The 3-PUU (Prismatic and Universal joints) translational parallel machine is one of the three-degree-of-freedom (DOF) parallel manipulators, which was previously introduced by Tsai & Joshi in 2002. Since it performs 3-DOF and pure translation motion, this mechanism can be used as a welding robot that only needs to move in a translation manner. This paper discusses the kinematics and singularity analysis of a 3-PUU parallel machine composed of three identical PUU limbs. A constraint equation is computed based on the algebraic geometry method, and it solves the forward kinematics solutions. Based on the computation, it reveals that there are four solutions of the forward kinematics problem. The workspace of each solution of forward kinematics is plotted. The singularity conditions are obtained by computing the zero set of the determinant of the Jacobian matrix, then mapped onto the workspace. The analysis shows that the singularities occur when the moving platform altitude is null and at the highest. Additionally, a numerical simulation with trapezoidal trajectory input has been proposed to verify the kinematics analysis results.