In this work, redundancy resolution has been employed to increase the Cartesian mechanical rigidity of 7 DOF robot manipulators during tasks requiring stiff interactions with the environment (e.g. milling or drilling). The Cartesian static stiffness of the end-effector for a given joint configuration is deduced from an identified joints stiffness model. The Cartesian reflected rigidity evolution over an analytically computed self-motion of the manipulator shows significant variations that clearly highlight the need to select the right set of joint angles among the possible ones. A global optimization scheme of the redundant DOF is proposed to determine the stiffest robot configurations for a given pose of the end-effector. An experimental study on 7 DOF KUKA LBR iiwa then shows the relevance of the proposed approach in finding the redundant robot joint angles that optimize this rigidity criteria.