**7. Stylus Tip Offsets Calculation**

On a five-axis machine tool, there will be a situation when measurements are taken at different rotary axes positions are combined to analyse particular geometric characteristics of the workpiece. In such cases, the stylus tip coordinates are needed. These coordinates can be obtained using different approaches yielding different quality of results. In addition, the machine's own geometry, as for a coordinate measuring machine, needs to be calibrated and compensated. However, most machine tools are not geometrically calibrated. In this section, various ways to calibrate the stylus tip offsets and the machine geometry are presented. The ball dome is then used as a reference to evaluate the

effectiveness of the calibrated models. One of the parameters studied is the effect of the stylus tip offsets. The term stylus tip offsets here stand for the coordinates of the stylus tip center of the touch trigger probe relative to last machine tool branch axis frame, in our case the Y-axis frame. The stylus tip offsets can either be the nominal value for the tool length or values estimated through the SAMBA algorithms by probing one or more balls at various positions of the machine rotary axes. Table 4 lists the various stylus tip offsets used and how they are obtained. The ball dome data was processed either using a nominal machine model with null error parameters or using error parameters estimated from the SAMBA method.

N1-Nominal machine model, nominal tool (tool item N1):

The nominal tool is assumed to have zero lateral offsets and the tool length, as measured by the machinist during tool setting, as a negative z value.

N2-Nominal machine, estimated tool from a single ball dome ball (tool item N2):

The other approach to determine the stylus tip offsets is to use a nominal machine to estimate the stylus tip offsets. The tool length (−z value) and lateral offsets in x and y are estimated by using a single ball on the ball dome, which is located close to the ring section; no machine error parameter is estimated, and the parameters are set to zero. The ball and the tool coordinates are the only estimated variables to explain the machine readings.

N3-Nominal machine:

A similar process as for N2 but using all ball dome balls at once.

S1-SAMBA estimated machine, the tool from machinist for ball dome probing (tool item S1):

The same tool as for item N1, reported by the machinist, is used in this case. However, the ball dome coordinates are calculated based on the machine estimated from the SAMBA process, from one year ago.

S2-SAMBA estimated machine, an estimated tool from a single ball dome ball (tool item S2):

The tool x, y and z coordinates are estimated in order to best explain the machine readings while using the machine error parameters estimated by the SAMBA process from one year ago. The calibrated ball dome coordinates are not used.

S3-SAMBA estimated machine, an estimated tool from all ball dome balls (tool item S3):

As for S2 but all the ball dome balls are used for the tool estimation.

S4-SAMBA estimated machine, manually estimated tool (tool item S4):

The tool is estimated during the machine calibration using the SAMBA method. However, the stylus tip used to measure the ball dome was different from that used for the SAMBA calibration. In addition, during the dome measurement, the spindle was not rotated so that the tool could not be estimated independently from the spindle position. The spindle location was estimated during the SAMBA calibration conducted a year earlier. A complete machine, tool and ball coordinates estimation are conducted to explain the machine probing readings but only the tool coordinates are further used here. Vector subtraction is used to extract the tool geometry from the two vectors as illustrated in Figure 4 resulting in the following equation,

$$total^a = total^{ball-dome} - spindle^{SAMBA} \tag{5}$$

$$total^a = \left(total^n + \delta tool^{ball - domc}\right) - spindle^{SAMBA} \tag{6}$$

where *tool<sup>n</sup>* is the nominal tool geometry used during ball-dome measurement and *δtoolball*−*dome* is the deviation of the tool geometry calculated by Equation (4) and

$$
\delta \text{s}^{\dagger} \\
in \\
dle^{SAMBA} = \begin{bmatrix} E\_{\text{YOS}} & E\_{\text{YOS}} & 0 \end{bmatrix} \tag{7}
$$

where *E*XOS and *E*YOS are the two spindle lateral offset errors obtained by SAMBA method defined in Table 2. The kinematics of the machine tool accompanied by the two offset errors contributing in the tool twist estimation are illustrated in Figure 2.

**Figure 4. Left**: For a nominal machine, the reference frame of the spindle coincides with the B- and C-axis crossing point, which is also the machine foundation frame. For case N1 the tool only has a non-zero z coordinate. **Right**: The non-nominal machine estimated by the SAMBA method has spindle offsets in x from the B-axis and in y from the C-axis. This right-side diagram also illustrates the cases of an imperfect tool with lateral offsets in x and y.


