Physical Implications and Physics-Based Design Conditions of Nominal Inertia Matrix for Operational Space Perturbation Observer

Abstract

This study investigated the physical implications and design conditions for a nominal inertia matrix for operational space perturbation observer-based control, including operational space disturbance observer-based control and operational space time delay control. Previous reports indicated that using a constant diagonal nominal inertia matrix deteriorated control performance and led to instability. However, the underlying physical implications of the constant diagonal nominal inertia matrix and its effect on control performance remained unclear for decades. We identified that the main problem arises from the loss of information on the robot's kinematics. The loss resulted in a joint space representation of the operational space nominal inertia matrix that is neither positive-definite nor uniformly bounded, with its principal inertia becoming zero at singular configurations. Moreover, uncompensated residual coupled dynamics of the robot are significantly amplified by the inverse of a principal inertia approaching zero, disturbing the closed-loop dynamics and causing dynamic coupling and even instability. In light of these findings, physics-based conditions for selecting the operational space nominal inertia matrix were proposed: operational space nominal inertia must be positive-definite, and its joint space representation must be positive-definite and uniformly bounded from below. Extensive experiments using a 2 degree-of-freedom robot verified these physical implications and physics-based conditions. Our findings are expected to provide insights into the role and selection criteria of the operational space nominal inertia matrix, thereby promoting the practical application of the operational space perturbation observer in various robot applications requiring robust, efficient, and straightforward control.