Two numerical issues in simulating constrained dynamics

A common approach to formulating the dynamics of closed-chain mechanisms requires finding the forces of constraint at the loop closures. However, there are indications that this approach leads to ill-conditioned systems that must be inverted and to numerically unstable differential equations of motion. The authors derive a sufficient condition for ill conditioning of augmented dynamical systems, namely, that the mechanism´s trajectory passes through, or very near, a kinematic singularity. In singular regions the equations of motion are also numerically stiff, and frequently require special numerical methods for computer solution. The authors propose a novel method of calculating closed-chain dynamics, based on the systematic elimination of variables that are both redundant and that may adversely affect the computations. This approach produces numerically stable solutions of the differential equations of motion, and the equations are apparently much less stiff than the equations produced by the traditional force-closure approach