## 4 Experiment
### Synchronized Motion Experiments  
#### 4.2.1 Verification of Peak Voltage Calculation  
The AFOAM calculates the theoretical peak voltage based on the spectrum of the input signal and designs the corresponding fractional-order FDC, aiming to avoid the saturation effect of the control voltage. Therefore, the correctness of the calculation expression for the theoretical peak voltage directly affects the effectiveness of the proposed method in improving the synchronization accuracy. The experiments verified the accuracy of the calculation under different accelerations and velocities. To accurately analyze the spectrum of the input signal, we modified the short-term input signal into a long-period signal through periodic processing and calculated the spectrum according to the discrete Fourier transform shown in Equation (42). \[s_{p}(\omega)=T_{s}^{2}\sum_{n=-\infty}^{+\infty}\Bigg{(}\sum_{k=-\infty}^{n} \Bigg{(}\sum_{i=-\infty}^{k}a(i)\Bigg{)}\Bigg{)}e^{-j\omega n}, \tag{42}\]  
where \(s_{p}(\omega)\) is the spectrum of the input signal, \(T_{s}\) is the sampling time, \(a(i)\) denotes the discrete acceleration sequence, \(k\) is the number of discrete acceleration sequences, and n is the number of discrete velocity sequences. The cubic curve shown in Equation (43) is a commonly used point-to-point motion planning that can realize the adjustment of acceleration, velocity, and displacement. Therefore, this curve was used for experimental analysis. Figure 9 shows the motion planning when the displacement is 0.05 m, maximum speed is 0.4 m/s, and maximum acceleration is 30 m/s\({}^{2}\). \[r=\left\{ \begin{array}{l}a_{0}+a_{1}t+a_{2}t^{2}+a_{3}t^{3},\ for\ t\leq t_{a},\\ a_{0}+a_{1}t_{a}+a_{2}t_{a}t+a_{3}t_{a}^{3}\,,\ for\ t\ \leq(t_{a}+t_{c}),\\ a_{0}+a_{1}(t-t_{c})+a_{2}t_{a}t_{c}+a_{2}(t-t_{c})^{2}+a_{3}(t-t_{c})^{3},\\ for\ (t_{a}+t_{c})\ <t\leq(2t_{a}+t_{c}).\end{array} \right., \tag{43}\]  
where \(a_{i}\) (i = 1, 2,..., \(k\)) is a parameter of the cubic curve, \(t_{a}\) is the acceleration time, \(t_{c}\) is the uniform motion time, and r is the displacement. Figure 8: Diagram of the friction compensation voltage. Figure 9: Motion planning of a cubic curve. Figure 10 shows the calculated amplitude spectrum of the motion planning with different accelerations under the same acceleration time, and Figure 11 shows the calculated amplitude spectrum of the motion planning with different velocities under the same acceleration. It can be seen from Figure 10 that, as the acceleration increases, the amplitude spectrum at the non-zero frequency increases, but all accelerations are at a similar frequency. The results in Figure 11 show that, under the same acceleration, with an increase in velocity, the amplitude spectrum at the non-zero frequency increases, but the frequencies are all retained within a small range. Based on the spectrum analysis of the input signal, the voltage amplitude characteristics of the motor system at the corresponding angular frequency were calculated. Table 1 shows the comparison results of the theoretical and experimental peak voltages under different motion planning conditions. The relative error \(\delta\) in the table is calculated as follows:  
\[\delta=|u_{cal}-u|/u\times 100\%, \tag{44}\]  
where \(u\) is the experimental peak voltage, and \(u_{cal}\) is the calculated theoretical peak voltage. In the experimental results shown in Table 1, the maximum relative error between the experimental and the calculated theoretical peak voltages was 3.86%, which proves the accuracy of the calculation of the theoretical peak voltage; thus, Equation (17) can guarantee the accurate fractional-order FDC planning for the adaptive fractional-order anti-saturation synchronous control method. Figure 11: Calculated amplitude spectrum of different velocities under the same acceleration time (50 m/s\({}^{2}\)): **(a)** displacement comparison; **(b)** amplitude spectrum comparison.