization formulations within the framework of nonlinear control theory. It is shown that two related viewpoints can yield quite different constraint stabilization theories. By introducing Lie algebraic feedback control theory, it is possible to interpret many constraint stabilization methods as nonlinear controllers. Baumgarte's method, as well as some variable structure and sliding mode controllers fall in this category. Within this theoretical framework, the constraint stabilization scheme can be interpreted rigorously as a controller that seeks to achieve asymptotic output tracking. When the measures of constraint violation are considered as output functions of the state variables, constraint violation is simply output error. Conditions that guarantee asymptotic output tracking can, in fact, be used to ensure asymptotic stabilization of constraint violation. Alternatively, somewhat simpler constraint stabilization methods can be derived by exploiting energy integrals associated with the nonlinear system. In this case, we can show that several stabilization methods correspond to nonlinear control methods that utilize conservation laws. The inertial penalty method and several of its variants including the stiffness and damping penalty method, fall in this category. Finally, we show that this interpretation yields a framework that is suitable for the development of simple and effective alternatives to existing stabilization methods. Numerical examples are presented that can be used to synthesize nonlinear multibody dynamics models from nonlinear subassembly models.