An unexpected path reversal is discovered for a serial robot with an offset-axis wrist over a finite range of proximity to the wrist singularity. This is replicated by a kinematic model. A prior spherical-wrist method transits the singularity with limited joint rate and acceleration under a constant rate of tool traversal. Accurate position is maintained by controlling a small deviation in orientation. Extensions to the method for an offset wrist (1) find the least-maximum deviation, (2) identify and locate where a path reversal occurs, and (3) use this point to control step size in a high-order predictor-correction path following procedure.