## 4 Results and Discussion
### Hysteresis Effect of 8 Axle
#### 4.1.1 Relationship between Pedal Opening and Delay Time  
The swing angle of the pedal arm or the linear displacement of the brake's master cylinder is usually used for evaluating the pedal opening [25]; the latter is adopted in this paper because of the easily measured moving displacement of the servo drive device. Thus, the pedal opening can be denoted as:  
Figure 9: Hysteresis characteristic of PBS under the testing condition of group 1. (**a**) Original data curves; (**b**) Delay time of eight loops in the brake process; (**c**) Delay time of eight loops in the brake release process. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multicolumn{11}{c}{**Testing Groups**} \\ \hline \multirow{2}{*}{**Loop**} & \multicolumn{3}{c}{**1**} & \multicolumn{3}{c}{**2**} & \multicolumn{3}{c}{**3**} & \multicolumn{3}{c}{**4**} & \multicolumn{3}{c}{**5**} & \multicolumn{3}{c}{**6**} \\ \cline{3-11}  & & **Up** & **Dw** & **Up** & **Dw** & **Up** & **Dw** & **Up** & **Dw** & **Up** & **Dw** & **Up** & **Dw** \\ \hline \multirow{5}{*}{**Delay**} & A & 0.38 & 0.48 & 0.42 & 0.62 & 0.48 & 0.72 & 0.58 & 0.88 & 0.67 & 1.02 & 2.49 & 2.66 \\  & B & 0.31 & 0.45 & 0.34 & 0.51 & 0.42 & 0.73 & 0.50 & 0.84 & 0.62 & 0.93 & 2.41 & 2.63 \\  & C & 0.37 & 0.47 & 0.36 & 0.55 & 0.45 & 0.72 & 0.52 & 0.83 & 0.61 & 0.95 & 2.42 & 2.67 \\  & D & 0.39 & 0.53 & 0.42 & 0.65 & 0.49 & 0.78 & 0.61 & 0.86 & 0.66 & 1.04 & 2.47 & 2.71 \\ \cline{1-1}  & E & 0.47 & 0.55 & 0.52 & 0.67 & 0.55 & 0.81 & 0.68 & 0.92 & 0.72 & 1.10 & 2.56 & 2.73 \\ \cline{1-1}  & F & 0.41 & 0.54 & 0.45 & 0.62 & 0.52 & 0.75 & 0.65 & 0.89 & 0.70 & 1.04 & 2.50 & 2.68 \\ \cline{1-1}  & G & 0.42 & 0.53 & 0.46 & 0.63 & 0.53 & 0.76 & 0.67 & 0.91 & 0.72 & 1.06 & 2.52 & 2.69 \\ \cline{1-1}  & H & 0.48 & 0.59 & 0.55 & 0.69 & 0.57 & 0.83 & 0.69 & 0.97 & 0.78 & 1.12 & 2.57 & 2.75 \\ \hline \hline \end{tabular} Note: In this table, \(\Delta_{dd\_up}\) is denoted as “up”, \(\Delta_{dd\_down}\) is denoted as “dw”.  
\end{table} 
Table 3: Delay time of eight loops under the testing condition of all the groups. \[0\leq\psi=\frac{l}{l_{max}}\leq 100\%, \tag{2}\]  
where \(l\) is the measured displacement and \(l_{max}\) is the rated moving displacement, here \(l_{max}=15\) mm, and \(\psi\) is the pedal opening. When analyzing the experimental results from group 1 to group 5 in Table 3, we know that the delay time of each loop was not the same due to different pedal opening though the pedal's moving speeds being approximately equal. For the full eight loops, the delay time of the rear four loops (E, F, G and H) are higher than the front four loops (A, B, C and D). For a certain loop, the higher the pedal opening, the longer the delay time. In order to explore the quantitative relationships between them, polynomial fitting is carried out for both the brake process and the brake release process using least squares method. \[\left\{ \begin{array}{l}\mathbf{\Delta_{del\_up}=K_{1\_up}\psi^{2}+K_{2\_up} \psi+W_{up}}\\ \mathbf{\Delta_{del\_down}=K_{1\_down}\psi^{2}+K_{2\_down}\psi+W_{down}}\end{array}  \right., \tag{3}\]  
where \(\mathbf{K_{1\_up}}\), \(\mathbf{K_{2\_up}}\), \(\mathbf{K_{1\_down}}\), \(\mathbf{K_{2\_down}}\) are coefficient matrixes and \(\mathbf{W_{up}},\mathbf{W_{down}}\) are constant matrixes. The fitting results are shown in Figure 10, and the corresponding fitting parameters are shown in Table 4.  
The fitting coefficient of determination \(R^{2}\) can be calculated as follows,  
\[R^{2}=1-\frac{\sum_{a=1}^{n}\left(y_{a}-y_{a}^{\prime}\right)^{2}}{\sum_{a=1}^ {n}\left(y_{a}-\overline{y}\right)^{2}}, \tag{4}\]  
where \(y_{a}\) is the original data, \(y_{a}^{\prime}\) is the obtained fitting data, \(\overline{y}\) is the average value of the original data, and \(n\) is the number of original data. Each calculated \(R^{2}\) is close to 1, indicating that the fitting is very good. Figure 10: Relationships between pedal opening and delay time of eight loops.