In this work, a family of two degrees of freedom (2-DOF) rotational parallel manipulators (RPMs) with an equal-diameter spherical pure rotation (ESPR) is presented and discussed systematically. The theoretical models of both kinematics and constraints inherited in the manipulators are analyzed through a graphical approach. Based on the established constraint model, these 2-DOF ESPR RPMs are classified into three types according to their compositions of constraint spaces and several novel parallel manipulators are illustrated correspondingly. Finally, two common necessary geometric conditions satisfied for these manipulators are discussed in details with examples. The two conditions will be helpful for engineers with designing ESPR RPMs. Moreover, as one characteristic existing in the ESPR RPMs, two cases of self-rotations accompanying revolutions around fixed axes are revealed. As a result, the corresponding loci of points in the moving platform are proved to be compositions of two subrotations, which are spatial curves and surfaces rather than spherical curves and surfaces.

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