## 3 Methods
### Road-Vehicle Interaction Model
#### 3.1.1 Features and Limitations of the Bicycle Model  
1. Features of the bicycle model  
The main criticism of the point-mass model is its failure to consider the per-axle force generation capabilities of a vehicle. This limitation prevents the model from accurately accounting for situations where one axle exceeds the friction limit while another remains below it, particularly on horizontal curves where braking affects the vertical tire load, as well as the longitudinal and side friction on each axle. To assess the impact of per-axle friction utilization, the bicycle model is employed, which idealizes the vehicle as a rigid beam and representing each axle as a single tire positioned at the vehicle's midline [48]. It is utilized to evaluate the per-axle friction demand and ascertain whether the friction supply generated by the tire-pavement interface is sufficient for cornering during braking. The bicycle model makes several basic assumptions [19] as follows: (1) the motion of the vehicle in the vertical direction (z-axis direction) is not considered, that is, the motion of the vehicle is assumed to be in a two-dimensional plane; (2) lateral load transfer is not taken into account; (3) the vehicle possesses right/left symmetry; (4) the suspension system of the vehicle is considered to be rigid and nonmoving throughout the curve; and (5) aerodynamics and rolling resistance of the tires are ignored. 2. Model analysis  
Lowercase x, y, and z are utilized to represent the coordinate system of the vehicle, while uppercase X, Y, and Z denote the coordinate system referenced to the Earth. The  
Figure 1: Flowchart of the research method. designations, as well as the forces acting along each axis, are illustrated in Figure 2. By employing small angle approximations, the following three governing equations can be derived separately for the local longitudinal (\(x\)-axis), lateral (\(y\)-axis), and vertical (\(z\)-axis) directions. The following Equations (1)-(19) refer to the research results of [PERSON] et al. (2017). Braking Equation:  
\[\Sigma F_{x}=F_{bf}+F_{br}=ma_{x}+mgi \tag{1}\]  
Cornering Equation:  
\[\Sigma F_{y}=F_{cf}+F_{cr}=mv^{2}/R-mi_{h} \tag{2}\]  
Weight Balance Equation:  
\[\Sigma F_{z}=F_{Nf}+F_{Nr}=mg \tag{3}\]  
where \(F_{bf}\) is the braking force on the front axle, N; \(F_{br}\) is the braking force on the rear axle, N; \(F_{cf}\) is the cornering force on the front axle, N; \(F_{cr}\) is the cornering force on the rear axle, N; \(F_{Nf}\) is the vertical force on the front axle, N; \(F_{Nr}\) is the vertical force on the rear axle, N; \(i\) is the downgrade rate; \(i_{h}\) is the superelevation rate; \(v\) is the velocity, m/s\({}^{2}\); \(R\) is the circular curve radius, m; \(m\) is the mass, kg; and \(g\) is the gravitational acceleration, m/s\({}^{2}\). The vertical forces acting on each axis can be expressed as (11)  
\[F_{Nf}=mgb/L+(m(a_{x}+gi))h_{g}/L \tag{4}\]  
\[F_{Nr}=mga/L-(m(a_{x}+gi))h_{g}/L \tag{5}\]  
Equations (4) and (5) demonstrate that the braking results in the weight transfer from the rear axle to the front axle. The magnitude of this weight transfer can be expressed as \((m(a_{x}+gi))h_{g}/L\). It is worth noting that a higher deceleration rate corresponds to a greater amount of weight transfer. The ratio of cornering forces between the front and rear axles can be determined by the following equation:  
\[F_{cf}/F_{cr}=b/a \tag{6}\]  
\[F_{cf}/\left(F_{cf}+F_{cr}\right)=b/(b+a)=b/L \tag{7}\]  
where \(a\) is the distance from the centroid to the front axle, m; \(b\) is the distance from the centroid to the rear axle, m; and \(h_{g}\) is the CG height, m.  
Figure 2: Forces acting on a vehicle on a horizontal–vertical combined curve: (**a**) a vehicle braking on a downgrade; (**b**) a vehicle cornering on a horizontal curve. Note: G is gravity, N; G = mg. The red dot represents the center of gravity.