Rotational Impedance Formulation in a Unified Viewpoint of Lie Algebra

Impedance control is essential in robotic manipulation tasks involving unpredictable interactions with the environment, as it ensures safety and stability during contact. While the concept of translational impedance is consistently understood, rotational impedance exhibits varied forms in scientific literature. This letter seeks to elucidate the fundamental nature of rotational impedance and presents a comprehensive, unified framework for formulating rotational impedance using Lie algebra and Noether's theorem. This approach facilitates the derivation of the various expressions of rotational impedance observed in the field. We utilized quaternions and rotation matrices to represent rotational motion within our proposed framework to ensure theoretical validity. This approach yielded a rotational impedance expression that aligns with existing literature. For empirical verification, we have conducted both simulated experiments using robots and actual trials on a UR5 robot. These assessments demonstrated the expected dynamic responses along all axes. These results showcase the efficacy and broad applicability of our approach.