A manipulator is at a singular configuration when the screws representing the instantaneous joint motions of the manipulator are linearly dependent, and the manipulator cannot be moved along an exact path with specified orientation in world coordinates. There are ∞3 singular configurations for a six-degree-of-freedom manipulator, and all these configurations constitute the singularity field. An algorithm is derived to trace the singularity field. Another algorithm presented in this paper finds all the joint screws reciprocal to a given wrench screw. Some new robotic applications are then possible using the principle of aligning a power tool with the screw of the reciprocal wrench.