This paper presents a closed-form solution for direct and inverse kinematic models of a four-degree-of-freedom parallel manipulator designed for weld torch positioning of orbital welding of metallic pipes. The mathematical model was developed, implemented and simulated considering the geometric parameters of the prototype. The proposed closed-form solution showed that the mathematical model matched the behavior of the manipulator, confirming that the motion was entirely defined through the mathematical and geometric relations, without the requirement of iterations or numerical solutions. The closed-form solution also offered a simpler approach for the mechanical behavior of the manipulator, making it feasible to implement a less complex-coded routine in the embedded microcontroller due to its lower computational requirements.