## 2 Friction Coefficient Estimation
### Calculation of the Friction Coefficient
#### 2.3.1 Analysis of the Tire Model  
The Magic tire model [38] is a commonly-used tire model in vehicle dynamics investigations, and it has also been found to be suitable for experimental applications. Thus, in this study, the Magic tire model is used to estimate the friction coefficient:  
\[ \begin{cases}F_{x}=D_{x}\sin\left\{C_{s}\arctan\left[B_{x}s-E_{x}\left(B_{x}s- \arctan\left(B_{x}s\right)\right)\right]\right\}\\ F_{y}=D_{y}\sin\left\{C_{y}\arctan\left[B_{y}\alpha-E\left(B_{y}\alpha- \arctan\left(B_{y}\alpha\right)\right)\right]\right\}\end{cases} , \tag{9}\]  
where \(B_{x}\), \(C_{x}\), \(D_{x}\), \(E_{x}\), \(B_{y}\), \(C_{y}\), \(D_{y}\), and \(E_{y}\) are the parameters of the tire model. The above functions constitute a common simplified description of the actual Magic tire model. The parameters of the tire model are mainly affected by the vertical force and the friction coefficient [38]. However, the friction coefficient does not explicitly appear in Equation (9), which makes this model difficult to use for friction coefficient estimation. The structure of the Magic tire model is visualized in Figure 4. This model consists of four parts [38], labeled from A-D. Parts A and C are used to calculate the longitudinal and lateral forces under conditions of pure longitudinal slip and pure side slip, respectively. Parts B and D are used to calculate the longitudinal and lateral forces, respectively, under combined slip conditions based on the calculation results from Parts A and C. The corresponding detailed calculations can be found in [38]. To facilitate the estimation of the friction coefficient, the Magic tire model can be further rewritten as follows:  
\[ \begin{cases}F_{x}=\text{Magic}_{x}\left(F_{z},s,\alpha,\mu\right)\\ F_{y}=\text{Magic}_{y}\left(F_{z},s,\alpha,\mu\right)\end{cases} , \tag{10}\]  
where \(F_{x}\) and \(F_{y}\) are the longitudinal and lateral tire forces, respectively. The functions defined above provide a simple way to represent the Magic tire model for the purpose of friction coefficient estimation. The structures of these functions are visualized in Figures 5 and 6, respectively. Figure 4: Structure of the Magic tire model. Figure 5: Structure of the Magic\({}_{x}\) function. For friction coefficient estimation, the vertical force, side-slip angle, and slip ratio are known and are viewed as parameters instead of inputs in the defined functions. Once the vertical force, side-slip angle, and slip ratio of a tire have been obtained, the relationships between \(F_{x}\), \(F_{y}\) and \(\mu\) can be further rewritten as follows:  
\[ \begin{cases}F_{x}=\text{Magic}_{x,i}\left(\mu\right)\\ F_{y}=\text{Magic}_{y,i}\left(\mu\right)\end{cases} , \tag{11}\]  
where \(i=fl\), \(fr\), \(rl\), and \(rr\) and the functions \(\text{Magic}_{x,i}\) and \(\text{Magic}_{y,i}\) represent the relationships between \(F_{x}\), \(F_{y}\), and \(\mu\) when the vertical force, side-slip angle, and slip ratio are known for tire \(i\). The structures of the functions defined above are illustrated in Figures 7 and 8, respectively. To facilitate the discussion of the estimation of the friction coefficient, the tire force use ratio is defined as follows:  
\[\mu_{u}=\frac{\sqrt{F_{x}^{2}+F_{y}^{2}}}{F_{z}}. \tag{12}\]  
The relationships between the tire force use ratio and the slip ratio and side-slip angle with different friction coefficients are plotted in Figures 9 and 10. Under conditions of a low slip ratio or low side-slip angle, the friction coefficient has only a small effect on the tire force use ratio, as shown in Figure 9. Thus, the friction coefficient also exerts only a small effect on the tire force under such conditions, which causes friction coefficient estimation based on the tire force to be unreliable. For instance, if the slip ratio and side-slip angle are both zero, then the longitudinal and lateral forces are also zero regardless of the friction coefficient. Indeed, the friction coefficient has no effect on the longitudinal and lateral forces under such conditions.