Singularity-free simulation of closed loop multibody systems by using null space of Jacobian matrix

Abstract

Numerical simulation of closed loop multibody systems is associated with numerical solution of equations of motion which are, in general, in the form of DAE’s index-3 systems. For assuring continuous simulation, one should overcome some difficulties such as stabilization of the constraint equations, singular configuration of the system. In this paper, the system equations of motion with the Lagrange multipliers is rewritten by introducing generalized reaction forces. The combination with the condition of ideality of constraints leads to the system of equations which can be solved by numerical techniques smoothly, even over singular positions. Based on the new criterion of ideality of constraints, which relates generalized reaction forces and the null space matrix of Jacobian matrix, it is possible also to remove reaction forces and use only the reduced system of equations with null space matrix for passing singular positions. In order to prevent the constraint equations from the accumulated errors of integral time, the method of position and velocity projection has been exploited. Some numerical experiments are carried out to verify the proposed approach.