## 3 Master Control System and Motion Control System
### Motion Control System
#### 3.3.2 Estimation of Accuracy of Motion Velocity Control  
The velocity control accuracy not only depends on the motor speed control accuracy and the measurement accuracy of the magnetic grating displacement transducer, but also on the machining accuracy and the tension strength of the synchronous belt and the elastic deformation property of the magnetic grating displacement transducer. Ignoring the deformation of the belt and the displacement transducer and based on the distance recorded by the reading head of the magnetic grating displacement transducer (with an accuracy of \(\pm 15\)\(\mu\)m) and the clock on the control board (the clock frequency is \(+/-100\) PPM, i.e., parts per million, with a time error of 100 units per million time units), an analysis of the carriage motion control accuracy can be presented. When the moving carriage moves a distance \(\Delta L\) in a given time period \(\Delta t\), the moving velocity can be calculated as  
\[V=\frac{\Delta L}{\Delta t}, \tag{5}\]  
The motion velocity control system feeds back the difference between this velocity and the set velocity to the servo motor controller, and then adjusts the speed of the servo motor to make the carriage velocity to approach a more accurate value, which is the outer-level closed-loop control of the moving carriage velocity. According to the error transfer  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & **First Time** & & & **Second Time** & \\ \hline
**Interferometer** & **Magnetic** & **Deviation** & **Interferometer** & **Magnetic** & **Deviation** \\  & **Transducer** & **Error** & **Transducer** & **Error** \\ \hline
0 & 0 & 0 & 0 & 0 & 0 \\
1001.02 & 1001.02 & 0.000\% & 1000.97 & 1001.01 & 0.004\% \\
2002.85 & 2002.86 & 0.0005\% & 2002.78 & 2002.85 & 0.003\% \\
3004.50 & 3004.58 & 0.0027\% & 3004.48 & 3004.59 & 0.004\% \\
4006.50 & 4006.42 & \(-\)0.002\% & 4006.39 & 4006.39 & 0.000\% \\
5008.25 & 5008.21 & \(-\)0.001\% & 5008.21 & 5008.2 & 0.000\% \\
6010.10 & 6010.05 & \(-\)0.001\% & 6010.03 & 6010.01 & 0.000\% \\
7011.75 & 7011.72 & 0.000\% & 7011.7 & 7011.71 & 0.000\% \\
8013.58 & 8013.56 & 0.000\% & 8013.52 & 8013.53 & 0.000\% \\ \hline \end{tabular}
\end{table}
Table 3: Check and verification result by dual-frequency laser interferometer.  
theory [31], the error \(\delta V\) in velocity \(V\) is jointly determined by the error \(\delta(\Delta L)\) in moving distance \(\Delta L\) and the error \(\delta(\Delta t)\) in moving time \(\Delta t\), and can be obtained from Equation (5)  
\[\delta V =\tfrac{1}{\Delta t}\delta(\Delta L)+\Delta L\Bigl{(}-\tfrac{1}{ \Delta t^{2}}\Bigr{)}\delta(\Delta t) \tag{6}\] \[=V\tfrac{\delta(\Delta L)}{\Delta L}-V\tfrac{\delta(\Delta t)}{ \Delta t}\]  
According to the check and verification in Section 3.3.1, the relative error of magnetic grating displacement transducer in measuring length is 0.0041%, and the clock frequency of 100 PPM (parts per million) gives a time error of 100 units per million units, i.e., 0.01%, from Equation (6), with each term taking the maximum value, the relative velocity error:  
\[\frac{\delta V}{V}\,\leq\,\left|\frac{\delta(\Delta L)}{\Delta L}\right|+\left| \frac{\delta(\Delta t)}{\Delta t}\right|=0.0041\%+0.1\%=0.0141\%.\]