Impedance Control Design Framework Using Commutative Map Between $SE(3)$ and $\mathfrak {se}(3)$

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 the exponential coordinate. Since the exponential coordinate can be treated as the Euclidean variable within the injectivity radius, it allows for the formulation of the impedance control more systematically and familiarly. In our framework, a detour strategy is utilized; the impedance is designed in the Lie group <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SE(3)$</tex-math></inline-formula>, and the control is designed in the Lie algebra <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathfrak {se}(3)$</tex-math></inline-formula>, which is isomorphic to the vector space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbb {R}^{6}$</tex-math></inline-formula>. The group structure of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SE(3)$</tex-math></inline-formula> can be maintained using the proposed conversion formula between the Lie group and the Lie algebra, called the differential of the exponential map and its time derivative, with a closed-form expression. Experiments with a 6-DoF robot manipulator verified that the proposed impedance control framework effectively reflects the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SE(3)$</tex-math></inline-formula> group structure and achieves the desired dynamic behavior as the functionality of the impedance control with minimal parameters.