# A Turning Brake System for Motorcycles via an Autoregulative Optimal Slip Ratio
## 2 Optimization Model for the Optimum Slip Ratio
Should the lateral force exerted by the ground on the tire prove insufficient to counteract the centrifugal force, the vehicle is at risk of lateral skidding or even overturning. The lateral force directly affects the directional stability. If the maximum lateral force \(F_{y\_max}\) exerted by \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Item** & **Nonmal** & **Free** & **Section** & **Tread** & **Aspect** & **Rim** \\  & **Load** & **Radius** & **Width** & **Height** & **Ratio** & **Radius** \\ \hline Value & 1100 N & 300 mm & 150 mm & 83 mm & 0.55 & 217 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Tire geometric parameters. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Item** & **Parameter** & **Value** & **Parameter** & **Value** & **Parameter** & **Value** \\ \hline \multirow{8}{*}{Longitudinal force} & PCX1 & 1.6064 & PDX1 & 1.3548 & PDX2 & \(-\)0.06029 \\  & PDX3 & 0.006 & PEX1 & 0.0263 & PEX2 & 0.27056 \\  & PEX3 & \(-\)0.07688 & PEX4 & 0 & PKX1 & 25.939 \\  & PKX2 & \(-\)4.2327 & PKX3 & 0.33686 & PHY1 & \(-\)0.0009 \\  & PHX2 & 0.0006 & PVX1 & 0 & PVX2 & 0 \\  & RBX1 & 13.476 & RBX2 & 11.354 & RCX1 & 1.1231 \\  & REX1 & 0 & REX2 & 0 & RHX1 & 0 \\ \hline \multirow{8}{*}{Lateral force} & PCY1 & 0.9 & PDY1 & 1.3 & PDY2 & 0 \\  & PDY3 & 0 & PEY1 & \(-\)2.2227 & PEY2 & \(-\)1.669 \\  & PEY3 & 0.05889 & PEY4 & \(-\)4.288 & PKY1 & \(-\)15.791 \\  & PKY2 & 1.6935 & PKY3 & 1.4604 & PHY1 & 0.0054 \\  & PHY2 & 0.0005 & PHY3 & 0.0036 & PVY1 & 0.0044 \\  & PY2 & \(-\)0.01329 & PVY3 & \(-\)0.3709 & PVY4 & \(-\)0.1221 \\  & RBY1 & 7.7856 & RPY2 & 8.1697 & RBY3 & \(-\)0.05914 \\  & RCY1 & 1.0533 & REY1 & 0 & REY2 & 0 \\  & RHY1 & 0.0008 & RHY2 & 0 & RVY1 & 0.042 \\  & RYY2 & \(-\)0.038 & RVY3 & \(-\)1.852 & RVY4 & 3.562 \\  & RYY5 & 1.5 & RVY6 & \(-\)2.357 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Pacejka tire model fitting parameters. the ground is greater than \(F_{C}\), the vehicle turns stably; otherwise, sideslip occurs. The tire is subjected to a longitudinal force and a lateral force. The longitudinal force, synonymous with the braking force, is a determinant of braking performance, dictating both the braking distance and the rate of deceleration. Concurrently, the maximum lateral force is a critical factor influencing the vehicle's stability. According to the study from [19], the slip ratio can affect the longitudinal and maximum lateral force. The nearer the slip ratio is to \(0\), the closer the tire is to a pure rolling state and the greater the maximum lateral force. For a slip ratio of \(-0.15\), the longitudinal force is greatest, and the maximum lateral force decreases to a center degree. For a slip ratio of \(-1\), the tire is locked and in a pure sliding state, and the ground exerts little maximum lateral force. The tire slip ratio must be maintained within a specific range so that the vehicle can maintain stability and braking performance. Therefore, the selection of the optimal slip ratio presents a constrained optimization problem, with the slip ratio itself serving as the design variable. The constraint stipulates that the maximum lateral force must exceed the centrifugal force to ensure stability. The constraint condition is that the maximum lateral force is greater than the centrifugal force. The objective of optimization is to obtain the best braking performance, that is, the maximum longitudinal force. Referring to [21], the mathematical model for constraint optimization is as follows:  
\[ \begin{cases}\kappa^{*}=\underset{\kappa}{\text{argmax}}||F_{x}(\kappa, \alpha,\gamma,F_{z})||\\ s.t.\,\,\,F_{y\_max}(\kappa^{*},\alpha,\gamma,F_{z})\geq F_{z}\text{tan}(| \gamma|)\end{cases}  \tag{3}\]  
where \(\kappa^{*}\) is the optimal slip ratio; \(\kappa\) is the current slip ratio; \(\alpha\) is the sideslip angle; \(\gamma\) is the camber angle; \(F_{z}\) is the vertical force of the tires. The longitudinal characteristics in [19] show that the longitudinal force exhibits a unimodal distribution across a slip ratio range from \(-1\) to \(0\).