This paper addresses the orientation-singularity analysis and the nonsingular orientation- workspace computation of the semi-regular Stewart-Gough platform manipulator. Based on the half-angle transformation, a polynomial expression that represents the orientation- singularity locus of the semi-regular Stewart-Gough platform manipulator when the mobile platform of the platform manipulator is fixed to a constant position is derived. It is shown that the orientation-singularity locus of the semi-regular Stewart-Gough platform manipulator at a fixed position is a polynomial expression of degree thirteen in the mobile platform orientation parameters and graphical representations of the orientation-singularity locus of the semi-regular Stewart-Gough platform manipulator are illustrated with examples to demonstrate the theoretical result. Using this half-angle transformation, a new discretization method is proposed for computing the nonsingular orientation-workspace of the semi-regular Stewart-Gough platform manipulator in consideration of singularities, limitations of active and passive joints and the link interference. Examples of a semi- regular Stewart-Gough platform manipulator are given to demonstrate the results