An RRR—UPRR—RPUR parallel manipulator which can achieve spherical motion around fixed point was presented. The manipulator has three rotational degree-of-freedom (DOF), of which two rotational DOFs are independent and could be actuated by single actuator. The screw formulations of kinematic joints under ordinary configuration were established, which was employed to determine the three continues rotational DOF of the manipulator. Then, the kinematics of moving-platform and adjacent part were deduced, and the analytical algorithm of direct kinematics of the mechanism was established. Finally, the theoretical results are verified by the motion simulation of the prototype mechanism.