On the Role of Robot Configuration in Cartesian Stiffness Control

The stiffness ellipsoid, i.e. the locus of task-space forces obtained corresponding to a deformation of unit norm in different directions, has been extensively used as a powerful representation of robot interaction capabilities. The size and shape of the stiffness ellipsoid at a given end-effector posture are influenced by both joint control parameters and- for redundant manipulators- by the chosen redundancy resolution configuration. As is well known, impedance control techniques ideally provide control parameters which realize any desired shape of the Cartesian stiffness ellipsoid at the end-effector in an arbitrary non-singular configuration, so that arm geometry selection could appear secondary. This definitely contrasts with observations on how humans control their arm stiffness, who in fact appear to predominantly use arm configurations to shape the stiffness ellipsoid. To understand this discrepancy, we provide a more complete analysis of the task-space force/deformation behavior of redundant arms, which explains why arm geometry also plays a fundamental role in interaction capabilities of a torque controlled robot. We show that stiffness control of realistic robot models with bounds on joint torques can't indeed achieve arbitrary stiffness ellipsoids at any given arm configuration. We first introduce the notion of maximum allowable Cartesian force/displacement ("stiffness feasibility") regions for a compliant robot. We show that different robot configurations modify such regions, and explore the role of different configurations in defining the performance limits of Cartesian stiffness controllers. On these bases, we design a stiffness control method that suitably exploits both joint control parameters and redundancy resolution to achieve desired task-space interaction behavior.