# A Study of a Cornering Braking Control System for a Motorcycle
## 3 Optimization Model for the Best Slip Ratio  
As shown in Figure 6, when a motorcycle is turning, there is a roll angle to offset the centrifugal force and maintain vehicle balance. This roll angle creates a camber angle for the tires. If a tire slides in the lateral direction, the vehicle becomes unstable and rolls, so there must be sufficient lateral force from the ground to allow a safe turn.  
When a motorcycle brakes during a turn, a lateral force and a longitudinal force act on two wheels to brake the vehicle, so the longitudinal and lateral forces act on the tires concurrently. The longitudinal force is the braking force, which defines the braking performance, and the lateral force is related to the vehicle's stability. The tire model for this study shows that the longitudinal and lateral forces are affected by the slip ratio, so these two forces are controlled by changing the slip ratio. An appropriate slip ratio provides the best braking efficiency and the vehicle is stable, so there is a constrained optimization problem for which the design variable is the slip ratio, the optimization object is to maximize the longitudinal braking force, and the constraint is that the lateral force must be sufficient to overcome the centrifugal force and ensure stability. In terms of the principle of force balance, the centrifugal force acting on a wheel is expressed as:  
\[F_{\text{C}}=F_{z}\tan(|\gamma|) \tag{7}\]  
where \(F_{\text{C}}\) is the centrifugal force and \(F_{z}\) is the vertical force. \(\gamma\) is the camber angle of the tire, and its value is approximately equal to the vehicle roll angle.  
Figure 5: Lateral characteristics of a tire.  
The mathematical expression for the constrained optimization problems is:  
\[\begin{array}{l}\kappa*=\arg\max_{\kappa}[F_{x}(\kappa,\alpha,\gamma,V,F_{z})] \\ \text{s.t. }\ F_{y}(x*,\alpha,\gamma,F_{z})\geq F_{z}\tan(|\gamma|)\end{array} \tag{8}\]  
where \(\kappa^{*}\) represents the best slip ratio and is the optimization result. \(\kappa\) is the current slip ratio. The possible value for the slip ratio \(\kappa\) is \(-1\) to \(0\) during braking. If the slip ratio \(\kappa\) is \(-1\), the wheel is locked. A slip ratio of zero denotes a freely rolling wheel. According to the longitudinal characteristic in Figure 4, the longitudinal force has unimodality in the slip ratio range of \(-1\) to \(0\).  
Given the unimodality, the gold section method was used to solve the optimization problem. The gold section method, or 0.618 method, is a tentative optimization method that uses the principle of interval elimination. By inserting two points to create an interval and computing these two function values, the interval is divided into three sections. By comparing these values, one of three sections is deleted so the region of the search is reduced in size. Two points are then inserted into the reduced interval and this process iterates until the interval is smaller than the convergence precision. The median of the final interval is the optimal solution. This study adds a step to determine whether the interval meets the constraints after inserting two points. The algorithm flow chart is shown in Figure 7.  
The optimal slip ratio depends on the vertical force, the camber angle, the travel velocity, and the sideslip angle. For a sideslip angle of \(-2^{\circ}\) and a wheel travel velocity of \(100\) km/h, the optimal slip ratio is shown in Figure 8. This value changes with the camber angle and the vertical force.  
During braking, these four parameters change quickly. We plugged many sets of the tire parameter values into the optimization model and tabulated the results. The lookup table is shown in Figure 9. The input parameters were the vertical force, the camber angle, the travel velocity, and the sideslip angle, and the output was the optimal slip ratio. This 4-D lookup table shows the optimal slip ratio for the cornering braking controller during braking.  
Figure 6: Force diagram for a motorcycle turning.  
Figure 7: Algorithm flow chart for the optimal slip ratio.