An optimized fingertip mapping (OFM) algorithm has been developed to transform human hand poses into robot hand poses. It has been implemented to teleoperate the Utah/MIT Dextrous Hand by a new hand master: the Utah Dextrous Hand Master. The keystone of the algorithm is the mapping of both the human fingertip positions and orientations to the robot fingers. Robot hand poses are generated by minimizing the errors between desired human fingertip positions and orientations and possible robot fingertip positions and orientations. Differences in the fingertip workspaces that arise from kinematic dissimilarities between the human and robot hands are accounted for by the use of a priority based mapping strategy. The OFM gives first priority to the human fingertip position goals and the second to orientation.