## 2 Principle and Model  
### Measurement Principle  
The measuring device includes a laser and fiber coupling unit, a measurement unit, an error-sensitive unit, and a reference rotary unit [2]. The measurement unit and the error-sensitive unit constitute the main optical path. The optical path is divided into a collimation measurement part and an interferometry measurement part. The collimation measurement part consists of two quadrant detectors (QDs) and two position-sensitive detectors (PSDs). As shown in Figure 1, QD1 receives the light reflected back from a retro-reflector (RR3) to detect the radial motion error along the \(y\) axis and the axial motion error of the target rotary axis. QD2 is similar to QD1, and the difference between the two QDs in the vertical direction corresponds to the tilt motion error around the \(x\) axis. PSD1 receives the light reflected from a mirror (a beam-splitting film, BS3) and detects the angular error around the \(z\) axis and tilt motion error around the \(y\) axis. The target rotary axis rotates a nominal angle (\(\theta_{\text{nominal}}\)), and the reference rotary axis rotates in the opposite direction (\(\theta_{\text{encode}}\)). The angular positioning error is determined by \(Y_{\text{PSD1}}\) (the \(y\) direction of PSD1), \(\theta_{\text{encode}}\), and \(\theta_{\text{nominal}}\). PSD2 detects the laser beam drift [2]. The interference part consists of the reflected light by RR1 (as the reference light) and the reflected light by RR3 (as the signal light). Detector D1 records the changes in the interference signals, which correspond to the radial motion error along the \(x\) axis. Combined with collimation and interferometry, the measuring device is capable of the simultaneous measurement of all six of the DOF motion errors of a rotary axis.  
To ensure the continuity of the interference signal, the reference rotary axis needs to rotate synchronously with the target rotary axis. The proportional-integral-derivative (PID) algorithm was developed to achieve the servo-tracking strategy, and this process is shown in Figure 2. The start and end thresholds are based on the horizontal data of QD1. After the target rotary axis starts to rotate, the light incident to QD1 will deviate from its initial position. When the light reaches the start threshold, the reference rotary axis starts to rotate in the opposite direction. According to the data in the horizontal direction of QD1, the PID algorithm continuously outputs the speed commands to the reference axis. By constantly adjusting the speed, the reference axis rotates at a relatively stable speed with the target axis. When approaching the measuring position, the target axis rapidly decelerates to a stop. When the QD data reaches the end threshold, the computer stops the reference axis, and the servo-tracking process is completed. In order to eliminate errors introduced by bi-directional rotation, the reference axis is set to rotate in only one direction. Thus, if the speed calculated using PID is negative, the speed command will be zero.  
In order to shorten the control delay of the host computer, separate threads for the data acquisition from the detectors and PID algorithm were opened. By changing the end threshold, the stop position of the spot on QD1 is controlled; thus, the linear range of the QD detector can be fully utilized. The proportion coefficient of PID determines the stable state of the motion. An excessively small proportion coefficient will cause interference signal interruption, while an excessively large proportion coefficient will cause unstable movement of the reference axis, which is not suitable for measurement. The differential coefficient is sensitive to a sudden change. Thus, it can cope with excessive acceleration when the target axis starts or stops. Through the servo-tracking strategy, the device not only realizes interferometry but also reduces the time of a full-circle measurement from about 30 min [12] to less than 15 min.