Examination of the ill-conditioning of the inertia matrix used in mechanical analyses

Abstract

In a state-space representation of the equations of motion for a system of rigid bodies one component of these equations is the so-called inertia matrix. This matrix can be used for inverse dynamics and its inversion is necessary to perform direct dynamics analyses, and to perform induced acceleration analyses. The contents of the inertia matrix are a function of the lengths of the segments, the locations of the centers of masses, segment masses, segment moments of inertia, and joint angles. It is demonstrated that the inertia matrix is an ill-conditioned matrix meaning that, for example, small errors in joint moments cause correspondingly larger errors in the joint accelerations computed using the matrix. The ill-condition of the matrix can be quantified by computing its condition number; the magnitude of the error is bounded by the condition number. It is demonstrated for a two-rigid body system representing the upper-limb that the configuration of the system influences the magnitude of the condition number, and that because the mass and moment of inertia of the distal segment is smaller than the proximal segment a relatively low condition number is produced. For a three-segment system representing the shanks, thighs, and HAT (head, arms, and trunk) the closer each segment rotated towards the adjacent segment the lower the condition number. The magnification of errors due to the inertia matrix arise from the inertial properties of the human body segments and their configuration, not from errors per se in the components of that matrix.