Robot impedance control with nondiagonal stiffness

Impedance control is a widely adopted strategy to execute tasks involving interaction of a robot manipulator with the environment. The goal is to impose an end-effector dynamic behavior described by a mechanical impedance. A crucial point is the definition of the elastic contribution in the impedance equation according to the task requirements; this is achieved by a proper choice of the equivalent stiffness matrix. In the paper an energy based argument is used to derive the dynamic equation of a mechanical impedance characterized by a translational part and a 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.