An empirical study of the joint space inertia matrix

The joint-space inertia matrix of a robot mechanism can be highly ill-conditioned. This phenomenon is not merely a numerical artifact; it is symptomatic of an underlying property of the mechanism itself that can make it more difficult to simulate or control. This paper investigates the problem by means of an empirical study of the eigenvalues, eigenvectors and condition number of the joint-space inertia matrix. It is shown that the condition number is typically large, and that it grows anywhere from O(N) to O(N⁴) with the number of bodies in the system. Several graphs are presented, showing how the condition number varies with configuration, the number of links, variations in link sizes, variations in connectivity, and fixed or floating bases. Explanations are offered for some of the observed effects.