Abstract: Lie group and Lie algebra theory have wide application in robot mechanism. Continuous motion of the rigid body in E³ corresponding to a continuous curve in SE (3) of Lie group. Motion screw is the Lie algebra element in SE (3) of Lie group and integrates rigid motion with SE (3) of Lie group. Constructing the forward motion index product formula of serial robot based on exponential map and rigid body motion superposition theorem, thereby resolving expression of virtual rigid body motion. The edge of rigid body sweeps a curved surface in E³ space with space motion of the rigid body. Finally, obtaining building geometry swept surface which can be expressed by nurbs and corresponding to rigid motion by using computer graphics and geometric modeling approaches.