This paper describes the development of a novel non-anthropomorphic robot hand with the ability to manipulate objects by means of articulated, actively driven rollers located at the fingertips. An analysis is conducted and systems of equations for two-finger and three-finger manipulation of a sphere are formulated to demonstrate full six degree of freedom nonholonomic spatial motion capability. A prototype version of the hand was constructed and used to grasp and manipulate a variety of objects. Tests conducted with the prototype confirmed the validity of the mathematical analysis. Unlike conventional approaches to within-hand manipulation using legacy robotic hands, the continuous rotation capability of our rolling fingertips allows for unbounded rotation of a grasped object without the need for finger gaiting.