QR Decomposition for State Space Representation of Constrained Mechanical Dynamic Systems

Abstract

This paper presents a numerical solution method for dynamic analysis of constrained mechanical systems. This method reduces a coupled set of differential and algebraic equations to state space form. The reduction uses an independent set of velocities which lie on the tangent plane of the constraint surface. The tangent plane is defined by the nullspace of constraint Jacobian matrix. The nullspace basis is found using QR decomposition of the constraint Jacobian matrix. Because the nullspace basis is not unique, directional continuity of the nullspace is difficult to preserve each time the Jacobiar is decomposed. This paper presents an updating algorithm that is used instead oj repeated decomposition. This preserves directional continuity of the Jacobian matrix and increases efficiency. State equations are then derived in terms of independent accelerations and therefore can efficiently be integrated. Generalized velocities are integrated with constraints to obtain positions. This method has demonstrated minimal constraint violations and improved efficiency. Numerical examples with singular configurations and redundant constraints are presented to demonstrate the effectiveness of the method.