Multibody dynamics with redundant constraints and singular mass matrix: existence, uniqueness, and determination of solutions for accelerations and constraint forces

This paper addresses some important issues for multibody dynamics; issues that are basic and really not too difficult to solve, but rarely considered in the literature. The aim of this paper is to contribute to the resolution and clarification of these topics in multibody dynamics. There are many formulations for determining the equations of motion in constrained multibody systems. This paper will focus on three of the most important methods: the Lagrange equations of the first kind, the null space method and the Maggi equations. In all cases we consider singular inertia matrices and redundant constraint equations. We assume that the inertia matrix is positive-semidefinite (symmetric) and that the constraint equations may be redundant but always consistent. It is demonstrated that the aforementioned dynamic formulations lead to the same three mathematical conditions of existence and uniqueness of solutions, conditions that have at the same time a clear physical meaning. We conclude that the mathematical problem always has a solution if the physical problem is well conditioned. This paper also addresses the problem of determining the constraint forces in the case of redundant constraints. This problem is examined from a broad perspective. We will present several examples and a simple method to find practical solutions in cases where the forces of constraint are undetermined. The method is based on the weighted minimum norm condition. A physical interpretation of this minimum norm condition is provided in detail for all examples. In some cases a comparison with the results obtained by considering flexibility is included.