# A Turning Brake System for Motorcycles via an Autoregulative Optimal Slip Ratio
## 2 Optimization Model for the Optimum Slip Ratio
The golden section search method is adeptly employed to tackle the optimization problem. The calculation flow chart is shown in Figure 2. First, the relevant motion parameters are specified, such as the vertical force and sideslip angle. Define initial interval \([a,b]\) and convergence accuracy \(\varepsilon\). Insert two points \(\kappa_{a}\) and \(\kappa_{b}\) in the initial interval, and the initial range \([a,b]\) is divided into three sections: \([a,\,\kappa_{a}]\), \([\kappa_{a},\,\kappa_{b}]\) and \([\kappa_{b},b]\). The maximum lateral force corresponding to the insertion point is calculated, respectively, and compared with the centrifugal force to determine whether the vehicle stability constraints are met. If the insertion point satisfies the stability constraint, the insertion point is reserved and assigned to the variable \(\kappa\), otherwise \(\kappa\) is set to \(0\). The longitudinal forces \(F_{xa}\) and \(F_{xb}\) at the insertion point are calculated and compared. Figure 1: Friction ellipse. If \(F_{xa}<F_{xb}\), the interval \([k_{b},b]\) can be eliminated, and the new search interval is \([a,\;\kappa_{b}]\). If \(F_{xa}\geq F_{xb}\), the interval \([a,\;\kappa_{a}]\) can be eliminated, and the new search interval is \([\kappa_{a},b]\). The golden section point continues to be inserted in the new interval, and the loop is repeated. The interval is continuously reduced until the interval is shortened to less than the convergence accuracy \(\varepsilon\), that is, \(b-a<\varepsilon\), and the median value is taken as the best slip ratio. During braking, speed decreases. In terms of the vertical forces, the load transfer between the front and rear wheels changes dynamically due to deceleration. The body roll angle also changes dynamically, so the optimal slip ratio changes with the movement of the vehicle. According to motorcycle tire model, the sideslip angle is small, and it ranges from \(-4^{\circ}\) to \(4^{\circ}\) in a steady driving process. Therefore, sideslip angle has little influence on optimal slip ratio, and it can be set to a fixed value of \(0^{\circ}\) in this optimization operation. The optimal slip ratio is mainly related to the roll angle and vertical load, and these two parameters are, respectively, set to different values to obtain the change curve of the optimal slip ratio with the camber angle, as shown in Figure 3. The optimal slip ratio curve shows that the greater the vertical load, the smaller the absolute value of the optimal slip rate. In  
Figure 2: Calculation for optimal slip ratio based on golden section method. the braking process, the vertical load of the front wheel increases, the vertical load of the rear wheel decreases, and the optimal slip ratio of the front wheel should be smaller than that of the rear wheel. The greater the absolute camber angle is, the closer the optimal slip ratio is to 0, which means that it is not appropriate to apply a large braking force when the roll angle is large. When the absolute value of camber angle is less than \(20^{\circ}\), the optimal slip ratio is between \(-0.15\) and \(-0.2\). To determine the optimal slip ratio quickly and reduce the amount of calculation, this study substitutes several groups of motion data into the constrained optimization model and tabulates the results to form an optimal slip ratio lookup table, as shown in Table 3.  
In real scenarios, the principle for obtaining the optimal slip ratio is shown in Figure 4, where the vertical load and camber angle are transmitted to the optimal slip ratio lookup table, and then the optimal slip ratio can be obtained either by table lookup or interpolation, thus quickly obtaining the optimal slip ratio. The optimal slip ratio will serve as the input for the brake system, which controls the motorcycle's brake by adjusting the tire's slip ratio. Figure 4: Decision model for optimal slip ratio. Figure 3: Optimal slip ratio curve.