A novel serial–parallel kinematic machine based on a serial–parallel manipulator is proposed and its kinematics and statics are studied systematically. First, the concept of serial–parallel kinematic machine formed by the 3-PRS parallel manipulator and the 2-UPR + SPR PM is proposed. Second, the displacement is derived in close form based on the geometrical and dimensional constraints existed in the serial–parallel kinematic machine. Third, the Jacobian, velocity and statics of the serial–parallel kinematic machine are derived in the explicit and compact form. Finally, the formulas for solving the acceleration of this serial–parallel kinematic machine are derived. This research will lay a good foundation for the development of the serial–parallel kinematic machine.