## 1 Introduction  
There are 21 terms of error motion effecting spatial positioning accuracy of machine tools [1]. To improve the positioning accuracy and repeatability of machine tools, calibration and geometric error compensation are necessary. However, it is time consuming, and not all controllers suppose geometric error compensation, such as squareness, straightness, angular error motion and so on. The most basic CNC (computer (or computerized) numerical control) controller only provides linear positioning error compensation for machine tool performance improvement, but the improvement is limited. The procedure and method for linear axis positioning accuracy and repeatability measurement can be referred to ISO 230-2 and ISO 230-6 [2,3]. According to the ISO 230-2 and -6 standards, the linear positioning accuracy and the repeatability of \(x\)-, \(y\)- and \(z\)-axes as well as \(xy\), \(xz\), \(yz\) and \(xyz\) diagonal lines can be found. Note that we called the above-mentioned test the \"seven lines test\" in this study. To perform the seven lines test, two instruments are preferred, a laser interferometer and an auto-tracking laser interferometer (ATLI). For the seven lines test, the spending time by means of a laser interferometer is about three times that of ATLI. To reduce the time required, multi-DOF (degree of freedom) measurement systems could be considered [4]. Although multi-DOF measurement systems can distinctly decrease the test time, the measurable range is usually smaller than a meter. Laser Tracker [5,6] and LaserTRACER [7, 8, 9] both belong to a kind of ATLI. The comparison of the measurement accuracy/uncertainty for different types of ATLIs is shown in Table 1. From the table, we can see that LaserTRACER is more suitable to apply to positioning accuracy and repeatability measurements for machine tools than others because of its smaller measurement uncertainty and smaller ranging errors [10]. For LaserTRACER, the target spatial position can be determined by multilateration [3; 11; 12]. To perform multilateration, multiple LaserTRACERs are needed. Combined with time sharing method [13] single LaserTRACER can also perform multilateration. Although the measurable distance of LaserTRACER (15 m) is smaller than Laser Tracker (over than 20 m), it is currently sufficient for general machine tools test. There are some other auto-tracking spatial measurement methods, such as that proposed by [PERSON] _et al._[14] using a Kinect to tracking a human body, as well as triangulation measurement method with dual modulated laser diodes and single detector [15; 16]. Based on the concept of sphere surface refection, [PERSON] _et al._[17] propose a steel sphere center alignment device according to Michelson interference fringe deviation. 1. To reduce the measurement error due to bearing run-out error and mirror offset, LaserTRACER uses a steel sphere instead [18]. As seen in Figure 2a, when the laser head is rotated, the bearing run-out error affects the length difference. When a lens and a steel sphere are used instead, the run-out error can be eliminated, as shown in Figure 2b. 2. The target spatial coordinate, which is measured by Laser Tracker, is determined by the following Equation [19]: \[\left\{ \begin{array}{l}d_{\mathrm{x}}+\Delta x=(d+\Delta d)\cdot\cos\left( \beta+\Delta\beta\right)\cdot\cos\left(\alpha+\Delta\alpha\right)\\ d_{\mathrm{y}}+\Delta y=(d+\Delta d)\cdot\cos\left(\beta+\Delta\beta\right) \cdot\sin\left(\alpha+\Delta\alpha\right)\\ d_{\mathrm{z}}+\Delta z=(d+\Delta d)\cdot\sin\left(\beta+\Delta\beta\right) \end{array} \right.\] (1) where \(d\) represents the measured distance between the target and the Laser Trackers; \(\alpha\) and \(\beta\) represent the angular position of \(\theta z\) and \(\theta y\), respectively; and \(\Delta d\), \(\Delta\alpha\) and \(\Delta\beta\) represent the deviations of \(d\), \(\alpha\) and \(\beta\) (_i.e._, the error sources), respectively. 3.