**4. Machine Tool Estimated Error Parameters**

The estimated machine tool error parameters obtained from the SAMBA method are listed in Table 2. Figure 2 illustrates the error terms in the machine kinematic chain.

**Figure 2.** Estimated error parameters in the machine kinematic chain (modified from reference [16]).


**Table 2.** Estimated error parameters for the 13-error model (nomenclature as per ISO230-1:2012).

#### **5. The Ball Dome Artefact**

In order to evaluate the machine metrology performance across the entire machine workspace, the maximum number of artefact balls should be accessible for probing for a broad range of angular axis positions. The ball dome artefact structure includes three semi-circular arcs attached together at their mid-point, with both ends fixed to a base ring. The result is a quasi-hemispherical structure holding 25 balls. In addition, there are three balls on the base ring and four balls on the base plate, which provide stable points to define a reference coordinate system. This design allows testing the machine for the full range of rotary axis motion. On this machine tool, the B-axis and C-axis rotation range are –90◦ to +90◦ and 0◦ to 360◦ respectively. The ball dome artefact is shown in Figure 3. To limit thermal effects the artefact structure is made of Invar. The measurement repeatability is affected by the clamping force that is applied to hold the artefact on the base plate, and by elastic deformation caused by a changing gravity vector. The reported measurement repeatability for clamping and gravity deflection was on average of the order of 0.6 and 6.5 μm respectively [13]. The measurement uncertainty for the artefact ball center once mounted on the machine tool is also reported at 5.3 μm (k = 2) [13].

**Figure 3.** (**a**) Ball dome artefact, (**b**) the ball dome holds on the machine tool work table.

#### **6. Ball Dome Probing**

The ball dome artefact is used to evaluate the machine tool metrology performance with and without the machine calibration. The artefact balls coordinates were measured to an uncertainty of 5.3 μm; Table 3 lists the measured coordinates. The artefact then is probed on the machine tool. The machine measured ball center coordinates in the machine table frame are calculated using the axis position readings and either the nominal model or the calibrated one, Equation (1). Then they are compared to the reference artefact coordinates. Because coordinate metrology generally requires accessing some features from different angles, the artefact should be probed at various machine indexations to ensure that different rotary axes positions are involved in the measurement process. The ball dome artefact is probed in 24 different machine rotary axes indexations, from −90◦ to +90◦ for the B axis and from 0◦ to 360◦ for the C axis. At each indexation, the maximum numbers of balls, which are accessible for the probing tool, are measured. A total of 613 ball centers were measured in about 15 h. All the other accessible balls centers are measured once at each pair of rotary axes indexation,


**Table 3.** Balls coordinates measured on the coordinate measuring machine (CMM) and used as calibrated values.
