The controllably dexterous capability of a class of seven-jointed serially connected robots is quantitatively confirmed, and the generic features of the inverse kinematics for their control are developed. The generic features are the specific motions of the outermost two joints of the wrist and the aggregates of the motions of the remaining rotary joints in each direction. Of the two global functions among the actuator-coordinates, one is operationally equivalent to a six-jointed substitute-manipulator and the other is a seventh-order polynomial that relates the motion at the last joint of the wrist to the task-motion. The robots in the class can provide a full rotation at constant speed to the end-effector about every task-axis (for a resolution of one degree of angle between adjacent task-axes) through each controllably dexterous fixture-point without encountering the mechanical effects of axis-dependence. For those robots with three P-joints in the arm, this dexterity of motion is accomplished with all joints operating at speeds within 2.0 times the speed of the task rotation. In this paper, the quantitative evaluation is restricted to manipulators with 3R spherical wrists having successively orthogonal axes, the first of which is mounted at right-angles to the adjacent joint-axis in the arm. The CPU-time for executing each computational step is short enough to be useful for on-line operations.