In this paper a new concept, named the Extended Operational Space (EXOS), has been proposed for the effective analysis and the real-time control of the robot manipulators with kinematic redundancy. The EXOS consists of the operational space (OS) and the optimal null space (NS): the operational space is used to describe manipulator end-effector motion; whereas the optimal null space, described by the minimum number of NS vectors, is used to express the self motion. Based upon the EXOS formulation, the kinematics, statics, and dynamics of redundant manipulators have been analyzed, and control laws based on the dynamics have been proposed. The inclusion of only the minimum number of NS vectors has changed the resulting dynamic equations into a very compact form, yet comprehensive enough to describe: not only the dynamic behavior or the end effector, but also that of the self motion; and at the same time the interaction of these two motions. The comprehensiveness is highlighted by the demonstration of the dynamic couplings between OS dynamics and NS dynamics, which are quite elusive in other approaches. Using the proposed dynamic controls, one can optimize a performance measure while tracking a desired end-effector trajectory with a better computational efficiency than the conventional methods. The effectiveness of the proposed method has been demonstrated by simulations and experiments.