## 4 Simulation and Analysis
### Constant Speed Analysis  
The analysis reveals that our dual linear-motor differential-drive system attains a critical creeping velocity of about 1.0 mm/s. The discussion here emphasizes the identification of output variation in case the velocities of both linear motors exceed the table critical creeping velocity for the linear-motor single-drive system, yielding a 1.0 mm/s resultant rate. The settings for the upper linear-motor rate are 3-, 4-, 5- and 6 mm/s, separately, while the corresponding command speeds of the under linear motor are 2-, 3-, 4-, and 5 mm/s, the simulation results are shown in Figures 15-18, respectively. Figure 16: The two motor speeds are 4 mm/s and 3 mm/s. Figure 15: The two motor speeds are 3 mm/s and 2 mm/s. From Figures 11 and 15, Figures 16-18, it can be seen that when the minimum command speed of the upper linear motor and the under linear motor are both greater than 5 mm/s, and the differential value exceeds the critical creeping velocity of our dual linear-motor differential-drive system by 1.0 mm/s. The adjustment time of the output speed of the dual linear-motor differential-drive system is almost unchanged. Therefore, to ensure the stable output speed of the dual linear-motor differential-drive system, it is required that the minimum command speed of the upper linear motor and under linear motor are not less than 5 mm/s. To further investigate the minimum speed difference between the upper linear motor and under linear motor, the output speed of the dual linear-motor differential-drive system will not produce over modulation. Set the command speed of the upper linear motor to 7.3 mm/s and the command speed of the under linear motor to 5 mm/s, the simulation result is shown in Figure 19. Similarly, set the command speed of the upper linear motor to 42.3 mm/s and the command speed of the under linear motor to 40 mm/s, the simulation result is shown in Figure 20. From Figures 19 and 20, it can be concluded that when the command speeds of the upper linear motor and under linear motor are both higher than 5 mm/s and the speed difference is not less than 3 mm/s, the output speed of the dual linear-motor differential-drive system will not produce over modulation. Figure 17: The two motor speeds are 5 mm/s and 4 mm/s. Figure 18: The two motor speeds are 6 mm/s and 5 mm/s. The authors also compare the differences in rate fluctuations between the dual linear-motor differential- and linear-motor single-drive systems when the former gives a feed rate exceeding its own critical creeping velocity to the latter. The feed velocity of the latter is 2.3 mm/s, and Figure 21 depicts the result. As shown in Figure 22, the linear-motor single-drive system does not experience over-modulation when the feed speed is 3.7 mm/s. The rate of the upper linear motor is set to 43.7 mm/s, the corresponding command speed of the under linear motor to 40 mm/s, and Figure 23 presents the simulation outcome. Comparing Figures 20 and 21 and Figures 22 and 23, the fluctuation of the dual linear-motor differential-drive system is smaller than the linear-motor single-drive system when the rate settings are low for both. In addition, the former consumes less adjustment time, attains quicker response, and superior low-velocity micro-feed performance. From Figures 20 and 21, it can be concluded that before the system reaches a steady state, the oscillation frequency of the dual linear-motor differential-drive system is 77.78% less than that of the linear-motor single-drive system. In addition, the response speed of the dual linear-motor differential-drive system is 6.89% higher than that of the linear-motor single-drive system. From Figures 22 and 23, it can be concluded that before the system reaches a steady state, the oscillation frequency of the dual linear-motor differential-drive system is 60% less than that of the linear-motor single-drive system. At the same time, the response speed of the dual linear-motor differential-drive system is 3.13% higher than that of the linear-motor single-drive system.