Quaternion-based impedance with nondiagonal stiffness for robot manipulators

In this paper an energy-based argument is used to derive the dynamic equation of a mechanical impedance at the end effector of a robot manipulator, both for its translational part and for its rotational part. The adoption of unit quaternions to describe orientation displacements leads to a geometrically consistent definition of the stiffness in the impedance equation. Remarkably, off-diagonal elements in the equivalent stiffness matrix are considered; namely, coupling forces with orientation displacements and coupling moments with position displacements. The equilibrium and the stability of the impedance equation are discussed as well as the geometric properties of the stiffness matrix