A set of ordinary differential equations of motion for constrained mechanical systems

A new theoretical approach is presented for deriving an appropriate set of equations of motion for a class of mechanical systems subjected to motion constraints. This approach is facilitated by employing some fundamental concepts of differential geometry and can treat both holonomic and nonholonomic constraints simultaneously. The main idea is to consider the equations describing the action of the constraints as an integral part of the overall process leading to the equations of motion of the system. Specifically, taking into account the form of the constraints, a suitable set of linear operators is first defined, acting between the tangent spaces of manifolds, where the motion of the system is viewed to evolve. This then provides the foundation for determining an associated set of operators, acting between the corresponding dual spaces as a vehicle of transferring Newton’s law of motion in an invariant form. As a consequence of this approach, a set of coupled second-order ordinary differential equations is derived for both the generalized coordinates and the dynamic Lagrange multipliers associated to the motion constraints automatically. This leads to considerable theoretical advantages, avoiding problems related to systems of differential–algebraic equations or penalty formulations. Apart from its theoretical value and elegance, this approach is expected to bring significant benefits in the field of numerical integration in multibody dynamics and control.