Augmented Lagrangian and mass-orthogonal projection methods for constrained multibody dynamics

This paper presents a new method for the integration of the equations of motion of constrained multibody systems in descriptor form. The method is based on the penalty-Augmented Lagrangian formulation and uses mass-orthogonal projections for the solution to satisfy the kinematic constraint conditions. The number of equations being solved is equal to the number of states, and does not depend on the number of constraint conditions. Therefore, the method is particularly suitable for systems with redundant constraints, singular configurations or topology changes. The major advantage of the new method relies on the fact that for a low computational cost, the constraints in positions, velocities and accelerations are satisfied to machine precision during the numerical integration. This process is efficiently done by means of a mass-orthogonal projection without the need for coordinate partitioning or reduction to a minimum set of coordinates. The projection scheme allows for a more accurate and robust integration of the equations of motion since constraint violations constitute one of the primary sources of numerical errors and instabilities during the integration process. The proposed projection is also applied to the classical Lagrangian approach, thus eliminating the need for further stabilization as well as the selection of parameters in Baumgarte's method.