| Solving the inverse kinematic-rate problem is an old and ever-present problem in robotics. Many methods of solution for this problem have been discussed over the past 30 years and, as computational speeds have improved, there has been greater expectation that this calculation will be performed online. This paper compares the merits of many of the methods already presented and describes a new approach that leads to a fast and numerically well-conditioned algorithm. This method is most advantageous for robots with a spherical wrist, which is a common feature among industrial robots. More important, we uncover some interesting properties of efficient solutions that can be applied to a variety of robot architectures. We apply this and other methods to an industrial robot with near to PUMA geometry and report calculation speeds indicative of the real-time cost of each of the several procedures.