Comparatively slow growth in power storage and generation makes power-efficient designs desirable for legged robot systems. One important cause of power losses in robotic systems is the mechanical antagonism phenomenon, i.e. one or more motors being used as brakes while the others exert positive energy. This two-part paper first develops a rigorous understanding of mechanical antagonism in multiactuator robotic limbs. We show that, for a 6-DoF robot arm, there exist 4096 distinct regions in the force-velocity space of the end effector (the regions are distinguishable by the sign of the actuator powers). Only sixty-four of these regions correspond with operating points where all actuators exert positive power into the system. In the second part of the paper, we formulate a convex optimization problem which minimizes mechanical antagonism in redundant manipulators. We solve the optimization problem which becomes the derivation for a new, power-optimal, pseudoinverse for non-square Jacobians. In fact, two such pseudoinverses are derived: one for statically determinate systems, such as serial manipulators, and one for statically indeterminate systems, such as parallel manipulators.