## 2 Calibration Setup Parameters of Multilateration Method
### Tracking Laser Setup Parameters  
In this study, a TL system was used [1]. Its distinctive characteristic is that it can accurately estimate the distance between a reference sphere bound to the appliance and another sphere (the reflector) attached to a mobile item. The form defect of these two spheres and the evolution of their position are the major factors that deteriorate length measurement. To minimize these parasitic effects, the form defect of the reference sphere was less than 50 nm, and an invar support avoids any displacement of the reference sphere due to thermal expansion. Temperature and air pressure were also recorded and used by the acquisition software (V2.3, ETALON, Braunschweig, Germany) to compensate laser beam deviation. The TL characterized the distance between the two spheres. At least four length measurements in four different TL locations were then required to determine the reflector position. This principle, shown in Figure 1, is called multilateration. TL\(j\), where \(j=1\) to 4, is the position of the TL reference spheres, and I1 is the reflector location.  
The use of four TLs is quite expensive, and few laboratories or companies can afford such investment. However, the reflector location can be determined using only one appliance, by simply shifting its position three times. This is called sequential multilateration. It is the method used in this study.  
Sequential multilateration is the main source of factors that influence machine tool calibration. The kind of pattern, which defines machine tool trajectory geometry, needs to be characterized. It leans on the outlines of the compact working volume (box pattern: \(200\times 200\times 280\) mm) as shown in Figure 2. The linear guide kinematic errors are greater at the start and end positions, which constitute the most critical cases. The distance between the two consecutive nodes (sampling step) of the box edges needs to be fixed. Ten millimeters is an acceptable distance to correctly sample the box pattern (Figure 2). Moreover, the dynamic variation of the machine tool engenders some reflector oscillations. Once the machine tool stops, the TL carries out a sampling of the distances measured during a predetermined  
Figure 1: Multilateration principle.  
acquisition time. The distance estimated is the mean of the sampled values. The acquisition time is a parameter taken into account in the optimization process.  
The location of the measurement appliances also strongly influences calibration quality [19]. The reflector (ETALON, Braunschweig, Germany) needs to be equally spread around the spindle axis. Machine tool compactedness makes this operation difficult to implement as shown in Figure 3. A fixing system was designed to place the reflector in different locations without reducing the working area of the machine tool. This self-made fixing system uses a metal strap. It surrounds the spindle of the machine tool and supports the reflector. The mechanical part has cylindrical bores that allows the reflector to be in different orientations and positions using a set of rods and connectors. In this study, the number of reflector locations was limited to four. An overly large offset size puts the reflector outside the working volume during measurement and may engender some collision with the machine tool bed. According to [PERSON]'s principle, the offset size amplifies the locating errors that are easier to estimate. It then needs to be optimized. Spreading the TL positions around the working volume is a factor to consider. In this study, machine tool compactedness made it necessary to position the TL in front of the machine tool. Figure 4 shows two different positions. As a consequence, this parameter had to be optimized. The minimum number of TL positions imposed by the multilateration technique is four.  
Figure 4: Two tracking laser (TL) positions.  
Figure 3: Fixing system of the reflector around the spindle.  
Figure 2: Box pattern.