# A Study of a Cornering Braking Control System for a Motorcycle
## 2 Motorcycle Tire Model  
ISO coordinates were used to define the model parameters, as shown in Figure 1. The origin of coordinate \(O\) was the intersection point of the wheel plane and the ground. The intersecting line between the wheel plane and the ground was the \(x\)-axis and the forward direction was positive. The \(z\)-axis was perpendicular to the ground and was positive in the upward direction. The right-hand rule was used to determine the \(y\) direction. The respective projections of the wheel travel velocity \(V\) on the \(x\) and \(y\)-axes were the longitudinal velocity \(V_{x}\) and the lateral velocity \(V_{y}\), respectively. There is slippage in the longitudinal and lateral directions when braking in a turn. \(V_{\mathrm{sx}}\) and \(V_{\mathrm{sy}}\) represent the slip velocity in two directions and \(V_{\mathrm{sy}}\) is equal to \(V_{y}\), as shown in Figure 2. The longitudinal slip ratio \(\kappa\) and the slip angle \(\alpha\) are defined as:  
\[\kappa=-\frac{V_{\mathrm{sx}}}{V_{x}} \tag{1}\]  
\[\alpha=\arctan(\frac{V_{\mathrm{sy}}}{|V_{x}|}) \tag{2}\]  
The tire kinematic parameters, ground forces, and torque are shown in Figures 1 and 2. The tire model describes the relationship between the kinematics and the forces that are applied by the ground, as shown in Figure 3. There are several types of tire model: the Brush model, Magic Formula, UniTire, and Flexible Ring Tire Model. Most are for car tires. Unlike a car tire, a motorcycle tire apex has a large camber angle and a small sideslip angle during turning. The Magic Formula tire model is widely used because it features high  
Figure 1: Motorcycle tire coordinates and parameters. precision and a uniform form. It is also suitable for motorcycle tires with a large camber angle [19; 20]. This study uses Magic Formula PAC2012 to establish the tire model. In Figure 3, longitudinal force directly determines the acceleration and braking deceleration and the lateral force significantly affects controllability and stability. When a motorcycle is braking on a curve, there is a longitudinal force and the road exerts a lateral force to counteract the centrifugal force, which is a feature of circular motion. This study uses the longitudinal and lateral characteristics of a motorcycle tire to determine the effect of specific parameters on the longitudinal and lateral forces. If there is only longitudinal slip, the sideslip angle \(\alpha\) is zero and the longitudinal force is expressed as:  
\[F_{x0}=D_{x}\sin[C_{x}\text{arctan}\{B_{x}(\kappa+S_{Hx})-E_{x}(B_{x}(\kappa+ S_{Hx})-\text{arctan}(B_{x}(\kappa+S_{Hx})))\}]+S_{Vx} \tag{3}\]  
In the above equation, \(B_{x}\) is the stiffness factor, \(C_{x}\) is the shape factor, \(D_{x}\) is the peak factor, and \(E_{x}\) is the curvature factor. The horizontal shift \(S_{Hx}\) and vertical shift \(S_{Vx}\) represent the effects of ply-steer, conicity, and the rolling resistance. If there is combined slip, a weighting function \(G_{x\alpha}\) is used to represent the interaction effect of the sideslip angle \(\alpha\) on \(F_{x}\). The longitudinal force for combined slip is:  
\[ \begin{array}{l}F_{x}(\kappa,\alpha,\gamma,V,F_{x},\dot{\phi})=G_{x\alpha} \cdot F_{x0}\\ G_{x\alpha}=\cos[C_{x}\text{arctan}\{B_{x\alpha}A_{\beta}-E_{x\alpha}(B_{x \alpha}A_{\beta}-\text{arctan}(B_{x\alpha}A_{\beta}))\}]/G_{x\alpha\alpha}\\ G_{x\alpha\alpha}=\cos[C_{x\alpha}\text{arctan}\{B_{x\alpha}S_{Hx\alpha}-E_ {x\alpha}\ (B_{x\alpha}S_{Hx\alpha}-\text{arctan}(B_{x\alpha}S_{Hx\alpha}))\}]\\ \alpha_{\beta}=\alpha+S_{Hx\alpha}\end{array}  \tag{4}\]  
Figure 3: Input/output variables for the tire model. Figure 2: Tire velocity diagram. If there is only sideslip, the longitudinal slip ratio \(\kappa\) is zero and the lateral force is expressed as:  
\[ \begin{array}{l}F_{y0}=D_{y}\sin[C_{y}\arctan\big{\{}B_{y}\alpha_{y}-E_{y}(B_ {y}\alpha_{y})-\arctan(B_{y}\alpha_{y}))\big{\}}]+S_{yy}\\ \alpha_{y}=\alpha+S_{Hy}\end{array}