Mathematical models of the industrial robots' control systems allow us to calibrate the instruments fixed on the installation of the robot flange. Calibration procedures are carried out when a new industrial robot tool is introduced. Instrument calibration is performed in two stages. The first phase is determination of the tool center point (TCP - Tool Center Point). The second stage includes the act of determination of the orientation of the Cartesian coordinate system, connected with the tool, which is placed at TCP. This article is devoted to the study of the second phase of the final calibration of the instrument. The mathematical models, which describe the transformation of the coordinates for the tool control system, are created automatically, when the instrument calibration procedures begins before introduction of a new instrument into operation. The mathematical models are stored in the permanent memory of the controller in the tool library. This paper introduces a mathematical model for calibration of the tool orientation of the industrial robots, the most common matching methods in practice of operation of the industrial robots. The obtained mathematical model can be used in the control systems of the industrial robots. The resulting mathematical model performs mapping of the TOOL coordinate system in relation to the WORLD coordinate system. The TOOL coordinate system is associated with the tool. The center of the TOOL coordinate system coincides with TCP. The coordinate system of the WORLD is connected with the stationary base of an industrial robot. The BASE coordinate system is connected with the manipulation object. In addition, the position of the BASE coordinate system must be calibrated.