## 4 Simulation and Analysis
### Critical Creeping Speed Analysis  
Through repeated computation of system parameters, the critical creeping velocity of the table is derived for a linear-motor single-drive system, whose value is about 2.0 mm/s. Figures 6-9 depict the simulations of this system when the feed rates are separately 1.9-, 2.0-, 2.1-, and 2.5 mm/s. The critical creeping velocity is considered to be attained when the table state is steady and the feed rate does not fluctuate pronouncedly. curve attains the steady state following approximately 2 s of adjustment when the table feed velocities are severally 2.1- and 2.5 mm/s. After overall analysis, the linear-motor single-drive system is considered to have a critical creeping velocity of 2.0 mm/s. In the case of a dual linear-motor differential-drive system, to identify its critical creeping velocity, the constant rate is introduced into the under and upper linear motors, respectively, whose value considerably exceeds (20 times at maximum) that derived with  
Figure 8: Linear-motor single-drive feed speed (2.1 mm/s). Figure 7: Linear-motor single-drive feed speed (2.0 mm/s). Figure 9: Linear-motor single-drive feed speed (2.5 mm/s). the single-drive system. A few simulations were conducted under identical parameters, and the dual linear-motor differential-drive system attains a critical creeping speed of about 1.0 mm/s. To find the critical creeping speed for the dual linear-motor differential-drive system, we input two constant rates separately to the upper and under linear motors via the control system, thereby guaranteeing the upper and under table instantaneous approximate synchronization. In this paper, the upper linear motor rate is set to 40.9-, 41-, 41.1-, and 41.5 mm/s, respectively, while the under linear motor rate is set to 40 mm/s. Simulation results for the dual linear-motor differential-drive system  
It is clear from Figure 10 that when the table feed velocity is below the critical creeping velocity, the system feed characteristics exhibit oscillation along with the limit-cycle phenomenon. Nonetheless, as displayed in Figure 11, when the table feed velocity exceeds the critical creeping velocity, the system is capable of attaining a steady state after a certain duration of adjustment (approximately 2.234 s at a 1.0 mm/s feed velocity), and our dual linear-motor differential-drive system boasts excellent feed characteristics at low velocity. According to Figure 10, when the dual linear-motor differential-drive system has a 0.9 mm/s feed velocity, evident vibration along with limit-cycle phenomena are present in the output velocity curve for the system. Nonetheless, as is clear from Figures 11-13, the system output velocity curve attains the steady state following adjustment severally at 1.0-, 1.1-, and 1.5 mm/s feed velocities. The dual linear-motor differential-drive system is considered to have a critical creeping velocity of 1.0 mm/s. Through repeated computation of system parameters, the foregoing four magnitudes (0.9, 1.0, 1.1, and 1.5 mm/s) are identified to mostly approximate the critical creeping velocity. Hence, critical creeping velocity for our dual linear-motor differential-drive system is explored by choosing these four magnitudes. As is clear from Figures 6-13, our dual linear-motor differential-drive system exhibits lower critical creeping velocity than the linear-motor single-drive system under identical parameters. As shown in Figure 14, the upper and under linear motor rates are set separately to 42- and 40 mm/s. The output result of table velocity is further studied when the constant feed velocity for the dual linear-motor differential-drive system is the critical creeping velocity for the linear-motor single-drive system as 2.0 mm/s. Figure 14: Differential-drive system feed speed (2.0 mm/s). Figure 13: Differential-drive system feed speed (1.5 mm/s).