# A Turning Brake System for Motorcycles via an Autoregulative Optimal Slip Ratio
## 5 The Design of a Turning Braking System  
### The Composition of a Turning Braking System  
The motorcycle turning brake system structure is shown in Figure 11, which mainly includes: a front and rear wheel optimal slip ratio lookup table, front and rear wheel braking control based on PID, a motorcycle dynamics model, a roll angle estimator that uses a Kalman filter, and the calculation for vertical force. The vehicle motion parameters are measured by an IMU and a wheel speed sensor. The roll angle and vertical force are estimated by the Kalman filter and Equation (4) according to vehicle motion parameters. The optimal slip ratio of the front and rear wheels is obtained by the lookup table for vehicle vertical load and roll angle. The PID controller forms a control command according to the current slip ratio and the optimal slip ratio, and controls the front and rear wheels, following the optimal slip ratio. The vehicle dynamics are fed back to the slip ratio lookup table, the control system and the estimation system to form a closed-loop control.  
A motorcycle is a nonlinear system that features static instability. The motorcycle turning braking system is part of the active safety system, which functions no longer simply to prevent tire lock, but also to take into account both braking and stability. During the turning and braking period, when the roll angle is too large, the vehicle has a greater risk of instability, the driver should appropriately reduce the braking force to ensure stability. When the roll angle is small, the rollover risk is small, and the braking can be appropriately increased. As the control target value, the selection of the optimal slip ratio is crucial. If the slip ratio is too small, the vehicle has good stability, but the braking performance is poor and the braking time is long. On the contrary, if the slip ratio is too large, the tire easily slips and the rollover risk is high. In the presented turning braking system, the optimal slip ratio is calculated by combining braking and stability. The lookup table can quickly provide the control target for the PID controller and adjust the control target immediately, according to the changes in vehicle roll angle and vertical load: when the roll angle is large, it generates small optimal slip ratio to ensure stability; when the roll angle is small and the vehicle is relatively stable, the tire is properly controlled to increase the slip ratio and improve braking. The cornering braking system designed in this paper can perform reasonable braking according to the stability of the vehicle, which is in accordance with the actual driving experience, and has good self-adjustment.  
Figure 11: Block diagram for a braking system for a motorcycle in a turn.  
In view of the advantages of the simple structure, easy implementation and good control effect, PID control mode has been widely used in the field of industrial engineering at present, and the development of technology is relatively perfect, so this paper adopts the PID controller to adjust the tire slip ratio. The control principle is shown in Figure 12.  
The optimal slip ratio \(\kappa_{1,2}^{*}\) and the current slip ratio \(\kappa_{1,2}\) constitute a deviation \(e_{1,2}(k)\). Proportional, integral and differential operations are carried out on the deviation value. The sum of the calculation results \(u(k)\) is used as the braking torque to adjust the tire slip ratio, as shown in Equation (19).  
\[u(k)=K_{p}e(k)+K_{i}\underset{j=0}{\overset{k}{\sum}}e(j)+K_{d}[e(k)-e(k-1)] \tag{19}\]  
where \(K_{p}\) is the proportional coefficient; \(K_{i}\) is the integral coefficient; \(K_{d}\) is the differential coefficient; \(k\) is the sampling sequence number.