A novel return map in non-flat configuration spaces οf multibody systems with impact

Abstract

This work presents a new return mapping, which can be an essential part in the numerical integration of the equations of motion of multibody systems involving impact events. For such systems, each unilateral constraint introduces a boundary hypersurface within the original configuration manifold, restricting the allowable motions on one side of this hypersurface only. When an impact is detected during a step of a time-stepping integration scheme, the position of the system may escape from the allowable range temporarily. Then, this return map is applied at the end of this step in order to bring the system position back to the constrained configuration manifold. The novelty of the new map is related to the fact that both the original and the constrained configuration manifolds of the class of systems examined are non-Euclidean. Its construction and application is based on geometric properties of Jacobi fields on a manifold, specialized for Lie groups. Specifically, once an impact is detected numerically, a sequence of appropriate projections is performed up until the converged position is located within the constrained manifold. Then, if the final position is sufficiently close to the boundary of this manifold and the generalized velocity has a direction pointing towards this boundary, the position and the velocity determined are considered to represent the pre-impact state of the system, which can be used to predict its post-impact state. Finally, the accuracy and efficiency of the new method is demonstrated by applying it to a selected set of mechanical examples.