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This paper mainly addresses the orientation-singularity and orientationability analyses of a special class of the Stewart-Gough parallel manipulators. Based on the half-angle transformation, an analytical polynomial expression of thirteen degree in the moving platform orientation parameters is derived, which represents the orientation-singularity locus of this class of the manipulator at a given position, graphical representations of the orientation-singularity locus of the manipulator are given to illustrate the result. It is shown that for this special class of the Stewart-Gough manipulators, it must exist a nonsingular orientation void in the orientation space around the orientation origin for each position in the position-workspace. Then a new performance index which describes the orientationability of the parallel manipulator at a given position is introduced. Moreover, effects of the design parameters of the moving platform on the orientationability of the manipulator are also presented in detailed in this paper. Based on the research on the orientation-singularity and nonsingular orientation void of the manipulator, another novel performance index representing the practical orientationability of this special class of the manipulator at a given position in the position-workspace is introduced, where singularities, the limitations of active and passive joints and the link interference are all taken into consideration. Furthermore, the practical orientationability of this special class of the Stewart-Gough parallel manipulators is also analyzed at length over the whole position-workspace of the manipulator.