ODE methods for solving the multibody dynamics with constraints

This paper deals with methods for solving the multibody dynamics with constraints. The problem is considered in the framework of solving the Lagrange multipliers in addition to the system coordinates in the differential and algebraic equation (DAE) governing the dynamics with holonomic or nonholonomic constraints. The proposed methods are originally based on Baumgarte's work for the holonomic constraints but its extensions. First,one considers a Lagrangian which includes the time-differentiated constraint equations in addition to the constraint equations themselves. Applying the Lagrange procedure se have the ordinary differential equations (ODE), not the DAE, including the differential equation with respect to the Lagrange multipliers. This paper also presents a numerically stable method for inverting the system matrix. The numerical solution for the differential equations with respect to the Lagrange multipliers as well as the system coordinates by using the ordinary numerical integration method, e.g. Runge-Kutta method, shows the excellent stability of the constraints, which is superior to the penalty method.