# A Geometric Error Measurement System for Linear Guideway Assembly and Calibration
## 2 Measurement Principles
### Pentaprism Module  
In addition to dual laser interferometer and PSD, a pentaprism module with a lifting mechanism was also developed for the geometric error measurement in this study. The pentaprism (BPP-12.7, Newport, Irvine, CA, USA), a five-sided reflecting prism, was utilized to deflect the incident laser beam by exactly 90\({}^{\circ}\), regardless of the incident angle. For keeping the maximum intensity of the reflected beam, the light receiver of the laser interferometer and a plane mirror were utilized to ensure the altitude angle between the pentaprism module and the laser source.  
The setup process of the pentaprism module is described as follows: The plane mirror placed above the pentaprism is first lifted up to receive the laser beam by adjusting the lifting mechanism of the pentaprism module for refracting the incident laser beam back to the optical field receiver of the laser source. The altitude angle between the laser source and the pentaprism module is ensured by keeping the maximum optical intensity of the backed beam. After ensuring the altitude angle with the maximum reflected beam intensity, the pentaprism is then lifted up to receive the laser beam by again adjusting the lifting platform of the pentaprism module.  
The optical arrangement of the beam module and the pentaprism module is shown in Figure 1.  
Figure 1: System optical arrangement.  
For the commercial laser interferometer, the coincidence between the reflected and the incident beams could be achieved by perpendicularly adjusting the altitude angle of the incident beam to the plane mirror by its light receiver according to the Michelson principle. However, with the feedback of calibration results, it was confirmed that the tolerance error of the \"100% coincidence\" measured by the commercial laser interferometer was about \(\pm\)5 arcsec, which could result in poor measuring repeatability, especially for the case of long-range measurements, as shown in Figure 2a. Therefore, an altitude angle adjustment module composed of a beam splitter, a convex lens, and a QD was proposed based on the autocollimator principle [18] for obtaining higher calibration repeatability, as shown in Figures 2b and 3. With the proposed module, the reflected beam was deflected by the beam splitter and passed through a convex lens onto QD, and the tolerance error of the \"100% coincidence\" could be reduced to \(\pm\)1 arcsec.  
Testing results showed that the perpendicularity repeatability (3\(\sigma\)) with a measuring distance of 2 m was greatly reduced from 5.7 arcsec (27.63 \(\upmu\)m/m) to 1.6 arcsec (7.56 \(\upmu\)m/m), proving the feasibility of the proposed altitude angle calibration module. Practically, the altitude angle between the laser source and the pentaprism could be calibrated using the first method based on the Michelson principle mentioned above for laboratories and factories owing to the Agilent 5529A interferometer, and the second method, based on autocollimator principle mentioned above, could be used for high calibration repeatability applications.  
Figure 2: The optical path of altitude angle calibration. (**a**) The tolerance error of the commercial laser interferometer and (**b**) the proposed module based on the autocollimator principle could reduce the tolerance error of the commercial laser interferometer.