Summary: Two Numerical Issues in Simulating Constrained Robot
R. E. Ellis S. L. Ricker
Department of Computing and Information Science
Queen's University, Kingston, Ontario, Canada K7L 3N6
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.
We derive a sufficient condition for ill conditioning of augmented dynamical systems 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.
We propose a new method of calculating closed-chain dynamics, based on the systematic elimination of vari-
ables 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.
Simulations of robot manipulators and other mechanisms rely on accurate dynamics models, and on the sub-
sequent solution via numerical integration techniques. As with any numerical endeavour efficiency, accuracy,
and stability of the method approximating a mathematically expressed problem are of concern. The efficiency of