Impedance Control Design Framework Using Commutative Map between SE(3) and Se(3)

Abstract

Impedance control is a widely adopted approach that ensures the compliant behavior of robot manipulators as they interact with their environment according to specifically designed dynamics. For tasks involving six degrees of freedom (DoF), it is crucial to appropriately manage the position and orientation of the end effector by controlling dynamic behavior. However, describing orientational displacement and designing the corresponding rotational impedance can be challenging, especially when we use a minimal representation. The well-known minimal representation for orientation, the Euler angle, suffers from representation singularity. As a remedy, the quaternion or dual quaternion can be an alternative but with non-minimal representations. This lack of minimal representation, which does not suffer from the representation singularity, often leads to handling the impedance design by directly defining the potential energy function in the matrix Lie group. This paper proposes a framework for the six-DoF impedance control design that takes advantage of Lie group theory with minimal representation, known as exponential coordinate. Since the exponential coordinate can be treated as the Eu