The object of research is a four-axis precision positioning system used in a complex for electron beam welding of metal parts. The purpose of this work is to develop a mathematical model of the motion of the four-axis precision positioning system, find solutions to the direct and inverse problems of the positioning system kinematics, and estimate positioning errors due to the discreteness of the link drive movements. Method. A mathematical model of the positioning system motion is developed based on mechanics methods using direction cosine matrices linking the coordinate systems located on each moving link. The solution to the inverse problem is found analytically. Estimates for positioning errors are constructed based on the linearization of the kinematic equations. Results. A mathematical model of the kinematics of the four-axis precision positioning system is constructed. An analytical solution to the inverse problem of the positioning system kinematics is found, which can be used to transmit control signals to the drive drivers of the system links. Based on the obtained solution of the direct problem of the kinematics of the positioning system, upper estimates were made for the maximum errors in the positioning of welded parts for given dimensions of the parts and the errors of angular and linear drives due to the discreteness of their movements and backlash.