# A Turning Brake System for Motorcycles via an Autoregulative Optimal Slip Ratio
## 1 Introduction
[12; 13] proposed a variable combined braking system that uses an adaptive control algorithm that can adjust the distribution of the braking force between the front and rear wheels. [PERSON] et al. [14] proposed an anti-lock braking system that uses fuzzy logic. The motorcycle parameters are calculated using an extended Kalman filter to determine the slip ratio and the road adhesion coefficient. The difference between the current slip ratio and the optimal slip rate, the deviation change rate, and the road adhesion coefficient are input to the fuzzy logic to generate a control command. To increase the performance of fuzzy logic control, a co-evolution algorithm is used to optimize the control parameters. [PERSON] et al. [15] proposed an optimal braking force distribution method for combined front and rear wheel braking that considers the influence of suspension. [PERSON] et al. [16; 17] analyzed the thermal properties of three different motorcycle brake disks through the finite element method, and the results show that the systems with higher cylinder capacity can guarantee better braking distance. Most studies [6; 7; 9; 10; 12; 13] design a motorcycle braking system that prevents skidding, and there is a lack of studies about tire models. The tire is a vital component of a vehicle, and the interaction between the tire and the ground through contact force plays a fundamental role in maintaining the vehicle's stability during braking. One study [8] directly defines the optimal slip ratio as \(-0.2\), which corresponds to the maximum longitudinal force for braking control for a specific tire slip rate. This braking strategy increases braking performance when driving in a straight line, but can cause instability if there is insufficient lateral force during turning. One study [11] proposes a braking system using the lateral force change rate to ensure the stability of vehicles, but there is a lack of correlation analysis on braking performance. One proposed braking system [14; 15] is only suitable for straight-line braking. [PERSON] [18] showed in a study of motorcycle tire dynamics that in addition to the longitudinal braking force, centrifugal force is offset by the lateral force that is exerted by the ground when the motorcycle is turning and braking. Sufficient lateral force ensures vehicle stability. However, longitudinal and lateral forces are affected by the friction ellipse. Onthis basis, the study [19] proposed constraint optimization to calculate the optimal slip ratio that serves as the input of the brake system. Though the input is autoregulative according to motorcycle dynamics, the optimal slip ratio depends on four kinematic parameters: vertical force, roll angle, velocity and sideslip angle. Most of these parameters, such as roll angle vertical force and sideslip angle, cannot be measured directly via sensor, which means an estimation or calculation for these parameters is necessary if this brake system is considered for actual application. Too many parameters increase the calculation time of the system, and then affect the brake reaction sensitivity. Roll angle is an important parameter, and it has a direct impact on motorcycle stability. The estimation algorithm based on the Kalman filter from [20] makes the acquisition of the roll angle possible. Motivated by the above discussion, this paper combines the result of [19; 20], and proposes a turning brake system that includes roll angle estimation and calculation for vertical force. Negligible parameters are ignored, and all parameters that are used in this system can be estimated or calculated by sensor signals, which is an improvement of the work in [19]. The proposed brake system is more pragmatic. The major contribution of this study is an increase in the braking performance and stability of motorcycles in a turn. This paper is organized as follows. In Section 2, a constrained optimization model that generates the optimal slip ratio is proposed and solved. Section 3 calculates the front and rear tire vertical force.