Impedance Control Design Framework Using Commutative Map Between SE(3) and se(3)
Jonghyeok Kim, Minchang Sung, Youngjin Choi, Jonghoon Park, Wan Kyun Chung
AI summary
Problem
Six-DoF impedance control lacks a minimal orientation representation that avoids singularities and topological obstructions, while existing Lie group methods often suffer from convergence issues and complex inertia shaping.
Approach
The authors map control design from the Lie group SE(3) to its isomorphic Lie algebra se(3) using exponential coordinates, enabling familiar Euclidean-space impedance formulation while maintaining group structure via closed-form differential mappings.
Key results
- Minimal-parameter 6-DoF impedance controller using exponential coordinates
- Closed-form commutative mapping between body twist and exponential coordinate derivatives
- Potential energy function derived directly in the Lie algebra se(3)
- Stability analysis and experimental validation on a 6-DoF manipulator
Why it matters
Enables robust, singularity-free 6-DoF robot compliance for contact-heavy tasks without requiring non-minimal orientation parameters or complex force sensors.
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 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 SE(3), and the control is designed in the Lie algebra se(3), which is isomorphic to the vector space R6. The group structure of SE(3) 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 SE(3) group structure and achieves the desired dynamic behavior as the functionality of the impedance control with minimal parameters.