# Calibration of a Hybrid Machine Tool from the Point of View of Positioning Accuracy
## 2 Materials and Methods
### Positioning Accuracy Test and Applications in Machine Tool Calibration  
The positioning accuracy test procedure, which is necessary for the condition analysis and calibration of machine tools, is defined by numerous standards and recommendations. ISO 230-2 [22], VDI/DGQ 3441 [23], ANSI B5.54 [24], JIS 6333 [25], and others are currently used. All procedures involve the exposure of machine elements to programmed movement along the measured axis while simultaneously measuring the achieved displacement with the machine's measuring system and reference based on instrumentation, including a laser interferometer with a compensator for environmental conditions. In this way, it is possible to compare the values measured on the machine and forward them to the machine's control system with reference values. The conducted research is based on the procedure defined by the ISO 230-2 standard with compensation for environmental conditions. The procedure includes placing the laser measuring head (1), interferometer (2), retroreflector (3), acquisitional device (4), compensator of external conditions (5), external condition measuring device (6), sensor of temperature of machine elements (7), and computer analyzing data (8), according to the scheme shown in Figure 4.  
The procedure for testing the positioning accuracy of machine tools implies that the machine is programmed to perform an incremental movement along the measured axis in an automated cycle. Stopping the movement is conducted with a pause at the appropriate points according to the standard's requirements (in the positive and negative direction of movement from the start to the end programmed point). Realization of the test procedure gives the values of \(\mathrm{x_{i}}\). The results are presented tabularly and in the form of a diagram. According to the standard, the processing of measurement results includes the calculation of the following statistical quantities. These are reversal error at a position in point \"i\" (B\({}_{i}\)), reversal error of an axis (B), unidirectional positioning repeatability of an axis (R), unidirectional systematic positioning error of an axis (E\({}_{+}\) and E\({}_{-}\)), bi-directional systematic positioning error of an axis (E), mean bi-directional positioning error of an axis  
Figure 4: Positioning accuracy test procedure. (M), unidirectional positioning error of an axis (A\({}_{+}\) and A\({}_{-}\)), and bi-directional positioning error of an axis (A). The obtained results can be displayed as a standardized diagram showing the mean and individual deviation values from the programmed position, which enables the axis to be calibrated. The analysis of the obtained results in the form of tables and diagrams enables the calibration of individual machine tool axes by introducing predictive error values at individual machine locations. Calibrating machine tools with non-linear kinematics represents an additional challenge since the error is measured experimentally and expressed in external machine coordinates (the position of the executive body of the machine in the MCS coordinate system). Compensation values are entered in internal coordinates (in individual axis coordinates). Bearing in mind that movement in the direction of the X axis is realized by constant movement in the direction of the u\({}_{1}\) and u\({}_{2}\) axes and the direction of the Z axis in opposite directions at the same speed, the compensation value is determined for each point of movement according to the expression obtained by forming the first differential of the equation for internal coordinates.