## 2 Friction Coefficient Estimation
### Calculation of the Friction Coefficient
#### 2.3.2 Process of Estimation  
To obtain a reliable estimate of the friction coefficient, the following estimation process is considered. The longitudinal tire force can be estimated from Equation (3), and the lateral force on each axle can be estimated from Equation (4). Because the lateral force on each axle is the total lateral force on both tires on the same axle, under ideal conditions, the following equations should be satisfied:  
\[ \begin{cases}\text{Magic}_{x,fl}\left(\hat{\mu}_{f}\right)=\hat{F}_{xfl}\\ \text{Magic}_{x,fr}\left(\hat{\mu}_{f}\right)=\hat{F}_{xfr}\end{cases}  \tag{14}\]  
and:  
Figure 10: Tire force use ratio as a function of the side-slip angle. Figure 9: Tire force use ratio as a function of the slip ratio. \[\text{Magic}_{y,fl}\left(\hat{\mu}_{f}\right)+\text{Magic}_{y,fr}\left(\hat{\mu}_ {f}\right)=\hat{F}_{yf}. \tag{15}\]  
To facilitate the calculation, the following optimization problem is defined to estimate the friction coefficient of each front tire:  
\[ \begin{split}\text{min}:\qquad F_{M}(\hat{\mu}_{f})=& w_{x}(\text{Magic}_{x,fl}\left(\hat{\mu}_{f}\right)-\hat{F}_{fl})^{2}+w_{x}( \text{Magic}_{x,fr}\left(\hat{\mu}_{f}\right)-\hat{F}_{fr})^{2}\\ &+w_{y}(\text{Magic}_{y,fl}\left(\hat{\mu}_{f}\right)+\text{Magic} _{y,fr}\left(\hat{\mu}_{f}\right)-\hat{F}_{yf})^{2}\end{split}  \tag{16}\]  
where \(\hat{\mu}_{f}\) is the estimated friction coefficient of each front tire. The meaning of Equation (16) is that if the estimated friction coefficient is equal to the actual value, then \(F_{M}(\hat{\mu}_{f})=0\). Thus, the friction coefficient of the tires on the front axle can be estimated by means of the optimization problem defined in Equation (16). The friction coefficient of the tires on the rear axle can also be estimated using the same method. At each sampling instant, two estimated friction coefficients and their corresponding effectiveness factors can be obtained according to Equations (13) and (16). Due to the uncertainty of the model and the error in the numerical computations, the values from several successive sampling instants are used for estimation. The pseudocode for obtaining a reliable estimate of the friction coefficient is presented in Algorithm 1. In this algorithm, \(n_{t}\) is the total number of sampling instants used for estimation in each sampling window, \(m_{e}\) is the total number of effective estimates, \(\mu_{eU}\)  
and the rear wheels are injected with equal braking torques. The slip ratios of the front and rear wheels become larger as the injected torques increase. Thus, the estimation of the friction coefficient becomes more reliable as the slip ratio of each tire increases. The torque injection method can be used to force tires to work at high slip ratios while having only a small effect on the longitudinal dynamics of the vehicle [25]. However, note that torque injection also leads to tire wear and power loss. Thus, the need for reliable estimation should be balanced with the disadvantages of torque injection. The principles of and procedure for the torque injection method are illustrated in Figures 12 and 13, respectively. In these figures, \(T_{c,f}\) and \(T_{c,r}\) denote the normal front and rear drive torques, respectively, without torque injection; \(F_{x,sum}\) denotes the resultant longitudinal drive force on the four tires; \(T_{inj}\) is the injected drive torque; and \(T_{f}\) and \(T_{r}\) are the actual drive torques on the front and rear wheels, respectively. As indicated in Figures 12 and 13, in the torque injection method, the front wheels are injected with drive torques, and the rear wheels are injected with equal braking torques. Thus, the total injected drive torque has no effect on the vehicle's motion, but higher slip ratios of the front and rear wheels can be obtained, which are essential for accurately estimating the friction coefficient. In this paper, an adaptive torque injection method is proposed for friction coefficient estimation.