## 3 Master Control System and Motion Control System
### Motion Control System
#### 3.3.1 Check and Verification of Distance Measurement Accuracy  
In order to check, test, and verify the accuracy of the magnetic grating displacement transducer in measuring distance used in the motion control system, a dual-frequency laser interferometer named Renishaw XL-80 (produced by Renishaw company located in Gloucestershire, west of London, UK) was used on the facility assembly site (Figure 7). The laser interferometer has a measurement accuracy of 0.001 mm with a resolution of 0.0001 mm when measuring a distance of meter order magnitude along a straight line. The magnetic grating displacement transducer has a measurement accuracy of \(\pm\)15 \(\upmu\)m, as mentioned above. The distances to be measured were selected as 1000 mm, 2000 mm, 3000 mm, 4000 mm, 5000 mm, 6000 mm, 7000 mm, and 8000 mm. The dual-frequency laser interferometer was used to measure these distances as a standard. At the same time, the magnetic grating displacement transducer measured the same distances. The deviation error of the magnetic grating displacement transducer from the dual-frequency laser interferometer was obtained by comparing the distances measured by the above two instruments. Each of the eight distances was measured seven times by the two instruments. According to the analysis of the 7 sets of data measured on site, the maximum relative deviation error of the magnetic grating displacement transducer from the dual-frequency laser interferometer is 0.004%, which occurred when measuring the distances of 1000 mm and 3000 mm. Only the results of first and second runs are shown in Table 3 for demonstration. Other runs are close in magnitude to those shown here.  
Figure 7: Check and verification of magnetic grating displacement transducer via dual-frequency laser interferometer. 1. Laser source; 2. Interference mirror; 3. Light-reflecting mirror; 4. Moving flat plate carriage; 5. Guide track; 6. Guide track pedestal.  
In error analysis theory [31], the measurement error of an instrument in measuring a physical quantity is defined as the difference between the value measured by the instrument and the true value (or best estimate) of the quantity. For the distance of \(L=1000\) mm, the measurement error of the magnetic grating displacement transducer can be expressed as:  
\[\delta_{\mathrm{g}}=L_{\mathrm{g}}-L_{\mathrm{true}}, \tag{1}\]  
while the measurement error of the dual-frequency laser interferometer should be  
\[\delta_{\mathrm{i}}=L_{\mathrm{i}}-L_{\mathrm{true}}, \tag{2}\]  
then  
\[\delta_{\mathrm{g}}=L_{\mathrm{g}}-L_{\mathrm{true}}=(L_{\mathrm{g}}-L_{ \mathrm{i}})+(L_{\mathrm{i}}-L_{\mathrm{true}})=(L_{\mathrm{g}}-L_{\mathrm{i} })+\delta_{\mathrm{i}}. \tag{3}\]  
Substituting the value of each term into Equation (3) gives  
\[\begin{array}{l}\delta_{\mathrm{g}}\leq\leq\leq\leq L_{\mathrm{g}}-L_{\mathrm{i}}\big{|}+\big{|}\delta_{\mathrm{i}} \big{|}=0.004\%\times 1000\,\mathrm{mm}\,+\,0.001\,\mathrm{mm}\\ =0.04\,\mathrm{mm}+0.001\,\mathrm{mm}=0.041\,\mathrm{mm}=41\mu\mathrm{m}\end{array}\quad,\]  
Which is greater than twice the claimed accuracy of 15 \(\mu\)m of the magnetic grating displacement transducer. The relative measurement error will then be:  
\[E_{\mathrm{g}}=\frac{\delta_{\mathrm{g}}}{L_{\mathrm{true}}}=\frac{0.041}{1000 }=0.0041\%. \tag{4}\]