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A new approach to 6-DOF impedance control is proposed, where the end-effector orientation displacement is derived from the rotation matrix expressing the mutual orientation between the compliant frame and the desired frame. An alternative Euler angles-based description is proposed which mitigates the effects of representation singularities. Then, a class of angle/axis representations are considered to derive the dynamic equation for the rotational part of a 6-DOF impedance at the end effector, using an energy-based argument. The unit quaternion representation is selected to further analyze the properties of the rotational impedance. The resulting impedance controllers are designed according to an inverse dynamics strategy with contact force and moment measurements, where an inner loop acting on the end-effector position and orientation error is adopted to confer robustness to unmodeled dynamics and external disturbances. Experiments on an industrial robot were carried out, and the results of case studies are discussed.
Caccavale et al. (1999) studied this question.