# A Range-Extension Method for an Indoor Standard Device for Large-Scale Length Measurement
## 3 Range Extension of the Indoor Standard Device for Large-Scale Length Measurement
### Influence of the Measurement Uncertainty  
The measurement uncertainty of the range-extension technique for the indoor standard device for large-scale length measurement rests primarily on the angular error \(\theta\) of the hollow corner reflector, the straightness error \(S\) of the long guide rail, the laser interferometer error \(I\), and the environmental error \(T\). The analysis of measurement uncertainty in this section is mainly used to analyze the feasibility of the range-extension method described in this paper. The overall measurement uncertainty of the entire installation was determined as follows:  
\[U=2\times\sqrt{u_{\theta}^{2}+u_{S}^{2}+u_{I}^{2}+u_{T}^{2}} \tag{1}\]  
Considering the current status and technological level of the indoor standard device for large-scale length measurement, and with the guide rail extended from 50 m to 100 m, we obtained the following analysis results:  
(1) Measurement uncertainty caused by the angular error \(\theta\) of the hollow corner reflector was:  
\[u_{\theta}=\frac{100,000}{\cos(3\theta)}-100,000 \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Nominal** & **Measured** & **Measurement** & **Nominal** & **Measured** & **Measurement** \\
**Value (mm)** & **Value (mm)** & **Error (\(\mu\)m)** & **Value (mm)** & **Value (mm)** & **Error (\(\mu\)m)** \\ \hline
41,999.8906 & 41,999.8913 & 0.7 & 41,999.7266 & 41,999.7270 & 0.4 \\
43,999.8633 & 43,999.8659 & 2.6 & 43,999.7384 & 43,999.7396 & 1.2 \\
45,999.8242 & 45,999.8207 & \(-\)3.5 & 45,999.7852 & 45,999.7923 & 7.1 \\
47,999.7500 & 47,999.7500 & 0.0 & 47,999.7930 & 47,999.7966 & 3.6 \\
49,999.6797 & 49,999.6794 & \(-\)0.3 & 49,999.8008 & 49,999.8087 & 7.9 \\ \hline \end{tabular}
\end{table}
Table 2: \(Cont.\)  
Figure 8: A diagram showing indication error before and after the range-extension experiment.  
In the above equation, \(\theta\) is the angular error of the hollow corner reflector (up to 0.2\({}^{\prime\prime}\)). Since there were three reflections involved on two hollow corner reflectors, the overall angular error tripled, and upon the substitution of the parameter into the equation, was as follows:  
\[u_{\theta}=\frac{100,000}{\cos(3\times 0.2^{\prime\prime})}-100,000\approx 0 \tag{3}\]  
Measurement uncertainty associated with the angular error of the hollow corner reflector could be considered negligible.  
(2) Measurement uncertainty influenced by the straightness error \(S\) of the long guide rail was:  
\[u_{S}=\sqrt{100,000^{2}+S^{2}}-100,000 \tag{4}\]  
The straightness error of the long guide rail was 0.42 mm. Substituting the parameter into the above equation, we obtained:  
\[u_{S}\approx 0 \tag{5}\]  
Measurement uncertainty arising from the straightness error \(S\) of the long guide rail could be ignored.  
(3) According to the specifications of the laser interferometer, its measurement uncertainty stood at 0.2 ppm. And its measuring range was extended to 100 m.  
\[u_{I}=0.02\text{ mm} \tag{6}\]  
Given that the indoor standard device for large-scale length measurement was placed in a controlled temperature and humidity environment, measurement uncertainty resulting from the environmental error \(T\) could be overlooked.  
\[u_{T}=0 \tag{7}\]  
(4) By putting the results from the above analyses into the underlying equation for the device's measurement uncertainty, we obtained the following:  
\[U=2\sqrt{0.00042^{2}+0+0.02^{2}+0}=0.040\text{ mm} \tag{8}\]  
Upon calculation, it was determined that the measurement uncertainty of the indoor standard device for large-scale length measurement after the range extension satisfied the requirements for calibrating the laser interferometer.