**8. Deviation Results**

Table 5 shows the maximum and average deviation calculated for different models. The deviation is between artefact reference probing coordinate from CMM measurements and artefact probing coordinate from the machine tool as a coordinate measuring system. A least square fitting algorithm is applied to best match the two sets of coordinates. For every single ball, the coordinate deviations in x, y and z and the deviation norm, R, are calculated. The maximum and average values of R considering all ball dome balls are calculated.


**Table 5.** Maximum and mean deviation for different models (μm).

Figure 5 shows the deviation between the calibrated artefact and machine tool measured coordinates while using SAMBA estimated machine and manually estimated stylus tip offsets (item S4), which are used for comparison. The vectors (arrows) are the 3D deviation for every single ball while the reference values are the artefact reference coordinate measured on a CMM and each color represents a specific machine tool axes indexation out of the 24 indexations. Each vector has three Cartesian components; the length of each vector is calculated by Equation (8):

$$\text{dR} = \sqrt{\left(\text{dx}^2 + \text{dy}^2 + \text{dz}^2\right)}\tag{8}$$

The maximum and average deviations (vector lengths) for the 25 balls at 24 indexations are presented in Table 4.

**Figure 5.** Plotted deviation between compensated artefact by SAMBA and calibrated artefact, manually estimated tool is applied (item S4); (**a**) 3D view; the legend of the arrows' colors is as on figure b. (**b**) Deviation for one ball in X-Y view for different machine axes and indexations (units are millimeter).

Figure 6 presents the deviation while using the stylus tip offsets calculated based on just one ball on the artefact (item S2). In this case, to lighten the plots, the deviation arrows just for seven selected balls are shown through the artefact.

**Figure 6.** Plotted deviation between compensated artefact by SAMBA while considering one ball to model the tool and calibrated artefact for seven balls, estimated tool by probing one ball (item S2) (units are millimetre). (**a**) Deviation arrows, X-Y view, (**b**) 3-D view, (**c**) X-Z view, (**d**) Y-Z view

On the other hand, as it mentioned in Table 5, while the non-calibrated machine and nominal stylus tip offsets are used, the maximum and average deviations are 176 and 70 μm respectively.

#### **9. Discussion**

The SAMBA calibrated machine tool errors parameters are used to compensate the machine for the purpose of on-machine coordinate metrology. The considered errors are eight axis location errors, two spindle lateral offsets, three linear axis positioning scale gain errors and the stylus tip center coordinates (the tool) relative to the spindle frame. The ball dome artefact is used to evaluate the accuracy of the compensated machine. The ball dome includes 25 balls on a quasi-hemispherical envelop fabricated of Invar, which is clamped on kinematic supports to reduce clamping distortion.

The machine measuring performance when no calibration is applied neither for the machine geometry nor for the stylus tip offsets, displays the maximum and average deviations equal to 176 and 70 μm respectively. Calibrating the machine geometry based on the SAMBA estimated error parameters improves the machine performance and reduces the maximum and average deviation to 56 and 30 μm, respectively, a 60% improvement. Another important error contributor is the stylus tip offsets. There are two options to estimate the stylus tip offsets; the first one is using just one ball on the artefact which leads to 34 and 19 μm as the maximum and average deviations. The results achieved by using the second option, which stands on using all balls, are 32 and 12 μm. The other choice for the stylus tip offsets is achieved by vector calculation between the estimated tool from ball dome data only and estimated spindle from the SAMBA process. For this last case the maximum and average deviation are 31 and 16 μm respectively, the lowest maximum value obtained. The deviation reduction achieved by using calibrated machine and estimated stylus tip offsets is figured in Figure 7.

**Figure 7.** Deviation for nominal and estimated machine.

#### **10. Conclusions**

In this paper, the SAMBA method is used to calibrate the machine tool and to estimate the stylus tip offsets and then the efficiency of the calibration process and estimated models are verified by using the ball dome artefact. Due to the large size of the ball dome, during its measurement by the machine on various rotary axes indexations, all the linear and rotary axes motions are covered and then all the measurement results are transferred to the same reference frame. A single ball measurement would be the preferred option to estimate the stylus tip offsets. This approach provides the lowest average deviation.

Therefore, using the SAMBA calibration method accompanied with an optimized machine and stylus tip offsets has reduced the machine tool maximum and average volumetric errors by 82% and 83% respectively, while all the linear and rotary axes are involved in the coordinate measurement process.

**Author Contributions:** Conceptualization, J.R.R.M. and H.H.; methodology, H.H.; software, J.R.R.M.; validation, H.H.; formal analysis, H.H., S.E.M. and K.X.; investigation, H.H., S.E.M. and K.X.; resources, H.H. and K.X.; data curation, H.H. and S.E.M.; writing—original draft preparation, H.H.; writing—review and editing, J.R.R.M.; visualization, H.H., S.E.M.; supervision, J.R.R.M.; project administration, J.R.R.M.; funding acquisition, J.R.R.M.

**Funding:** This research was supported by the Natural Sciences and Engineering Research Council of Canada NSERC Canadian Network for Research and Innovation in Machining Technology–Phase 2: CANRIMT2.

**Acknowledgments:** The authors would like to thank the valuable support of CNC machine technicians, Guy Gironne and Vincent Mayer during the experimental tests.

**Conflicts of Interest:** The authors declare no conflict of interest.
