In this paper we introduce the concept of grasp redundancy, as a new method for relaxing the full-constrained hand-object relationships that are normally considered in robotic manipulation. Grasp redundancy refers to the cartesian degrees of freedom (DOF) that are not fully constrained by the grasp, and, thus, can used be to perform relative hand-object motion during a compliant motion task. We study how grasp redundancy can be exploited for improving the grasping and interaction capabilities of robots. For that, a sensor-based control approach exploiting this concept is defined, and some simulation experiments are presented with a 7 DOF and a 8 DOF arm. Results show how the use of grasp redundancy allows these robots to perform much better than with the classical approaches.