Six-DOF impedance control based on angle/axis representations

Impedance control is a well-established framework to manage the interaction of the end effector of a robot manipulator with the environment. For the execution of six-degree-offreedom (DOF) tasks, both the end-effector position and orientation must be handled. The operational space control schemes typically utilize minimal representations of end-effector orientation; however, such representations do not lead to a physically meaningful definition of the rotational part of the impedance equation, and they suffer from the occurrence of representation singularities. In this work a new approach to six-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 six-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 with open control architecture and force/torque sensor have been carried out, and the results in a number of case studies are discussed.