This paper proposes an analytical solution for a 5-DOF manipulator to follow a given trajectory while keeping the orientation of one axis in the end-effector frame. The forward kinematics and inverse kinematics for a 5-DOF manipulator are analyzed systemically. The singular problem is discussed after the forward kinematics is provided. For any given reachable position and orientation of the end-effector, the derived inverse kinematics will provide an accurate solution. In other words,there exists no singular problem for the 5-DOF manipulator, which has wide application areas such as welding, spraying,and painting. Experiment results verify the effectiveness of the methods developed in this paper.