Numerical stability of the constraints near singular positions in the dynamics of multibody systems

Abstract

It is common in mechanical systems that the constraint equations may become linearly dependent at instantaneous singular configurations. This occurs when the direction of a constraint coincides with the direction of the lost degree of freedom. This results in singularities in the conventional simulation algorithms. In this paper first a method based on the uptriangular decomposition of the constraint matrix will be utilized to reduce the equations of motion. The linear dependencies will be identified in the uptriangular decomposition process and the linear dependency relations will be determined. The reduction process will be modified by introducing a methodology which yields the constraint matrix to be of a full rank by using higher order derivatives in the constraint equations. Furthermore the constraint equations will be modified for use in the neighborhood of the singular configuration to yield the system inertia matrix to be nonsingular and to take the corresponding generalized constraint forces into account. Simulations of a three link planar manipulator at a singular configuration will be performed to illustrate the methods developed.